Snells Law
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- Refraction Introduction
- Snells Law Introduction
- Snells Law Density of Medium
- Snells Law Small Angles
- Exercise 1
- 2 Exercise
- Total Internal Reflection and Critical Angle
- Optical Fiber
- Exercise - Total Internal Reflection
- Wave Equations Solutions and Equations
- Deriving the Wave Equations Using Maxwell
- Deriving the Equation for the Dispersion Relationship
- Exercise
- Exercise - Calculate the Corresponding Magnetic Field, New Equation
- Exercise - Wave Equations
- Refraction Through a Pane of Glass
- Refraction in a Semicircle
- Dispersion Prism
- Refraction in a Cylinder

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In this lesson,"},{"Start":"00:01.875 ","End":"00:04.530","Text":"we\u0027re going to be learning about refraction."},{"Start":"00:04.530 ","End":"00:10.740","Text":"Up until now, we\u0027ve spoken in previous chapters"},{"Start":"00:10.740 ","End":"00:18.195","Text":"about different things that light does when it comes into contact with some medium."},{"Start":"00:18.195 ","End":"00:22.960","Text":"We\u0027ve spoken about light being absorbed."},{"Start":"00:26.870 ","End":"00:32.925","Text":"Here the light enters the medium and the energy is absorbed,"},{"Start":"00:32.925 ","End":"00:39.790","Text":"and this can be seen, let\u0027s say if the medium heats up, for example."},{"Start":"00:39.790 ","End":"00:44.170","Text":"The next thing that we saw was reflection."},{"Start":"00:44.170 ","End":"00:47.900","Text":"This, I\u0027m sure we all know what this is,"},{"Start":"00:47.900 ","End":"00:51.320","Text":"when a light ray reaches a medium and it is"},{"Start":"00:51.320 ","End":"00:56.290","Text":"reflected back which is also how mirrors work."},{"Start":"00:56.290 ","End":"01:01.235","Text":"Finally, the next thing that light can do is refraction,"},{"Start":"01:01.235 ","End":"01:05.960","Text":"which is of course what we\u0027re going to be speaking about in this lesson."},{"Start":"01:05.960 ","End":"01:13.049","Text":"Refraction is when light passes from 1 medium into another."},{"Start":"01:14.470 ","End":"01:19.835","Text":"Light passes from 1 medium into another and the light,"},{"Start":"01:19.835 ","End":"01:22.850","Text":"or the light ray refracts,"},{"Start":"01:22.850 ","End":"01:25.930","Text":"which means that it bends."},{"Start":"01:25.930 ","End":"01:28.055","Text":"Of course, like everything,"},{"Start":"01:28.055 ","End":"01:32.735","Text":"physics has come up with an equation to express this."},{"Start":"01:32.735 ","End":"01:37.170","Text":"This is what we\u0027re going to be studying in this chapter."},{"Start":"01:39.080 ","End":"01:47.675","Text":"This happens when the light ray is transmitted through 2 different mediums."},{"Start":"01:47.675 ","End":"01:49.940","Text":"Let\u0027s give an example."},{"Start":"01:49.940 ","End":"01:56.250","Text":"Let\u0027s say that we have an aquarium like so."},{"Start":"01:56.250 ","End":"02:01.030","Text":"Let\u0027s say that it\u0027s filled with water."},{"Start":"02:01.250 ","End":"02:07.325","Text":"Now if we have a light ray,"},{"Start":"02:07.325 ","End":"02:12.935","Text":"I\u0027ll draw it in red, that comes like so in this direction."},{"Start":"02:12.935 ","End":"02:17.525","Text":"As it hits the surface,"},{"Start":"02:17.525 ","End":"02:24.439","Text":"the light ray will be refracted because it is passing from 1 medium,"},{"Start":"02:24.439 ","End":"02:26.960","Text":"the air, into another,"},{"Start":"02:26.960 ","End":"02:30.340","Text":"the water, and so it changes direction."},{"Start":"02:30.340 ","End":"02:33.665","Text":"Instead of the light ray continuing like so,"},{"Start":"02:33.665 ","End":"02:39.185","Text":"it will bend, it will change direction to go like so."},{"Start":"02:39.185 ","End":"02:42.360","Text":"The initial part of"},{"Start":"02:42.360 ","End":"02:51.025","Text":"the ray in the first medium is called the incidence ray."},{"Start":"02:51.025 ","End":"02:54.135","Text":"That is this."},{"Start":"02:54.135 ","End":"02:59.110","Text":"The incident or incidence ray."},{"Start":"03:00.080 ","End":"03:08.850","Text":"The ray over here that is bent is the refracted ray."},{"Start":"03:09.620 ","End":"03:17.900","Text":"The incident ray is the light ray which is propagating through the first medium,"},{"Start":"03:17.900 ","End":"03:20.535","Text":"so here it\u0027s the air,"},{"Start":"03:20.535 ","End":"03:23.580","Text":"until it reaches the interface."},{"Start":"03:23.580 ","End":"03:25.790","Text":"As we said, it\u0027s this section,"},{"Start":"03:25.790 ","End":"03:29.180","Text":"so I\u0027ll just draw it in green up until it reaches here,"},{"Start":"03:29.180 ","End":"03:34.490","Text":"where of course the interface is this line"},{"Start":"03:34.490 ","End":"03:39.755","Text":"over here which separates the first medium from the second medium."},{"Start":"03:39.755 ","End":"03:48.430","Text":"It\u0027s the line that is exactly where the air touches the water."},{"Start":"03:48.440 ","End":"03:54.760","Text":"We can say that this is the interface."},{"Start":"03:56.870 ","End":"04:02.340","Text":"The next term that we have is the normal."},{"Start":"04:02.340 ","End":"04:12.760","Text":"The normal is this line that I\u0027m drawing as a black dotted line over here like so."},{"Start":"04:12.760 ","End":"04:16.690","Text":"The black dotted line is called the normal,"},{"Start":"04:16.690 ","End":"04:21.465","Text":"and it is perpendicular to the interface."},{"Start":"04:21.465 ","End":"04:27.460","Text":"The interface again is the separation between the 2 mediums,"},{"Start":"04:27.460 ","End":"04:31.210","Text":"case of between in this example, air and water."},{"Start":"04:31.210 ","End":"04:37.290","Text":"The black dotted line is the normal."},{"Start":"04:37.290 ","End":"04:42.340","Text":"The normal is perpendicular to the interface."},{"Start":"04:42.340 ","End":"04:47.810","Text":"This is important because the angles that we measure in order to"},{"Start":"04:47.810 ","End":"04:54.110","Text":"make the various calculations are dependent on the normal."},{"Start":"04:54.110 ","End":"05:00.440","Text":"The next term that we have is the angle of incidence."},{"Start":"05:00.440 ","End":"05:08.100","Text":"The angle of incidence is the angle between the normal and the incident ray."},{"Start":"05:08.100 ","End":"05:10.350","Text":"Here we have our normal,"},{"Start":"05:10.350 ","End":"05:12.935","Text":"in the green, we have our incident ray."},{"Start":"05:12.935 ","End":"05:17.689","Text":"This over here is our angle of incidence,"},{"Start":"05:17.689 ","End":"05:20.270","Text":"and we can call it whatever we want, right now,"},{"Start":"05:20.270 ","End":"05:23.910","Text":"let\u0027s call it Theta 1."},{"Start":"05:24.670 ","End":"05:28.610","Text":"This is our angle of incidence."},{"Start":"05:28.610 ","End":"05:34.470","Text":"The next definition we have is of course the refracted ray."},{"Start":"05:35.030 ","End":"05:40.265","Text":"The refracted ray is what we have over here."},{"Start":"05:40.265 ","End":"05:45.890","Text":"We said that this is the light ray propagating through the second medium."},{"Start":"05:45.890 ","End":"05:53.615","Text":"In our example, it\u0027s the light ray that we measure that goes only through"},{"Start":"05:53.615 ","End":"06:01.635","Text":"the water from the interface until the bottom of the aquarium tank."},{"Start":"06:01.635 ","End":"06:04.595","Text":"If we had the angle of incidence,"},{"Start":"06:04.595 ","End":"06:10.380","Text":"then of course we\u0027re also going to have the angle of refraction."},{"Start":"06:11.080 ","End":"06:18.620","Text":"The angle of refraction is the angle between the normal and the refracted ray."},{"Start":"06:18.620 ","End":"06:22.220","Text":"As we said, this black dotted line is the normal,"},{"Start":"06:22.220 ","End":"06:26.480","Text":"this red arrow over here is the refracted ray."},{"Start":"06:26.480 ","End":"06:31.110","Text":"Then we\u0027ll draw this angle over here."},{"Start":"06:33.110 ","End":"06:35.580","Text":"Let\u0027s call it Theta 2."},{"Start":"06:35.580 ","End":"06:37.725","Text":"We could also call it Beta, whatever,"},{"Start":"06:37.725 ","End":"06:46.200","Text":"and it is the angle of refraction."},{"Start":"06:46.200 ","End":"06:50.450","Text":"In our example over here,"},{"Start":"06:50.450 ","End":"06:58.175","Text":"we can already see that Theta 1 is bigger than Theta 2."},{"Start":"06:58.175 ","End":"07:04.250","Text":"But this we\u0027ll explain in the later videos during this chapter."},{"Start":"07:04.250 ","End":"07:08.585","Text":"What I want you to take away from this video is, 1,"},{"Start":"07:08.585 ","End":"07:12.325","Text":"what is refraction, and 2,"},{"Start":"07:12.325 ","End":"07:17.405","Text":"all of these definitions and just know them."},{"Start":"07:17.405 ","End":"07:20.795","Text":"In the next video we\u0027re going to be going over"},{"Start":"07:20.795 ","End":"07:25.520","Text":"the mathematical equation which links everything together,"},{"Start":"07:25.520 ","End":"07:29.140","Text":"including Theta 1 and Theta 2."},{"Start":"07:29.140 ","End":"07:32.290","Text":"That\u0027s the end of this lesson."}],"ID":22423},{"Watched":false,"Name":"Snells Law Introduction","Duration":"24m 10s","ChapterTopicVideoID":21571,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21571.jpeg","UploadDate":"2020-05-07T18:01:42.9630000","DurationForVideoObject":"PT24M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.965","Text":"Hello. In this lesson,"},{"Start":"00:01.965 ","End":"00:05.445","Text":"we\u0027re going to be looking at Snell\u0027s law."},{"Start":"00:05.445 ","End":"00:07.170","Text":"In the previous lesson,"},{"Start":"00:07.170 ","End":"00:10.770","Text":"we learned a little bit about refraction."},{"Start":"00:10.770 ","End":"00:15.270","Text":"We saw that there is a connection between the angle of"},{"Start":"00:15.270 ","End":"00:20.280","Text":"incidence and the angle of refraction,"},{"Start":"00:20.280 ","End":"00:28.470","Text":"possibly due to the change in the medium that the light ray is propagating through."},{"Start":"00:28.470 ","End":"00:30.795","Text":"There is a connection,"},{"Start":"00:30.795 ","End":"00:33.330","Text":"and this connection is called Snell\u0027s law."},{"Start":"00:33.330 ","End":"00:40.035","Text":"In this lesson, we\u0027re basically going to be discussing and deriving Snell\u0027s law."},{"Start":"00:40.035 ","End":"00:44.600","Text":"First of all, you should remember the definitions from"},{"Start":"00:44.600 ","End":"00:50.057","Text":"the previous video and I\u0027m just going to write here a reminder,"},{"Start":"00:50.057 ","End":"00:56.100","Text":"so the angle of incidence."},{"Start":"00:56.840 ","End":"01:01.040","Text":"Here, we\u0027re going to call it Alpha."},{"Start":"01:01.040 ","End":"01:03.770","Text":"In the previous video, I called it Theta 1,"},{"Start":"01:03.770 ","End":"01:06.650","Text":"but of course, that doesn\u0027t matter in this video,"},{"Start":"01:06.650 ","End":"01:08.105","Text":"we\u0027ll call it Alpha."},{"Start":"01:08.105 ","End":"01:15.210","Text":"The angle of refraction in the previous video,"},{"Start":"01:15.210 ","End":"01:16.425","Text":"I called Theta 2,"},{"Start":"01:16.425 ","End":"01:17.940","Text":"but in this video,"},{"Start":"01:17.940 ","End":"01:20.055","Text":"we\u0027ll call it Beta."},{"Start":"01:20.055 ","End":"01:24.010","Text":"This is what you need to remember here."},{"Start":"01:25.400 ","End":"01:30.545","Text":"Let\u0027s discuss how something like this is measured."},{"Start":"01:30.545 ","End":"01:33.240","Text":"Let\u0027s say that we have"},{"Start":"01:33.580 ","End":"01:42.495","Text":"2 media or 2 mediums and here,"},{"Start":"01:42.495 ","End":"01:46.680","Text":"we\u0027re going to draw our interface."},{"Start":"01:46.680 ","End":"01:52.650","Text":"Then what we can do is we can take a protractor."},{"Start":"01:54.380 ","End":"02:02.900","Text":"Imagine this is a protractor and then we\u0027ll mark out our normal,"},{"Start":"02:02.900 ","End":"02:10.530","Text":"where the normal on the protractor corresponds to over here the 0 angle,"},{"Start":"02:10.530 ","End":"02:13.610","Text":"and over here, either 180 or 0."},{"Start":"02:13.610 ","End":"02:15.425","Text":"It doesn\u0027t make a difference."},{"Start":"02:15.425 ","End":"02:17.450","Text":"Let\u0027s just write over here 0,"},{"Start":"02:17.450 ","End":"02:21.145","Text":"but probably on your protractor will say 0,"},{"Start":"02:21.145 ","End":"02:25.885","Text":"and then here we have our 90 degrees."},{"Start":"02:25.885 ","End":"02:28.046","Text":"Then over here we have 10,"},{"Start":"02:28.046 ","End":"02:29.713","Text":"20, 30, 40,"},{"Start":"02:29.713 ","End":"02:34.270","Text":"50, 60, 70, 80 degrees."},{"Start":"02:34.530 ","End":"02:38.067","Text":"Excuse the mess. 10,"},{"Start":"02:38.067 ","End":"02:40.141","Text":"20, 30,"},{"Start":"02:40.141 ","End":"02:42.215","Text":"40, 50,"},{"Start":"02:42.215 ","End":"02:44.980","Text":"60, 70, 80."},{"Start":"02:44.980 ","End":"02:48.440","Text":"All of these degrees, of course."},{"Start":"02:48.590 ","End":"02:54.215","Text":"The same thing over here so depends on the protractor you\u0027re using."},{"Start":"02:54.215 ","End":"03:02.170","Text":"Usually, this is 180 degrees and then here we\u0027ll have 190, 200, 210, 220, 230, 240."},{"Start":"03:08.050 ","End":"03:10.400","Text":"There\u0027s not really much space,"},{"Start":"03:10.400 ","End":"03:16.050","Text":"but here it\u0027s up until 270 degrees."},{"Start":"03:16.720 ","End":"03:20.520","Text":"This is 270."},{"Start":"03:21.550 ","End":"03:30.355","Text":"This will be190, 200, etc."},{"Start":"03:30.355 ","End":"03:34.910","Text":"up until 270 degrees over here."},{"Start":"03:34.910 ","End":"03:38.475","Text":"Again, excuse the mess."},{"Start":"03:38.475 ","End":"03:43.715","Text":"Then if we draw a table where we have Alpha,"},{"Start":"03:43.715 ","End":"03:46.625","Text":"the angle of incidence, and Beta,"},{"Start":"03:46.625 ","End":"03:50.150","Text":"the refracted angle or the angle of refraction,"},{"Start":"03:50.150 ","End":"03:56.930","Text":"and let\u0027s say we have a light ray going in at 10 degrees like so."},{"Start":"03:56.930 ","End":"04:02.038","Text":"This is the incidence ray and what we\u0027ll see in the second medium,"},{"Start":"04:02.038 ","End":"04:05.190","Text":"it will bend slightly."},{"Start":"04:05.190 ","End":"04:07.369","Text":"Then over here,"},{"Start":"04:07.369 ","End":"04:09.230","Text":"we have our Alpha angle,"},{"Start":"04:09.230 ","End":"04:13.010","Text":"and over here, we have our Beta angle."},{"Start":"04:13.010 ","End":"04:16.610","Text":"The angle between the normal and the ray,"},{"Start":"04:16.610 ","End":"04:20.495","Text":"whichever 1 it is, incidents refracted."},{"Start":"04:20.495 ","End":"04:29.010","Text":"Let\u0027s say that our incidence ray is 10 degrees."},{"Start":"04:29.010 ","End":"04:33.212","Text":"Then our Beta for this medium,"},{"Start":"04:33.212 ","End":"04:36.755","Text":"that is an experiment that was done so I\u0027m just quoting data,"},{"Start":"04:36.755 ","End":"04:38.840","Text":"doesn\u0027t really matter what this is."},{"Start":"04:38.840 ","End":"04:40.820","Text":"Potentially it was water."},{"Start":"04:40.820 ","End":"04:44.285","Text":"Beta will be 7.5 degrees."},{"Start":"04:44.285 ","End":"04:48.529","Text":"Then if we erase this,"},{"Start":"04:48.529 ","End":"04:54.410","Text":"and now we have a ray at 20 degrees,"},{"Start":"04:54.410 ","End":"04:55.670","Text":"and of course,"},{"Start":"04:55.670 ","End":"04:57.170","Text":"this ray is a laser."},{"Start":"04:57.170 ","End":"04:59.420","Text":"Sorry, I should have mentioned that."},{"Start":"04:59.420 ","End":"05:01.790","Text":"When you\u0027re doing this experiment,"},{"Start":"05:01.790 ","End":"05:03.080","Text":"this is a laser ray,"},{"Start":"05:03.080 ","End":"05:09.270","Text":"and then you can really just see through your protractor the angle."},{"Start":"05:10.250 ","End":"05:15.200","Text":"Then for 20 degrees over here,"},{"Start":"05:15.200 ","End":"05:17.780","Text":"for an incidence ray of 20 degrees,"},{"Start":"05:17.780 ","End":"05:23.090","Text":"will get 14.5 degrees over here approximately."},{"Start":"05:23.090 ","End":"05:25.670","Text":"Of course, we\u0027re starting from a 180."},{"Start":"05:25.670 ","End":"05:35.060","Text":"Just remember that this 14.5 just basically means 14.5 degrees away from 180."},{"Start":"05:35.060 ","End":"05:44.100","Text":"Here specifically, it will be over here, the 194.5."},{"Start":"05:44.100 ","End":"05:47.000","Text":"But of course, when we\u0027re measuring Beta,"},{"Start":"05:47.000 ","End":"05:52.610","Text":"the angle of refraction is just this section between the normal and"},{"Start":"05:52.610 ","End":"06:00.810","Text":"the ray of refraction or the refracted ray."},{"Start":"06:00.810 ","End":"06:08.370","Text":"Just remember to subtract 180 from it. That\u0027s that."},{"Start":"06:08.370 ","End":"06:13.280","Text":"I\u0027m going to just continue with the experimental results."},{"Start":"06:13.280 ","End":"06:19.055","Text":"Now, the laser ray shines through at an angle of 30 pointed."},{"Start":"06:19.055 ","End":"06:23.540","Text":"Then I measure the refraction on my protractor."},{"Start":"06:23.540 ","End":"06:28.339","Text":"In this experiment, I got 22.1 degrees."},{"Start":"06:28.339 ","End":"06:33.215","Text":"For 40 degree angle over here,"},{"Start":"06:33.215 ","End":"06:39.349","Text":"I got 28.9 degrees."},{"Start":"06:39.349 ","End":"06:44.120","Text":"For 50 degrees, I got"},{"Start":"06:44.120 ","End":"06:51.540","Text":"35.2 degrees and now I\u0027m just going to write down all the rest."},{"Start":"06:52.900 ","End":"06:58.790","Text":"What we have over here is all the data and we can see that there\u0027s"},{"Start":"06:58.790 ","End":"07:04.350","Text":"no common difference between the terms."},{"Start":"07:04.350 ","End":"07:10.280","Text":"Here, we can see very clearly the difference is 7.5 degrees."},{"Start":"07:10.280 ","End":"07:16.940","Text":"But over here, we can see very clearly that the difference is 2.8 degrees."},{"Start":"07:16.940 ","End":"07:19.655","Text":"From the beginning until the end,"},{"Start":"07:19.655 ","End":"07:26.280","Text":"we can see that the difference between each subsequent term decreases."},{"Start":"07:26.280 ","End":"07:29.555","Text":"What we can see is that if we were to draw a graph,"},{"Start":"07:29.555 ","End":"07:31.595","Text":"the graph wouldn\u0027t be a straight line,"},{"Start":"07:31.595 ","End":"07:35.450","Text":"but rather, it would be some curve."},{"Start":"07:35.450 ","End":"07:40.175","Text":"If here, I have my Alpha in degrees,"},{"Start":"07:40.175 ","End":"07:43.070","Text":"and here I have my Beta in degrees,"},{"Start":"07:43.070 ","End":"07:48.600","Text":"my angle of incidence and my angle of refraction."},{"Start":"07:48.600 ","End":"07:54.420","Text":"What I\u0027d have is some curve like so."},{"Start":"07:54.420 ","End":"07:57.859","Text":"I forgot to mention at the beginning,"},{"Start":"07:57.859 ","End":"08:09.122","Text":"if I erase this beautiful drawing that I\u0027ve done over here, like so."},{"Start":"08:09.122 ","End":"08:14.245","Text":"If I were to shine my laser straight"},{"Start":"08:14.245 ","End":"08:21.475","Text":"on perpendicular to the interface or in other words,"},{"Start":"08:21.475 ","End":"08:23.170","Text":"along the normal line,"},{"Start":"08:23.170 ","End":"08:27.130","Text":"so at an angle of 0 degrees,"},{"Start":"08:27.130 ","End":"08:32.710","Text":"so it\u0027s parallel to the normal line,"},{"Start":"08:32.710 ","End":"08:42.760","Text":"my refraction ray or my ray of refraction would continue in a straight line as well."},{"Start":"08:42.760 ","End":"08:50.540","Text":"Here, this would also be 0 like so."},{"Start":"08:50.700 ","End":"08:54.625","Text":"If it\u0027s perfectly perpendicular,"},{"Start":"08:54.625 ","End":"08:56.665","Text":"my laser to the interface,"},{"Start":"08:56.665 ","End":"09:04.675","Text":"we\u0027re just going to get this straight line of light at an angle of 0 to the normal."},{"Start":"09:04.675 ","End":"09:09.400","Text":"However, once we start putting the laser at"},{"Start":"09:09.400 ","End":"09:15.200","Text":"an angle we get these different refracted values."},{"Start":"09:15.500 ","End":"09:17.550","Text":"Let\u0027s return to this."},{"Start":"09:17.550 ","End":"09:18.810","Text":"We see that we have a curve."},{"Start":"09:18.810 ","End":"09:20.835","Text":"It\u0027s not a straight line."},{"Start":"09:20.835 ","End":"09:25.120","Text":"After many scientists have been working on"},{"Start":"09:25.120 ","End":"09:29.695","Text":"this trying to figure out this relationship, finally,"},{"Start":"09:29.695 ","End":"09:32.440","Text":"we have a mathematical law,"},{"Start":"09:32.440 ","End":"09:35.770","Text":"which is called Snell\u0027s law,"},{"Start":"09:35.770 ","End":"09:41.080","Text":"the name of the chapter and of the video."},{"Start":"09:41.080 ","End":"09:47.590","Text":"Snell\u0027s Law describes this curve for different materials."},{"Start":"09:47.590 ","End":"09:50.155","Text":"For Snell\u0027s law, of course,"},{"Start":"09:50.155 ","End":"09:52.825","Text":"we have different definitions."},{"Start":"09:52.825 ","End":"09:54.580","Text":"First of all, we have"},{"Start":"09:54.580 ","End":"10:04.083","Text":"a refractive index called n,"},{"Start":"10:04.083 ","End":"10:06.910","Text":"where n is equal to c,"},{"Start":"10:06.910 ","End":"10:16.460","Text":"where c is the speed of light."},{"Start":"10:18.870 ","End":"10:21.160","Text":"Of course, this is c,"},{"Start":"10:21.160 ","End":"10:22.765","Text":"the speed of light in a vacuum,"},{"Start":"10:22.765 ","End":"10:25.885","Text":"3 times 10 to the 8 approximately."},{"Start":"10:25.885 ","End":"10:30.955","Text":"All of this is divided by v,"},{"Start":"10:30.955 ","End":"10:37.390","Text":"which is the velocity of the light"},{"Start":"10:37.390 ","End":"10:45.750","Text":"in the specific medium that we are using."},{"Start":"10:45.750 ","End":"10:48.825","Text":"If it\u0027s air, if it\u0027s water,"},{"Start":"10:48.825 ","End":"10:54.190","Text":"if it\u0027s any other thing that we\u0027re looking at glass, for instance."},{"Start":"10:55.290 ","End":"10:58.090","Text":"This is the refractive index."},{"Start":"10:58.090 ","End":"11:04.030","Text":"It\u0027s basically the relationship of the speed of light in a vacuum relative"},{"Start":"11:04.030 ","End":"11:10.945","Text":"to the speed of light in the specific medium that we\u0027re measuring."},{"Start":"11:10.945 ","End":"11:17.170","Text":"What\u0027s important to note is that always speed of light in"},{"Start":"11:17.170 ","End":"11:24.925","Text":"a vacuum is going to be bigger than the velocity of the light in the medium."},{"Start":"11:24.925 ","End":"11:28.897","Text":"This leads us to know that n which is equal to c"},{"Start":"11:28.897 ","End":"11:36.235","Text":"divided by v is always going to be greater or equal to 1."},{"Start":"11:36.235 ","End":"11:43.360","Text":"Let\u0027s say we\u0027re trying to calculate the refractive index of a vacuum."},{"Start":"11:43.360 ","End":"11:46.825","Text":"If we\u0027re in space and there\u0027s no particles there."},{"Start":"11:46.825 ","End":"11:52.264","Text":"The refractive index would be equal to n which is equal to c, always,"},{"Start":"11:52.264 ","End":"11:55.870","Text":"divided by the velocity of light in this medium,"},{"Start":"11:55.870 ","End":"11:58.135","Text":"where the medium is the vacuum."},{"Start":"11:58.135 ","End":"12:01.480","Text":"Of course, the velocity will be"},{"Start":"12:01.480 ","End":"12:05.260","Text":"c because it\u0027s light traveling through the medium where the medium is vacuum."},{"Start":"12:05.260 ","End":"12:11.830","Text":"We can see that the refractive index in a vacuum is equal to 1.This is"},{"Start":"12:11.830 ","End":"12:12.831","Text":"the"},{"Start":"12:23.280 ","End":"12:24.760","Text":"refractive"},{"Start":"12:24.760 ","End":"12:26.260","Text":"index of a vacuum."},{"Start":"12:26.260 ","End":"12:30.595","Text":"The refractive index of water is going to be equal to c because it\u0027s always c"},{"Start":"12:30.595 ","End":"12:35.220","Text":"divided by the speed of light in water,"},{"Start":"12:35.220 ","End":"12:39.120","Text":"which is equal to, from the Internet,"},{"Start":"12:39.120 ","End":"12:47.465","Text":"2.256 times 10^8 approximately."},{"Start":"12:47.465 ","End":"12:49.510","Text":"C is of course,"},{"Start":"12:49.510 ","End":"12:53.725","Text":"as we said equal to 3 times 10^8."},{"Start":"12:53.725 ","End":"13:01.550","Text":"What we get is that the refractive index of water is equal to approximately 1.33."},{"Start":"13:03.950 ","End":"13:07.860","Text":"Now that we have this definition and we understand"},{"Start":"13:07.860 ","End":"13:12.490","Text":"this we can finally get to Snell\u0027s law."},{"Start":"13:13.550 ","End":"13:15.885","Text":"It goes like this."},{"Start":"13:15.885 ","End":"13:18.320","Text":"Snell says that n_1,"},{"Start":"13:18.320 ","End":"13:24.445","Text":"the refractive index in the first medium or of the first medium"},{"Start":"13:24.445 ","End":"13:32.035","Text":"multiplied by sine of Alpha or of Theta 1."},{"Start":"13:32.035 ","End":"13:34.487","Text":"You know what? Let\u0027s just call it Theta 1,"},{"Start":"13:34.487 ","End":"13:37.140","Text":"because it goes with the 1 over here."},{"Start":"13:37.140 ","End":"13:40.470","Text":"Sine of Theta_1, where of course,"},{"Start":"13:40.470 ","End":"13:47.860","Text":"Theta 1 is the angle of incidence is equal to n_2 which is"},{"Start":"13:47.860 ","End":"13:53.035","Text":"the refractive index of medium number 2 of the second medium"},{"Start":"13:53.035 ","End":"13:59.185","Text":"multiplied by sine of Theta_2 or Beta,"},{"Start":"13:59.185 ","End":"14:01.495","Text":"where Theta_2 is of course,"},{"Start":"14:01.495 ","End":"14:04.910","Text":"the angle of refraction."},{"Start":"14:07.110 ","End":"14:13.150","Text":"Let\u0027s give this equation a little wall."},{"Start":"14:13.150 ","End":"14:20.890","Text":"Let\u0027s say, I want to take air as my first medium and water as"},{"Start":"14:20.890 ","End":"14:24.970","Text":"my second medium and let\u0027s say I want to put"},{"Start":"14:24.970 ","End":"14:31.510","Text":"my laser for the light to travel through air and to be refracted in the water,"},{"Start":"14:31.510 ","End":"14:37.730","Text":"and I want to know what the angle of refraction is going to be."},{"Start":"14:37.860 ","End":"14:45.100","Text":"N_1, so this is the refractive index of air,"},{"Start":"14:45.100 ","End":"14:50.665","Text":"is very, very similar to the refractive index in a vacuum."},{"Start":"14:50.665 ","End":"14:54.080","Text":"We\u0027re just going to say that it\u0027s 1."},{"Start":"14:54.840 ","End":"15:01.710","Text":"I know that the refractive index of my second substance,"},{"Start":"15:01.710 ","End":"15:06.260","Text":"which is water is equal to 1.33."},{"Start":"15:09.060 ","End":"15:18.745","Text":"Now, let\u0027s say that I shine my laser at an angle of 10 degrees."},{"Start":"15:18.745 ","End":"15:21.880","Text":"My incident ray is 10 degrees."},{"Start":"15:21.880 ","End":"15:25.720","Text":"Then what I have is n_1,"},{"Start":"15:25.720 ","End":"15:30.580","Text":"which is equal to 1 multiplied by sine of Theta_1,"},{"Start":"15:30.580 ","End":"15:34.035","Text":"which is 10 which is equal to n_2,"},{"Start":"15:34.035 ","End":"15:40.675","Text":"which is 1.33 multiplied by sine of Theta_2."},{"Start":"15:40.675 ","End":"15:50.035","Text":"Then what I get is that sine after I simplify this equation,"},{"Start":"15:50.035 ","End":"15:51.693","Text":"I calculate this,"},{"Start":"15:51.693 ","End":"15:53.980","Text":"and I divide both sides by 1.33,"},{"Start":"15:53.980 ","End":"16:02.110","Text":"I get the sine of Theta_2 is equal to 0.131, which therefore,"},{"Start":"16:02.110 ","End":"16:11.048","Text":"means that my Theta_2 is equal to 7.5 degrees."},{"Start":"16:11.048 ","End":"16:13.930","Text":"7.5, there we go."},{"Start":"16:13.930 ","End":"16:19.495","Text":"What we can see if I scroll up is that over here,"},{"Start":"16:19.495 ","End":"16:26.665","Text":"this 10 degree angle corresponds to Theta_2 of 7.5."},{"Start":"16:26.665 ","End":"16:31.360","Text":"What we can see is that in my experiment at the beginning of the lesson,"},{"Start":"16:31.360 ","End":"16:35.200","Text":"I was probably measuring 2 mediums where"},{"Start":"16:35.200 ","End":"16:39.955","Text":"my first medium was air and my second medium was water."},{"Start":"16:39.955 ","End":"16:46.090","Text":"1 of the uses of this equation is I can discover what material I\u0027m using"},{"Start":"16:46.090 ","End":"16:48.550","Text":"via knowing the refractive index of"},{"Start":"16:48.550 ","End":"16:53.240","Text":"different materials and shining a laser light through it."},{"Start":"16:55.020 ","End":"17:03.505","Text":"Now, I want us to scroll up back to this graph over here."},{"Start":"17:03.505 ","End":"17:08.420","Text":"I\u0027ll just add in here that 0 corresponds with 0."},{"Start":"17:08.610 ","End":"17:11.245","Text":"What I want to do is now,"},{"Start":"17:11.245 ","End":"17:13.795","Text":"instead of this graph like so,"},{"Start":"17:13.795 ","End":"17:15.550","Text":"let\u0027s scroll a bit."},{"Start":"17:15.550 ","End":"17:19.569","Text":"I want to do a graph that shows,"},{"Start":"17:19.569 ","End":"17:22.240","Text":"instead of as a function of the angles,"},{"Start":"17:22.240 ","End":"17:26.140","Text":"I want it as a function of sine of the angles."},{"Start":"17:26.140 ","End":"17:30.385","Text":"Here, I want sine of Theta_1,"},{"Start":"17:30.385 ","End":"17:33.190","Text":"where Alpha is equal to Theta_1,"},{"Start":"17:33.190 ","End":"17:37.195","Text":"and here, I want sine of Theta_2,"},{"Start":"17:37.195 ","End":"17:41.485","Text":"where Beta is equal to Theta_2."},{"Start":"17:41.485 ","End":"17:45.560","Text":"Let\u0027s take a look."},{"Start":"17:47.220 ","End":"17:53.170","Text":"If I look at Snell\u0027s law that says that n_1 multiplied by"},{"Start":"17:53.170 ","End":"17:59.770","Text":"sine of Theta_1 is equal to n_2 multiplied by sine of Theta_2."},{"Start":"17:59.770 ","End":"18:08.755","Text":"Then if I divide both sides by n_2 and I\u0027ll just reverse the order."},{"Start":"18:08.755 ","End":"18:13.585","Text":"I\u0027ll have sine of Theta_2 is equal to"},{"Start":"18:13.585 ","End":"18:21.100","Text":"n_1 divided by n_2 multiplied by sine of Theta_1."},{"Start":"18:21.100 ","End":"18:24.100","Text":"I just reversed the order over here."},{"Start":"18:24.100 ","End":"18:30.505","Text":"What I can see is that if this is the y value and this is my x value,"},{"Start":"18:30.505 ","End":"18:32.095","Text":"as I\u0027ve done over here."},{"Start":"18:32.095 ","End":"18:35.740","Text":"Of course, this is the x-axis sine of Theta_1,"},{"Start":"18:35.740 ","End":"18:39.295","Text":"and this is the y-axis sign of Theta_2,"},{"Start":"18:39.295 ","End":"18:43.795","Text":"then this over here is going to be my gradient,"},{"Start":"18:43.795 ","End":"18:46.495","Text":"my m or my gradient,"},{"Start":"18:46.495 ","End":"18:54.910","Text":"which means that we\u0027re going to expect a straight line, something like so."},{"Start":"18:54.910 ","End":"18:58.660","Text":"Just imagine that this is perfectly straight."},{"Start":"18:58.660 ","End":"19:03.370","Text":"The gradient is going to be the refractive index of medium"},{"Start":"19:03.370 ","End":"19:08.720","Text":"1 divided by the refractive index of medium 2."},{"Start":"19:09.510 ","End":"19:14.170","Text":"Of course, if you want to try this and compare the 2 graphs,"},{"Start":"19:14.170 ","End":"19:17.710","Text":"you\u0027re more than welcome to plot these out"},{"Start":"19:17.710 ","End":"19:24.235","Text":"accurately and then calculate the sign of each angle,"},{"Start":"19:24.235 ","End":"19:31.720","Text":"and plot on your x-axis the sine of Theta_1 or the Alpha,"},{"Start":"19:31.720 ","End":"19:36.940","Text":"against sine of Theta_2 on the y-axis,"},{"Start":"19:36.940 ","End":"19:40.075","Text":"over here and you\u0027ll see that you get"},{"Start":"19:40.075 ","End":"19:43.630","Text":"these shapes and that this over here is exactly a straight line,"},{"Start":"19:43.630 ","End":"19:47.870","Text":"which shows us that this equation is correct."},{"Start":"19:48.570 ","End":"19:53.800","Text":"Now, another thing that I\u0027m just going to plot over here."},{"Start":"19:53.800 ","End":"20:00.745","Text":"Let\u0027s say that I did plot the sine of Theta_1 and sine of Thet_2."},{"Start":"20:00.745 ","End":"20:06.400","Text":"This value over here, and over here,"},{"Start":"20:06.400 ","End":"20:12.170","Text":"let\u0027s just calculate the gradient m. This is,"},{"Start":"20:12.170 ","End":"20:14.695","Text":"of course, at 0,0."},{"Start":"20:14.695 ","End":"20:19.300","Text":"As we saw, sine of 0 is 0 and sine of 0 is 0."},{"Start":"20:19.300 ","End":"20:20.830","Text":"This is a 0,0,"},{"Start":"20:20.830 ","End":"20:22.915","Text":"and this point over here."},{"Start":"20:22.915 ","End":"20:29.995","Text":"What we have is sine of 80,"},{"Start":"20:29.995 ","End":"20:36.110","Text":"which is 0.925,"},{"Start":"20:36.120 ","End":"20:45.920","Text":"and sine of 47.8 is 0.7."},{"Start":"20:46.470 ","End":"20:51.145","Text":"Now, if I try to work out m,"},{"Start":"20:51.145 ","End":"20:53.425","Text":"I\u0027ll just do it in black."},{"Start":"20:53.425 ","End":"20:58.870","Text":"I want to work out m. This is going to be equal to,"},{"Start":"20:58.870 ","End":"21:08.240","Text":"so we have y_2"},{"Start":"21:08.240 ","End":"21:14.530","Text":"minus y_1 divided by x_2 minus x_1."},{"Start":"21:16.600 ","End":"21:20.860","Text":"Our y_2 is this over here,"},{"Start":"21:20.860 ","End":"21:26.215","Text":"0.7 minus y_1, which is 0,"},{"Start":"21:26.215 ","End":"21:31.570","Text":"divided by x_2,"},{"Start":"21:31.570 ","End":"21:39.685","Text":"which is 0.925 minus y_2, which is 0."},{"Start":"21:39.685 ","End":"21:45.650","Text":"What we get is 0.7 divided by 0.925."},{"Start":"21:46.680 ","End":"21:50.470","Text":"If you plug that into a calculator,"},{"Start":"21:50.470 ","End":"21:52.450","Text":"you\u0027ll get that this is equal to"},{"Start":"21:52.450 ","End":"22:02.200","Text":"0.757 approximately."},{"Start":"22:02.200 ","End":"22:06.895","Text":"We know that m is equal to n_1 divided by n_2."},{"Start":"22:06.895 ","End":"22:10.030","Text":"This is equal to n_1,"},{"Start":"22:10.030 ","End":"22:11.800","Text":"which we said over here, if you remember,"},{"Start":"22:11.800 ","End":"22:17.500","Text":"was 1 divided by n_2, which was 1.33."},{"Start":"22:17.500 ","End":"22:20.000","Text":"We can just see it over here."},{"Start":"22:20.970 ","End":"22:27.560","Text":"If I do 1 divided by 1.33 and plug that into the calculator,"},{"Start":"22:27.960 ","End":"22:34.640","Text":"I get approximately 0.752."},{"Start":"22:37.140 ","End":"22:40.390","Text":"Every time you measure, of course,"},{"Start":"22:40.390 ","End":"22:45.190","Text":"you know you have measuring error or so due to the eyesight and also due to"},{"Start":"22:45.190 ","End":"22:50.050","Text":"the accuracy of the measuring tool that you are using."},{"Start":"22:50.050 ","End":"22:53.605","Text":"The measuring instrument which was here, the protractor."},{"Start":"22:53.605 ","End":"22:58.430","Text":"If we see that we got 0.752,"},{"Start":"22:59.190 ","End":"23:05.530","Text":"which is this,"},{"Start":"23:05.530 ","End":"23:10.520","Text":"this was n_1 and this was n_2."},{"Start":"23:12.420 ","End":"23:16.945","Text":"They are approximately equal to 1 another."},{"Start":"23:16.945 ","End":"23:21.710","Text":"We can see that this equation does work."},{"Start":"23:23.580 ","End":"23:29.290","Text":"The 2 rules that I want you to take from this lesson is 1,"},{"Start":"23:29.290 ","End":"23:34.720","Text":"this, Snell\u0027s law with the equation."},{"Start":"23:34.720 ","End":"23:39.940","Text":"The second rule that I want you to take away from"},{"Start":"23:39.940 ","End":"23:45.430","Text":"this is that all the different lines that we have,"},{"Start":"23:45.430 ","End":"23:48.235","Text":"so we have the incident ray."},{"Start":"23:48.235 ","End":"23:50.260","Text":"The incident ray, the refracted ray,"},{"Start":"23:50.260 ","End":"23:55.280","Text":"and the normal are all on the same plane."},{"Start":"23:55.860 ","End":"23:58.540","Text":"This is what I want us to remember."},{"Start":"23:58.540 ","End":"24:01.240","Text":"The angles will be different between them,"},{"Start":"24:01.240 ","End":"24:05.305","Text":"but they\u0027re all on the same x-y plane."},{"Start":"24:05.305 ","End":"24:07.510","Text":"If you want to think of it like that."},{"Start":"24:07.510 ","End":"24:10.760","Text":"That is the end of this lesson."}],"ID":22424},{"Watched":false,"Name":"Snells Law Density of Medium","Duration":"15m 17s","ChapterTopicVideoID":21372,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21372.jpeg","UploadDate":"2020-04-06T22:29:50.2000000","DurationForVideoObject":"PT15M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.115","Text":"Hello. In the previous lesson,"},{"Start":"00:02.115 ","End":"00:08.520","Text":"we introduced Snell\u0027s law and proved it as well via a graphical method."},{"Start":"00:08.520 ","End":"00:13.740","Text":"Now we\u0027re just going to continue speaking about Snell\u0027s law."},{"Start":"00:13.740 ","End":"00:20.760","Text":"As we remember the equation for Snell\u0027s law is n1 multiplied by"},{"Start":"00:20.760 ","End":"00:28.380","Text":"sine of Theta 1 is equal to n2 multiplied by sine of Theta 2,"},{"Start":"00:28.380 ","End":"00:35.100","Text":"where n1 is the refractive index of medium 1 and 2 is"},{"Start":"00:35.100 ","End":"00:39.190","Text":"the refractive index of medium 2 and Theta 1"},{"Start":"00:39.190 ","End":"00:45.390","Text":"is the angle of incidence and Theta 2 is the refracted angle."},{"Start":"00:45.830 ","End":"00:50.660","Text":"Here we have all the definitions again, the incident ray,"},{"Start":"00:50.660 ","End":"00:53.300","Text":"the light ray propagating through the first meet him until it"},{"Start":"00:53.300 ","End":"00:56.975","Text":"reaches the interface and its angle,"},{"Start":"00:56.975 ","End":"00:59.695","Text":"as we said is Theta 1,"},{"Start":"00:59.695 ","End":"01:03.920","Text":"the normal which is perpendicular to the interface and of course,"},{"Start":"01:03.920 ","End":"01:06.650","Text":"the interface is what separates the 2 mediums."},{"Start":"01:06.650 ","End":"01:08.300","Text":"The angle of incidence,"},{"Start":"01:08.300 ","End":"01:10.460","Text":"oh sorry, this is what really we needed."},{"Start":"01:10.460 ","End":"01:14.645","Text":"This is Theta 1, the refracted ray,"},{"Start":"01:14.645 ","End":"01:17.840","Text":"the light ray propagating through the second medium and"},{"Start":"01:17.840 ","End":"01:22.115","Text":"the angle of refraction is the angle between the normal and the refracted ray."},{"Start":"01:22.115 ","End":"01:26.990","Text":"This is Theta 2 and we have the refractive index,"},{"Start":"01:26.990 ","End":"01:33.650","Text":"the various ends where here I\u0027m reminding you the equation to calculate n at c,"},{"Start":"01:33.650 ","End":"01:35.300","Text":"the speed of light in a vacuum,"},{"Start":"01:35.300 ","End":"01:40.800","Text":"divided by the velocities of the light through the mediums."},{"Start":"01:40.800 ","End":"01:43.970","Text":"If we\u0027re looking at the refractive index of medium 1,"},{"Start":"01:43.970 ","End":"01:53.010","Text":"we will divide c by the velocity of the light ray in medium 1 and the same for n2."},{"Start":"01:55.790 ","End":"02:00.030","Text":"Let\u0027s scroll down and let\u0027s take a look."},{"Start":"02:00.030 ","End":"02:06.170","Text":"Let\u0027s imagine that we have our light ray or from our laser"},{"Start":"02:06.170 ","End":"02:13.890","Text":"traveling through a less dense medium into a more dense or denser medium."},{"Start":"02:14.140 ","End":"02:17.045","Text":"This is our first case we\u0027re going,"},{"Start":"02:17.045 ","End":"02:20.860","Text":"the light ray is traveling from a less dense medium into a more dense medium,"},{"Start":"02:20.860 ","End":"02:24.470","Text":"which means that we have this increased density."},{"Start":"02:24.470 ","End":"02:28.480","Text":"What happens, the less than some medium is,"},{"Start":"02:28.480 ","End":"02:31.900","Text":"the easier it is for the light ray to travel"},{"Start":"02:31.900 ","End":"02:36.175","Text":"through and the easier it is for the light ray to travel through,"},{"Start":"02:36.175 ","End":"02:39.400","Text":"the faster its velocity will be,"},{"Start":"02:39.400 ","End":"02:41.915","Text":"the greater its velocity."},{"Start":"02:41.915 ","End":"02:45.955","Text":"Of course we know that light travels fastest through"},{"Start":"02:45.955 ","End":"02:54.055","Text":"a vacuum and the vacuum is the least dense medium that we can have."},{"Start":"02:54.055 ","End":"02:56.715","Text":"Any medium that has denser,"},{"Start":"02:56.715 ","End":"03:00.320","Text":"the light ray will travel slower."},{"Start":"03:03.080 ","End":"03:09.125","Text":"That\u0027s also how we get that n is always greater or equal to 1,"},{"Start":"03:09.125 ","End":"03:13.295","Text":"because we\u0027re always going to have a bigger number divided by"},{"Start":"03:13.295 ","End":"03:18.200","Text":"either or so c if we\u0027re looking for the refractive index of a vacuum"},{"Start":"03:18.200 ","End":"03:22.330","Text":"as we did in the previous lesson or divide it c like a bigger number in"},{"Start":"03:22.330 ","End":"03:27.470","Text":"the numerator divided by the smaller number in the denominator,"},{"Start":"03:27.470 ","End":"03:32.220","Text":"which means that we\u0027re always going to get a value greater or equal to 1."},{"Start":"03:37.940 ","End":"03:42.210","Text":"Let\u0027s write this out."},{"Start":"03:42.210 ","End":"03:50.510","Text":"The denser the medium,"},{"Start":"03:50.510 ","End":"03:55.895","Text":"the slower the wave and that means"},{"Start":"03:55.895 ","End":"04:03.940","Text":"therefore that the velocity is reduced."},{"Start":"04:06.380 ","End":"04:09.650","Text":"Now what we can see is that if we\u0027re going from"},{"Start":"04:09.650 ","End":"04:12.935","Text":"a less dense medium to a more dense medium,"},{"Start":"04:12.935 ","End":"04:19.265","Text":"so that means our v1 in the less dense medium is going to be"},{"Start":"04:19.265 ","End":"04:27.230","Text":"greater than v2 in our more dense medium because v2 will be reduced,"},{"Start":"04:27.230 ","End":"04:29.540","Text":"because the medium is denser,"},{"Start":"04:29.540 ","End":"04:32.640","Text":"which means that the wave is slower."},{"Start":"04:33.530 ","End":"04:38.365","Text":"If v1 is greater than v2,"},{"Start":"04:38.365 ","End":"04:42.155","Text":"given this equation that the refractive index n is"},{"Start":"04:42.155 ","End":"04:45.844","Text":"equal to the speed of light divided by the velocity,"},{"Start":"04:45.844 ","End":"04:48.320","Text":"because the velocity is in the denominator,"},{"Start":"04:48.320 ","End":"04:54.425","Text":"that means that the greater the denominator,"},{"Start":"04:54.425 ","End":"04:57.619","Text":"the smaller the fraction overall."},{"Start":"04:57.619 ","End":"05:07.380","Text":"That means that n1 will be smaller than n2 because the denominator of"},{"Start":"05:07.380 ","End":"05:11.210","Text":"n1 is bigger than the denominator of n2 which means that"},{"Start":"05:11.210 ","End":"05:17.640","Text":"the fraction will be smaller for n1 and for n2."},{"Start":"05:20.240 ","End":"05:23.685","Text":"Let\u0027s rewrite our equation."},{"Start":"05:23.685 ","End":"05:33.675","Text":"We had n1 sine of Theta 1 is equal to n2 sine of Theta 2."},{"Start":"05:33.675 ","End":"05:39.450","Text":"If we want to isolate out sine of Theta 2 as our y value."},{"Start":"05:39.450 ","End":"05:48.000","Text":"What we had was n1 divided by n2 of sine of Theta 1."},{"Start":"05:48.000 ","End":"05:53.070","Text":"This way of course saw in the previous lesson."},{"Start":"05:53.960 ","End":"06:01.059","Text":"Then as we said, this is going to be our gradient."},{"Start":"06:01.460 ","End":"06:09.735","Text":"If we plot this as a function where this is y and this is equal to x,"},{"Start":"06:09.735 ","End":"06:13.400","Text":"so n1 divided by 2 is m the gradient."},{"Start":"06:13.400 ","End":"06:17.600","Text":"Now because we have n1 is smaller than n2,"},{"Start":"06:17.600 ","End":"06:20.240","Text":"that means that n1 divided by"},{"Start":"06:20.240 ","End":"06:29.020","Text":"n2 is going to be smaller than 1."},{"Start":"06:30.500 ","End":"06:33.855","Text":"Let\u0027s scroll down a little bit further."},{"Start":"06:33.855 ","End":"06:37.800","Text":"If n1 divided by n2 is smaller than 1,"},{"Start":"06:37.800 ","End":"06:43.980","Text":"that therefore means that sine of Theta 2 is going to"},{"Start":"06:43.980 ","End":"06:50.640","Text":"be smaller than sine of Theta 1,"},{"Start":"06:50.640 ","End":"06:54.015","Text":"because we have sine of Theta 2"},{"Start":"06:54.015 ","End":"07:03.635","Text":"is equal to something smaller than 1 multiplied by sine of Theta 1,"},{"Start":"07:03.635 ","End":"07:08.320","Text":"which means that sine of Theta 2 is always going to be less"},{"Start":"07:08.320 ","End":"07:13.670","Text":"than sine of Theta 1 because it\u0027s being multiplied by a number smaller than 1."},{"Start":"07:13.670 ","End":"07:20.195","Text":"Then that therefore means that Theta 2 is smaller"},{"Start":"07:20.195 ","End":"07:27.465","Text":"than Theta 1 because we can [inaudible] sine both side."},{"Start":"07:27.465 ","End":"07:34.700","Text":"What does that mean if we plot our graph?"},{"Start":"07:34.700 ","End":"07:38.320","Text":"As we know, let\u0027s plot the graphs."},{"Start":"07:38.320 ","End":"07:45.550","Text":"Here, we\u0027ll have sine of Theta 2 and here sine of Theta 1."},{"Start":"07:45.550 ","End":"07:53.980","Text":"We know that sine goes something like this and over here we have 90 degrees."},{"Start":"07:53.980 ","End":"07:57.860","Text":"As we can see,"},{"Start":"07:58.080 ","End":"08:02.395","Text":"for every Theta 1 that we have,"},{"Start":"08:02.395 ","End":"08:08.950","Text":"we\u0027re going to have a smaller value for Theta 2."},{"Start":"08:08.950 ","End":"08:12.700","Text":"If we\u0027re always going to have for any value of Theta 1,"},{"Start":"08:12.700 ","End":"08:15.910","Text":"we\u0027re going to have a smaller value for Theta 2."},{"Start":"08:15.910 ","End":"08:24.145","Text":"Then what we can see is that if we\u0027re looking at this,"},{"Start":"08:24.145 ","End":"08:27.770","Text":"this is the interface"},{"Start":"08:33.990 ","End":"08:38.030","Text":"and this is the normal."},{"Start":"08:38.520 ","End":"08:45.190","Text":"If our incident ray is coming like so,"},{"Start":"08:45.190 ","End":"08:48.745","Text":"and this is the angle for Theta 1,"},{"Start":"08:48.745 ","End":"08:50.800","Text":"so for every Theta 1,"},{"Start":"08:50.800 ","End":"08:53.530","Text":"Theta 2 is going to be smaller,"},{"Start":"08:53.530 ","End":"08:55.585","Text":"it\u0027s going to be less than."},{"Start":"08:55.585 ","End":"09:01.135","Text":"That means that it will be coming out like this."},{"Start":"09:01.135 ","End":"09:05.170","Text":"This over here is Theta 2."},{"Start":"09:05.170 ","End":"09:15.130","Text":"In other words, we can see that our refracted ray is closer to the normal."},{"Start":"09:15.130 ","End":"09:16.855","Text":"Because that\u0027s Theta 2."},{"Start":"09:16.855 ","End":"09:19.855","Text":"Theta 2 is going to be smaller than Theta 1."},{"Start":"09:19.855 ","End":"09:24.010","Text":"Our refracted ray is going to be closer to the"},{"Start":"09:24.010 ","End":"09:28.165","Text":"normal than our incident ray or in other words,"},{"Start":"09:28.165 ","End":"09:30.595","Text":"the denser the medium is,"},{"Start":"09:30.595 ","End":"09:37.510","Text":"the more the ray bends towards the normal."},{"Start":"09:37.510 ","End":"09:41.485","Text":"This is what I really want you to remember."},{"Start":"09:41.485 ","End":"09:47.890","Text":"If the ray travels from a less dense medium into a more dense medium,"},{"Start":"09:47.890 ","End":"09:54.950","Text":"the refracted ray will bend towards the normal."},{"Start":"09:54.950 ","End":"09:57.270","Text":"Now let\u0027s look at the opposite."},{"Start":"09:57.270 ","End":"09:59.405","Text":"We looked at increasing density,"},{"Start":"09:59.405 ","End":"10:03.520","Text":"now let\u0027s look at decreasing density."},{"Start":"10:04.040 ","End":"10:08.219","Text":"It\u0027s going to be exactly the opposite."},{"Start":"10:08.219 ","End":"10:10.260","Text":"As we said, the denser the medium,"},{"Start":"10:10.260 ","End":"10:12.060","Text":"the slower the wave."},{"Start":"10:12.060 ","End":"10:14.145","Text":"The less dense the medium,"},{"Start":"10:14.145 ","End":"10:16.350","Text":"the faster the wave."},{"Start":"10:16.350 ","End":"10:23.140","Text":"Then that will mean that V_1 will be less than V_2,"},{"Start":"10:23.140 ","End":"10:28.510","Text":"because V_1 is the more dense and V_2 is the less dense."},{"Start":"10:28.510 ","End":"10:37.900","Text":"Then we can say that because n is equal to C divided by V,"},{"Start":"10:37.900 ","End":"10:43.270","Text":"what we\u0027ll have is that the denominator"},{"Start":"10:43.270 ","End":"10:49.984","Text":"V_1 is smaller so"},{"Start":"10:49.984 ","End":"10:55.310","Text":"n_1 is going to be bigger than n_2."},{"Start":"10:58.920 ","End":"11:07.670","Text":"Then that therefore means that when we have this equation over here,"},{"Start":"11:07.860 ","End":"11:11.050","Text":"let\u0027s rather write it over here."},{"Start":"11:11.050 ","End":"11:16.149","Text":"When we have sine of Theta 2 is equal to n_1 divided by"},{"Start":"11:16.149 ","End":"11:22.285","Text":"n_2 multiplied by sine of Theta 1,"},{"Start":"11:22.285 ","End":"11:29.180","Text":"because n_1 is greater than n_2, this over here,"},{"Start":"11:32.490 ","End":"11:39.520","Text":"n_1 divided by n_2 is going to be greater than 1,"},{"Start":"11:39.520 ","End":"11:42.100","Text":"the opposite to over here,"},{"Start":"11:42.100 ","End":"11:46.195","Text":"which means that sine of Theta 2,"},{"Start":"11:46.195 ","End":"11:48.490","Text":"which is equal to something greater than 1,"},{"Start":"11:48.490 ","End":"11:50.560","Text":"multiplied by sine of Theta 1."},{"Start":"11:50.560 ","End":"11:56.065","Text":"That means that sine of Theta 2 is going to be bigger than sine of Theta 1,"},{"Start":"11:56.065 ","End":"12:04.400","Text":"which means that Theta 2 is going to be for sure bigger than Theta 1."},{"Start":"12:06.960 ","End":"12:13.900","Text":"What does that mean? Let\u0027s say we\u0027re looking at again, our aquarium."},{"Start":"12:13.900 ","End":"12:16.060","Text":"Let\u0027s draw it big."},{"Start":"12:16.060 ","End":"12:19.945","Text":"Here we have the interface of water."},{"Start":"12:19.945 ","End":"12:24.250","Text":"Then I put my laser inside the water."},{"Start":"12:24.250 ","End":"12:27.890","Text":"Let\u0027s draw this in pink."},{"Start":"12:31.500 ","End":"12:34.960","Text":"This is now my incident ray over here."},{"Start":"12:34.960 ","End":"12:41.750","Text":"This is my normal line."},{"Start":"12:44.070 ","End":"12:48.010","Text":"We have the incident ray, the normal, and the interface."},{"Start":"12:48.010 ","End":"12:57.830","Text":"As we know, this over here is going to be my Theta 1."},{"Start":"12:58.230 ","End":"13:04.480","Text":"My Theta 1 is the angle that corresponds to the incident ray,"},{"Start":"13:04.480 ","End":"13:07.970","Text":"the angle between the incident ray and the normal."},{"Start":"13:08.250 ","End":"13:15.100","Text":"Now I know that Theta 2 is going to be in this case because I\u0027m going from water to air,"},{"Start":"13:15.100 ","End":"13:18.130","Text":"Theta 2 is always going to be bigger than"},{"Start":"13:18.130 ","End":"13:22.105","Text":"Theta 1 when we have a case of decreasing density."},{"Start":"13:22.105 ","End":"13:28.160","Text":"That means that my refracted ray will come out somewhere like so."},{"Start":"13:28.530 ","End":"13:32.229","Text":"This is the refracted ray, and of course,"},{"Start":"13:32.229 ","End":"13:34.990","Text":"this over here is my Theta 2,"},{"Start":"13:34.990 ","End":"13:38.065","Text":"which is bigger than my Theta 1."},{"Start":"13:38.065 ","End":"13:45.110","Text":"In other words, what we have over here is that the refracted ray,"},{"Start":"13:45.660 ","End":"13:50.410","Text":"when we\u0027re going from a more dense medium to a less dense medium,"},{"Start":"13:50.410 ","End":"13:54.130","Text":"or when we\u0027re going with decreasing density,"},{"Start":"13:54.130 ","End":"14:03.470","Text":"my refracted ray bends away from the normal."},{"Start":"14:04.620 ","End":"14:10.330","Text":"A little note, notice that Theta 1 is always on"},{"Start":"14:10.330 ","End":"14:16.060","Text":"1 side of the normal and Theta 2 is on the other side of the normal."},{"Start":"14:16.060 ","End":"14:18.010","Text":"This is important to note."},{"Start":"14:18.010 ","End":"14:19.735","Text":"We can see it also over here."},{"Start":"14:19.735 ","End":"14:24.070","Text":"If we look at the normal Theta 1 is on the left side"},{"Start":"14:24.070 ","End":"14:28.975","Text":"and Theta 2 is on the right side of the normal. Just note that."},{"Start":"14:28.975 ","End":"14:31.195","Text":"This is the end of the lesson."},{"Start":"14:31.195 ","End":"14:37.195","Text":"What I really want you to take away from this is that when you\u0027re looking at Snell\u0027s law,"},{"Start":"14:37.195 ","End":"14:41.785","Text":"when you\u0027re moving from a less dense medium to a more dense medium,"},{"Start":"14:41.785 ","End":"14:44.770","Text":"such as from air to water,"},{"Start":"14:44.770 ","End":"14:48.850","Text":"the refracted ray is going to bend towards the"},{"Start":"14:48.850 ","End":"14:54.685","Text":"normal as in Theta 2 is smaller than Theta 1."},{"Start":"14:54.685 ","End":"14:59.950","Text":"Conversely, when we\u0027re going from a more dense medium to a less dense medium,"},{"Start":"14:59.950 ","End":"15:01.720","Text":"we have decreasing density,"},{"Start":"15:01.720 ","End":"15:03.430","Text":"such as in this example,"},{"Start":"15:03.430 ","End":"15:05.365","Text":"going from water to air,"},{"Start":"15:05.365 ","End":"15:10.585","Text":"our refracted ray bends away from the normal or in other words,"},{"Start":"15:10.585 ","End":"15:15.130","Text":"Theta 2 will be greater than Theta 1."},{"Start":"15:15.130 ","End":"15:18.110","Text":"That\u0027s the end of this lesson."}],"ID":21462},{"Watched":false,"Name":"Snells Law Small Angles","Duration":"14m 24s","ChapterTopicVideoID":21572,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21572.jpeg","UploadDate":"2020-05-07T18:06:38.5330000","DurationForVideoObject":"PT14M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this lesson,"},{"Start":"00:01.830 ","End":"00:05.265","Text":"we\u0027re going to be continuing with Snell\u0027s law"},{"Start":"00:05.265 ","End":"00:09.960","Text":"and with refraction and what we\u0027re going to do is, in this lesson,"},{"Start":"00:09.960 ","End":"00:13.380","Text":"we\u0027re going to see another conclusion that we can arrive"},{"Start":"00:13.380 ","End":"00:17.340","Text":"at from Snell\u0027s law and that conclusion is"},{"Start":"00:17.340 ","End":"00:25.560","Text":"that the trajectory of a light ray in refraction is reversible."},{"Start":"00:25.560 ","End":"00:32.290","Text":"Now, we\u0027re going to put that rule over here and we\u0027re going to explain this."},{"Start":"00:34.820 ","End":"00:39.590","Text":"Let\u0027s write out our equation for Snell\u0027s law."},{"Start":"00:39.590 ","End":"00:41.855","Text":"We have n_1,"},{"Start":"00:41.855 ","End":"00:45.065","Text":"the refractive index of the first medium,"},{"Start":"00:45.065 ","End":"00:51.030","Text":"multiplied by sine of the array in that medium."},{"Start":"00:51.030 ","End":"00:53.180","Text":"This is usually the incident ray."},{"Start":"00:53.180 ","End":"00:54.755","Text":"If this is the first medium,"},{"Start":"00:54.755 ","End":"00:57.335","Text":"is equal to n_2,"},{"Start":"00:57.335 ","End":"01:00.170","Text":"the refractive index of the second medium,"},{"Start":"01:00.170 ","End":"01:04.850","Text":"multiplied by sine of the refracted angle;"},{"Start":"01:04.850 ","End":"01:09.150","Text":"the angle in the second medium of the ray of light."},{"Start":"01:11.060 ","End":"01:15.830","Text":"Let\u0027s imagine that we have over here 2 materials."},{"Start":"01:15.830 ","End":"01:21.050","Text":"This line is the interface and here we have air,"},{"Start":"01:21.050 ","End":"01:29.435","Text":"and here we have glass and air has a refractive index of"},{"Start":"01:29.435 ","End":"01:40.115","Text":"1 and glass has a refractive index of 1.5, approximately, of course."},{"Start":"01:40.115 ","End":"01:44.210","Text":"Now, here we have our normal."},{"Start":"01:44.210 ","End":"01:50.210","Text":"Remember, the normal is perpendicular to the interface and now"},{"Start":"01:50.210 ","End":"01:56.565","Text":"I\u0027m going to shine my incident light at this angle over here,"},{"Start":"01:56.565 ","End":"02:01.980","Text":"where the angle over here is 45 degrees."},{"Start":"02:01.980 ","End":"02:08.230","Text":"This is Theta 1 is equal to 45 degrees."},{"Start":"02:08.930 ","End":"02:13.880","Text":"We\u0027re going to get this light ray that is refracted"},{"Start":"02:13.880 ","End":"02:18.380","Text":"coming out of the other side, like so."},{"Start":"02:18.380 ","End":"02:19.790","Text":"Now, from the previous lesson,"},{"Start":"02:19.790 ","End":"02:25.565","Text":"we saw that when we go from a less dense medium air into a more dense medium,"},{"Start":"02:25.565 ","End":"02:30.450","Text":"the refracted ray bends towards the normal."},{"Start":"02:30.450 ","End":"02:36.975","Text":"We already know that Theta_2 is going to be smaller than Theta_1."},{"Start":"02:36.975 ","End":"02:42.800","Text":"Now, what we want to do is we want to calculate what the Theta_2 is."},{"Start":"02:42.890 ","End":"02:46.100","Text":"In our first medium we\u0027re dealing with air."},{"Start":"02:46.100 ","End":"02:53.520","Text":"We have n_1, the refractive index is 1 multiplied by sine of the angle Theta_1,"},{"Start":"02:53.520 ","End":"02:55.585","Text":"which is the angle in the first medium."},{"Start":"02:55.585 ","End":"02:58.890","Text":"That is 45 degrees,"},{"Start":"02:58.890 ","End":"03:01.105","Text":"which is equal to n_2,"},{"Start":"03:01.105 ","End":"03:06.275","Text":"which is 1.5, multiplied by sine of the angle,"},{"Start":"03:06.275 ","End":"03:09.540","Text":"which is what we\u0027re trying to find Theta_2."},{"Start":"03:09.830 ","End":"03:14.750","Text":"If we plug all of this into"},{"Start":"03:14.750 ","End":"03:19.730","Text":"the calculator and divide both sides by 1.5 and then arc sine,"},{"Start":"03:19.730 ","End":"03:21.760","Text":"just normal algebra,"},{"Start":"03:21.760 ","End":"03:29.290","Text":"we\u0027ll get that Theta_2 is equal to 28.1 degrees."},{"Start":"03:30.260 ","End":"03:33.915","Text":"Now, let\u0027s take a look at the same question,"},{"Start":"03:33.915 ","End":"03:35.420","Text":"but from the other side."},{"Start":"03:35.420 ","End":"03:39.430","Text":"Again, we have the interface."},{"Start":"03:39.430 ","End":"03:44.380","Text":"Again, here we have air with n as 1,"},{"Start":"03:44.380 ","End":"03:51.140","Text":"and here we have glass with n as 1.5 and"},{"Start":"03:51.140 ","End":"04:00.525","Text":"the only difference is this time we\u0027re shining a light first from here, from the glass."},{"Start":"04:00.525 ","End":"04:11.160","Text":"We shine a light like so and glass is our first medium."},{"Start":"04:11.160 ","End":"04:18.540","Text":"That means that this angle is Theta_1 and let\u0027s say that this is equal to 28.1 degrees."},{"Start":"04:18.540 ","End":"04:21.350","Text":"Then again from the previous lesson,"},{"Start":"04:21.350 ","End":"04:27.335","Text":"we saw that when we\u0027re going from a more dense medium into a less dense medium,"},{"Start":"04:27.335 ","End":"04:29.000","Text":"we have decreasing density."},{"Start":"04:29.000 ","End":"04:31.390","Text":"We\u0027re going from glass to air."},{"Start":"04:31.390 ","End":"04:36.870","Text":"Our refracted ray bends away from the normal."},{"Start":"04:36.870 ","End":"04:40.410","Text":"Then it goes like so and this is of"},{"Start":"04:40.410 ","End":"04:44.950","Text":"course our Theta_2 and now this is what we\u0027re trying to find."},{"Start":"04:45.200 ","End":"04:48.975","Text":"Again, we set up our equation."},{"Start":"04:48.975 ","End":"04:55.285","Text":"Now, the ray in the glass is the incident ray and the glass is our first medium."},{"Start":"04:55.285 ","End":"04:59.480","Text":"N_1 is the refractive index of the glass."},{"Start":"04:59.480 ","End":"05:06.130","Text":"We have 1.5 multiplied by sine of Theta_1."},{"Start":"05:06.130 ","End":"05:08.345","Text":"The angle of the incident ray,"},{"Start":"05:08.345 ","End":"05:12.170","Text":"which is equal to 28.1 degrees,"},{"Start":"05:12.170 ","End":"05:14.510","Text":"and this is equal to n_2,"},{"Start":"05:14.510 ","End":"05:17.300","Text":"the refractive index of the second medium,"},{"Start":"05:17.300 ","End":"05:18.445","Text":"which here is air,"},{"Start":"05:18.445 ","End":"05:27.860","Text":"so 1 multiplied by sine of the refractive angle,"},{"Start":"05:27.860 ","End":"05:30.620","Text":"which is Theta_2, which is what we\u0027re trying to find."},{"Start":"05:30.620 ","End":"05:35.450","Text":"Again, we plug everything into our calculator and we get that"},{"Start":"05:35.450 ","End":"05:41.100","Text":"Theta_2 is equal to 45 degrees."},{"Start":"05:41.650 ","End":"05:46.465","Text":"Notice, we have the exact same angles,"},{"Start":"05:46.465 ","End":"05:49.739","Text":"45 degrees, 45 degrees,"},{"Start":"05:49.739 ","End":"05:53.820","Text":"28.1 degrees, 28.1 degrees."},{"Start":"05:53.820 ","End":"06:01.035","Text":"In other words, the trajectory of light in refraction is reversible."},{"Start":"06:01.035 ","End":"06:07.355","Text":"It doesn\u0027t matter if I shine the light first"},{"Start":"06:07.355 ","End":"06:15.030","Text":"from air into glass or the exact reversal from glass,"},{"Start":"06:15.230 ","End":"06:19.240","Text":"like so into air."},{"Start":"06:19.240 ","End":"06:23.540","Text":"My angles are going to be the same."},{"Start":"06:23.540 ","End":"06:26.510","Text":"This is a notice the exact same image."},{"Start":"06:26.510 ","End":"06:33.515","Text":"It doesn\u0027t matter that here I began the question whether ray is coming from the glass."},{"Start":"06:33.515 ","End":"06:35.915","Text":"Here, the ray is coming from the air."},{"Start":"06:35.915 ","End":"06:40.700","Text":"If I look at a snapshot of the line that is made due to refraction,"},{"Start":"06:40.700 ","End":"06:43.960","Text":"it\u0027s going to be exactly the same,"},{"Start":"06:43.960 ","End":"06:49.955","Text":"given that I keep this ratio between the angles the same."},{"Start":"06:49.955 ","End":"06:53.270","Text":"If I start from the air,"},{"Start":"06:53.270 ","End":"06:57.245","Text":"it\u0027s going to look like this and if I start from the glass,"},{"Start":"06:57.245 ","End":"07:00.335","Text":"it\u0027s going to look exactly the same."},{"Start":"07:00.335 ","End":"07:03.290","Text":"This is the line that we\u0027re going to see."},{"Start":"07:03.290 ","End":"07:06.110","Text":"The trajectory is the same."},{"Start":"07:06.110 ","End":"07:10.820","Text":"It just matters what I put where in the equation."},{"Start":"07:10.820 ","End":"07:17.190","Text":"We can see that the equations are just mirror images of one another as well."},{"Start":"07:18.270 ","End":"07:22.810","Text":"This rule over here is the second rule or"},{"Start":"07:22.810 ","End":"07:27.760","Text":"the second conclusion rather that we can get from Snell\u0027s law,"},{"Start":"07:27.760 ","End":"07:33.640","Text":"where the first conclusion was in the previous video when we were speaking about what"},{"Start":"07:33.640 ","End":"07:36.820","Text":"happens to the refracted ray when we move from"},{"Start":"07:36.820 ","End":"07:40.660","Text":"a more dense medium or from a less dense medium."},{"Start":"07:40.660 ","End":"07:43.870","Text":"That was the first 1. This is the second conclusion,"},{"Start":"07:43.870 ","End":"07:48.970","Text":"and now I want to talk about the third conclusion."},{"Start":"07:48.970 ","End":"07:57.910","Text":"The third conclusion is what the equation for Snell\u0027s law looks like in small angles."},{"Start":"07:57.910 ","End":"08:00.730","Text":"I\u0027m going to write this out in"},{"Start":"08:00.730 ","End":"08:04.405","Text":"small angles and soon we\u0027re going to speak about what this is."},{"Start":"08:04.405 ","End":"08:06.010","Text":"Small angles."},{"Start":"08:06.010 ","End":"08:12.235","Text":"Snell\u0027s law becomes n_1 Theta_1"},{"Start":"08:12.235 ","End":"08:16.705","Text":"is equal to n_2 Theta_2."},{"Start":"08:16.705 ","End":"08:19.000","Text":"Where does this come from?"},{"Start":"08:19.000 ","End":"08:24.250","Text":"If you remember, we may have spoken previously about small angles."},{"Start":"08:24.250 ","End":"08:29.785","Text":"Small angles are considered"},{"Start":"08:29.785 ","End":"08:35.604","Text":"anything that is smaller than either in degrees,"},{"Start":"08:35.604 ","End":"08:44.260","Text":"14 degrees, or in radians 1 quarter of a radian."},{"Start":"08:44.260 ","End":"08:53.785","Text":"Because 1 quarter of a radian is approximately equal to 14.3 degrees."},{"Start":"08:53.785 ","End":"08:55.855","Text":"Where does this come from?"},{"Start":"08:55.855 ","End":"08:58.270","Text":"When we\u0027re using small angles,"},{"Start":"08:58.270 ","End":"09:02.920","Text":"which are angles smaller than 14 degrees or smaller than,"},{"Start":"09:02.920 ","End":"09:05.650","Text":"if we\u0027re working on the calculator in radians,"},{"Start":"09:05.650 ","End":"09:08.274","Text":"1 quarter of a radian,"},{"Start":"09:08.274 ","End":"09:12.768","Text":"which corresponds to approximately 14 degrees,"},{"Start":"09:12.768 ","End":"09:20.875","Text":"then what we see in this is that sine of Theta is approximately equal to Theta."},{"Start":"09:20.875 ","End":"09:24.610","Text":"Let\u0027s say if we plug into our calculator,"},{"Start":"09:24.610 ","End":"09:26.530","Text":"so first we set it to radians,"},{"Start":"09:26.530 ","End":"09:32.605","Text":"and then if we write sine of and let\u0027s say we take 0.1"},{"Start":"09:32.605 ","End":"09:41.110","Text":"radians so this is going to be equal to 0.09983,"},{"Start":"09:42.390 ","End":"09:47.755","Text":"which is approximately equal to 0.1."},{"Start":"09:47.755 ","End":"09:51.130","Text":"Similarly, if we take sine of,"},{"Start":"09:51.130 ","End":"09:53.770","Text":"let\u0027s say 0.2 radians,"},{"Start":"09:53.770 ","End":"10:01.730","Text":"what we\u0027re going to get is 0.1987,"},{"Start":"10:03.750 ","End":"10:08.870","Text":"which is approximately equal to 0.2."},{"Start":"10:08.940 ","End":"10:17.110","Text":"Or let\u0027s say if we take sine of 0.04 we\u0027re"},{"Start":"10:17.110 ","End":"10:27.260","Text":"going to get 0.03998,"},{"Start":"10:33.810 ","End":"10:38.990","Text":"which is approximately equal to 0.04."},{"Start":"10:39.870 ","End":"10:46.899","Text":"However, if we take sine of 1 radian, for instance,"},{"Start":"10:46.899 ","End":"10:51.400","Text":"what we\u0027re going to get is 0.84,"},{"Start":"10:51.400 ","End":"10:54.370","Text":"which is already quite a big difference."},{"Start":"10:54.370 ","End":"10:57.415","Text":"These examples over here,"},{"Start":"10:57.415 ","End":"11:03.370","Text":"the difference between 0.1 and 0.09983."},{"Start":"11:03.370 ","End":"11:05.680","Text":"Or this over here,"},{"Start":"11:05.680 ","End":"11:15.400","Text":"0.2 and 0.1987 or 0.04 and 0.03998 is fractions,"},{"Start":"11:15.400 ","End":"11:18.170","Text":"fractions of a percent."},{"Start":"11:20.160 ","End":"11:22.825","Text":"Over here, that\u0027s the difference."},{"Start":"11:22.825 ","End":"11:28.435","Text":"Whereas over here, the difference is approximately 16 percent,"},{"Start":"11:28.435 ","End":"11:36.790","Text":"which is already a large difference between 1 and 0.84 or exactly 60 percent."},{"Start":"11:36.790 ","End":"11:40.315","Text":"That is why in small angles,"},{"Start":"11:40.315 ","End":"11:46.057","Text":"angles less than 14 degrees or less than 1 quarter of a radian,"},{"Start":"11:46.057 ","End":"11:49.975","Text":"sine Theta is equal to Theta."},{"Start":"11:49.975 ","End":"11:55.195","Text":"In that case we can rewrite instead of n_1 sine Theta_1,"},{"Start":"11:55.195 ","End":"12:00.505","Text":"we can just write n_1 Theta_1 and instead of n_2 sine of Theta_2,"},{"Start":"12:00.505 ","End":"12:04.310","Text":"we could just write n_2 Theta_2."},{"Start":"12:05.250 ","End":"12:14.245","Text":"This we can also see when we drew a graph of Theta_2 by Theta_1."},{"Start":"12:14.245 ","End":"12:18.020","Text":"Let\u0027s say this is Theta_1 and this is Theta_2."},{"Start":"12:18.020 ","End":"12:24.555","Text":"We really saw that the graph goes up according to something like so."},{"Start":"12:24.555 ","End":"12:27.635","Text":"When we\u0027re looking at small angles,"},{"Start":"12:27.635 ","End":"12:31.949","Text":"less than 14 degrees or less than 1 quarter of a radian,"},{"Start":"12:31.949 ","End":"12:35.270","Text":"we\u0027re looking at this section over here."},{"Start":"12:36.060 ","End":"12:41.605","Text":"We can make this off and what we can see is that over here,"},{"Start":"12:41.605 ","End":"12:44.410","Text":"the line is pretty much straight,"},{"Start":"12:44.410 ","End":"12:47.995","Text":"we have a straight line gradient."},{"Start":"12:47.995 ","End":"12:56.230","Text":"Then that really does make sense if we look at this equation over here and we make"},{"Start":"12:56.230 ","End":"13:04.570","Text":"Theta_2 the subject so we get n_1 divided by n_2 of Theta_1."},{"Start":"13:04.570 ","End":"13:08.185","Text":"Again, this is like y equals mx."},{"Start":"13:08.185 ","End":"13:12.730","Text":"Where this n_1 divided by n_2 is the gradient,"},{"Start":"13:12.730 ","End":"13:14.545","Text":"which is a straight line."},{"Start":"13:14.545 ","End":"13:17.200","Text":"We have this constant over here."},{"Start":"13:17.200 ","End":"13:21.650","Text":"It makes sense that over here we have this straight line,"},{"Start":"13:22.650 ","End":"13:26.560","Text":"and as we see when we get to bigger angles, greater angles,"},{"Start":"13:26.560 ","End":"13:32.080","Text":"then this relationship of a straight line doesn\u0027t work"},{"Start":"13:32.080 ","End":"13:40.090","Text":"anymore and that\u0027s how we got the bend in 1 of the previous lessons."},{"Start":"13:40.090 ","End":"13:43.795","Text":"It makes sense that we have this straight line over here,"},{"Start":"13:43.795 ","End":"13:49.375","Text":"which means again that this equation in this region works."},{"Start":"13:49.375 ","End":"13:53.110","Text":"The 2 conclusions that we have from this lesson is that"},{"Start":"13:53.110 ","End":"13:56.725","Text":"the trajectory of the light ray and refraction is reversible."},{"Start":"13:56.725 ","End":"13:59.170","Text":"It\u0027s always going to be in the shape,"},{"Start":"13:59.170 ","End":"14:04.190","Text":"whether we\u0027re coming from a more dense medium or coming from a less dense medium."},{"Start":"14:04.380 ","End":"14:09.463","Text":"The second is the issue with small angles."},{"Start":"14:09.463 ","End":"14:12.310","Text":"We can see that our equation changed with"},{"Start":"14:12.310 ","End":"14:16.360","Text":"small angles and our definition for small angles."},{"Start":"14:16.360 ","End":"14:20.965","Text":"This please remember and write down in your equation sheets."},{"Start":"14:20.965 ","End":"14:24.260","Text":"That is the end of this lesson."}],"ID":22425},{"Watched":false,"Name":"Exercise 1","Duration":"5m 52s","ChapterTopicVideoID":21573,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21573.jpeg","UploadDate":"2020-05-07T18:07:57.2200000","DurationForVideoObject":"PT5M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.905","Text":"Hello. In this lesson,"},{"Start":"00:01.905 ","End":"00:04.995","Text":"we\u0027re going to be answering a question."},{"Start":"00:04.995 ","End":"00:08.220","Text":"Here we have a laser beam that is propagating through"},{"Start":"00:08.220 ","End":"00:12.840","Text":"water where the refractive index of water is equal to 1.33,"},{"Start":"00:12.840 ","End":"00:18.840","Text":"and it hits glass where the refractive index of glass is 1.5."},{"Start":"00:18.840 ","End":"00:21.870","Text":"Notice a higher refractive index,"},{"Start":"00:21.870 ","End":"00:26.040","Text":"means that the material is of higher density."},{"Start":"00:26.040 ","End":"00:29.070","Text":"In other words, we have something going from"},{"Start":"00:29.070 ","End":"00:33.150","Text":"a less dense medium into a more dense medium."},{"Start":"00:33.150 ","End":"00:38.150","Text":"Part of the beam is refracted and part is reflected."},{"Start":"00:38.150 ","End":"00:44.090","Text":"The angle between the water surface and the incident ray is 60 degrees."},{"Start":"00:44.090 ","End":"00:46.940","Text":"Number 1, is to calculate the angle of refraction,"},{"Start":"00:46.940 ","End":"00:51.270","Text":"and number 2 is to draw a diagram illustrating the case."},{"Start":"00:54.050 ","End":"00:59.175","Text":"I think we should start by drawing a diagram,"},{"Start":"00:59.175 ","End":"01:03.090","Text":"and then we can answer the question."},{"Start":"01:03.090 ","End":"01:06.045","Text":"Here we have the interface,"},{"Start":"01:06.045 ","End":"01:09.370","Text":"and over here,"},{"Start":"01:09.370 ","End":"01:12.745","Text":"let\u0027s say that we have water at the bottom,"},{"Start":"01:12.745 ","End":"01:16.120","Text":"and glass at the top,"},{"Start":"01:16.120 ","End":"01:20.295","Text":"and here we have our normal."},{"Start":"01:20.295 ","End":"01:26.140","Text":"Very important. We\u0027re beginning with the water."},{"Start":"01:26.140 ","End":"01:36.485","Text":"What we have is that the angle between the water surface and the incident ray is 60."},{"Start":"01:36.485 ","End":"01:42.545","Text":"If I draw that this is my incident ray,"},{"Start":"01:42.545 ","End":"01:45.170","Text":"let\u0027s draw it in blue."},{"Start":"01:45.170 ","End":"01:49.595","Text":"If this is my incident ray going like so,"},{"Start":"01:49.595 ","End":"01:53.865","Text":"don\u0027t forget to draw the arrow."},{"Start":"01:53.865 ","End":"01:58.850","Text":"The angle between the water surface and the incident ray is 60."},{"Start":"01:58.850 ","End":"02:02.885","Text":"In other words, this angle over here is"},{"Start":"02:02.885 ","End":"02:08.630","Text":"60 degrees because this is the interface or the water surface."},{"Start":"02:08.630 ","End":"02:12.740","Text":"In other words, the angle that we need is this angle."},{"Start":"02:12.740 ","End":"02:19.910","Text":"We have 90 because the normal is at right angles to the interface, always."},{"Start":"02:19.910 ","End":"02:21.740","Text":"We have 90 minus 60,"},{"Start":"02:21.740 ","End":"02:23.755","Text":"which means that this is 30."},{"Start":"02:23.755 ","End":"02:31.335","Text":"In other words, we have that Theta_1 is equal to 30 degrees."},{"Start":"02:31.335 ","End":"02:33.240","Text":"Now as we said,"},{"Start":"02:33.240 ","End":"02:37.320","Text":"we\u0027re going from water to glass,"},{"Start":"02:37.320 ","End":"02:38.820","Text":"where glass is more dense."},{"Start":"02:38.820 ","End":"02:42.020","Text":"We\u0027re going from a less dense medium to a more dense medium,"},{"Start":"02:42.020 ","End":"02:47.880","Text":"so we know that the ray is going to bend towards the normal."},{"Start":"02:49.150 ","End":"02:53.195","Text":"The refracted ray is going to look something like this,"},{"Start":"02:53.195 ","End":"02:56.520","Text":"where this is Theta_2,"},{"Start":"02:56.520 ","End":"02:59.220","Text":"and Theta_2 is going to be less than"},{"Start":"02:59.220 ","End":"03:04.380","Text":"Theta_1 because we\u0027re going from a less dense medium to a more dense medium."},{"Start":"03:04.850 ","End":"03:07.650","Text":"This is partly the diagram,"},{"Start":"03:07.650 ","End":"03:10.430","Text":"we\u0027ll get back to this later because I don\u0027t want to confuse"},{"Start":"03:10.430 ","End":"03:14.630","Text":"the diagram with stuff that isn\u0027t relevant to the calculation."},{"Start":"03:14.630 ","End":"03:16.385","Text":"Now let\u0027s do the calculation."},{"Start":"03:16.385 ","End":"03:18.860","Text":"Snell\u0027s law is n1,"},{"Start":"03:18.860 ","End":"03:21.005","Text":"and now we want to look if we\u0027re dealing with"},{"Start":"03:21.005 ","End":"03:24.560","Text":"small angles because that will change the equation."},{"Start":"03:24.560 ","End":"03:31.370","Text":"Now we remember that small angles is anything less than 14 degrees or 1/4 of a radian."},{"Start":"03:31.370 ","End":"03:33.410","Text":"But we\u0027re dealing with 30 degrees,"},{"Start":"03:33.410 ","End":"03:37.340","Text":"which is bigger, so we\u0027re not in the region of small angles,"},{"Start":"03:37.340 ","End":"03:42.215","Text":"so we use our regular equation and 1 sine of Theta_1 is"},{"Start":"03:42.215 ","End":"03:47.500","Text":"equal to n_2 sine of Theta_2."},{"Start":"03:47.500 ","End":"03:51.305","Text":"So n_1 is the refractive index of water,"},{"Start":"03:51.305 ","End":"03:57.730","Text":"so 1.33 multiplied by sine of Theta_1,"},{"Start":"03:57.730 ","End":"03:59.760","Text":"which was 30 degrees,"},{"Start":"03:59.760 ","End":"04:01.885","Text":"which is equal to n_2,"},{"Start":"04:01.885 ","End":"04:05.045","Text":"which is of gloss of the second medium,"},{"Start":"04:05.045 ","End":"04:10.290","Text":"1.5 multiplied by sine of Theta_2,"},{"Start":"04:10.290 ","End":"04:13.690","Text":"which is what we\u0027re trying to calculate."},{"Start":"04:13.760 ","End":"04:18.665","Text":"Now, if we plug all of this into our calculator,"},{"Start":"04:18.665 ","End":"04:24.170","Text":"divide both sides by 1.5 and I.S sine everything will get that"},{"Start":"04:24.170 ","End":"04:31.850","Text":"Theta_2 is equal to 26.3 degrees."},{"Start":"04:33.140 ","End":"04:40.530","Text":"This is therefore our answer to question number 1."},{"Start":"04:40.530 ","End":"04:42.840","Text":"As we can see as expected,"},{"Start":"04:42.840 ","End":"04:46.220","Text":"Theta_2 is less than Theta_1,"},{"Start":"04:46.220 ","End":"04:49.130","Text":"which was 30 degrees as expected"},{"Start":"04:49.130 ","End":"04:52.490","Text":"because we were going from a less dense medium to a more dense medium,"},{"Start":"04:52.490 ","End":"04:54.965","Text":"so we knew that Theta_2 would be smaller."},{"Start":"04:54.965 ","End":"04:58.880","Text":"Now let\u0027s just finish the diagram."},{"Start":"04:58.880 ","End":"05:05.495","Text":"In the question, we said that part of the beam was refracted and part was reflected."},{"Start":"05:05.495 ","End":"05:12.020","Text":"This we drew the refracted part and now we have to draw the reflected part."},{"Start":"05:12.500 ","End":"05:15.395","Text":"The reflected part, as we know,"},{"Start":"05:15.395 ","End":"05:19.310","Text":"if the ray is coming in at 30 degrees to the normal,"},{"Start":"05:19.310 ","End":"05:22.460","Text":"the incident ray is 30 degrees to the normal."},{"Start":"05:22.460 ","End":"05:30.200","Text":"Then that means that the reflected ray is going to be a mirror image in the normal,"},{"Start":"05:30.200 ","End":"05:33.845","Text":"so it\u0027s going to be also 30 degrees to the normal,"},{"Start":"05:33.845 ","End":"05:35.200","Text":"but in the opposite direction,"},{"Start":"05:35.200 ","End":"05:37.205","Text":"and of course if it\u0027s reflected,"},{"Start":"05:37.205 ","End":"05:40.525","Text":"don\u0027t forget ever to draw the direction,"},{"Start":"05:40.525 ","End":"05:45.400","Text":"and it\u0027s going back away from the glass."},{"Start":"05:46.100 ","End":"05:50.600","Text":"That is it. That\u0027s the diagram and this is on."},{"Start":"05:50.600 ","End":"05:53.040","Text":"That\u0027s the end of this lesson."}],"ID":22426},{"Watched":false,"Name":"2 Exercise","Duration":"10m 55s","ChapterTopicVideoID":21373,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21373.jpeg","UploadDate":"2020-04-06T22:33:55.3770000","DurationForVideoObject":"PT10M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.025","Text":"Hello. In this lesson,"},{"Start":"00:02.025 ","End":"00:04.635","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.635 ","End":"00:08.430","Text":"A student shone a laser at different angles through air and"},{"Start":"00:08.430 ","End":"00:12.390","Text":"another transparent medium with unknown refractive index."},{"Start":"00:12.390 ","End":"00:17.400","Text":"The student recorded her findings in the table below."},{"Start":"00:17.400 ","End":"00:21.210","Text":"Here we have Theta 1, the incident ray,"},{"Start":"00:21.210 ","End":"00:25.275","Text":"and Theta 2, the refracted ray."},{"Start":"00:25.275 ","End":"00:34.035","Text":"Question number 1 is the graph of Theta 2 as a function of Theta 1 expected to be linear."},{"Start":"00:34.035 ","End":"00:38.885","Text":"All right, so everything we\u0027ve learned from Snell\u0027s Law"},{"Start":"00:38.885 ","End":"00:45.020","Text":"will tell us that the answer to Question 1 is no."},{"Start":"00:45.020 ","End":"00:50.735","Text":"This is already the answer."},{"Start":"00:50.735 ","End":"00:54.470","Text":"It will not be a linear graph. Why is this?"},{"Start":"00:54.470 ","End":"00:55.835","Text":"Let\u0027s write out Snell\u0027s Law."},{"Start":"00:55.835 ","End":"01:05.970","Text":"We have n1 multiplied by Sine of Theta 1 is equal to n2 multiplied by Sine of Theta 2."},{"Start":"01:06.050 ","End":"01:13.860","Text":"If I want to isolate out Theta 2 to make this a function,"},{"Start":"01:13.860 ","End":"01:15.810","Text":"Theta 2 as a function of Theta 1."},{"Start":"01:15.810 ","End":"01:21.970","Text":"What I\u0027ll have is that Theta 2 is equal to Sine to the minus 1,"},{"Start":"01:21.970 ","End":"01:32.030","Text":"Arc Sine of n1 divided by n2 multiplied by Sine of Theta 1."},{"Start":"01:32.030 ","End":"01:34.490","Text":"As we can see,"},{"Start":"01:34.490 ","End":"01:39.020","Text":"this is a very complicated equation that clearly is not linear."},{"Start":"01:39.020 ","End":"01:42.740","Text":"I\u0027m reminding you a linear equation is where we have"},{"Start":"01:42.740 ","End":"01:46.820","Text":"an equation that\u0027s some variable y is equal to mx,"},{"Start":"01:46.820 ","End":"01:48.230","Text":"where m is a constant,"},{"Start":"01:48.230 ","End":"01:54.545","Text":"x is another variable plus c some constant if it doesn\u0027t go through the origin."},{"Start":"01:54.545 ","End":"02:01.770","Text":"This equation over here does not resemble this."},{"Start":"02:01.770 ","End":"02:06.664","Text":"This is 1 explanation of why we can see this isn\u0027t going to be a linear function,"},{"Start":"02:06.664 ","End":"02:12.660","Text":"because we don\u0027t have a constant multiplied by Theta 1."},{"Start":"02:13.190 ","End":"02:17.540","Text":"The next thing that we can look at to explain this is we can see"},{"Start":"02:17.540 ","End":"02:23.310","Text":"that the jumps in Theta 2 on constant."},{"Start":"02:23.310 ","End":"02:26.580","Text":"The jumps in Theta 1 stay the same and jumps of 10."},{"Start":"02:26.580 ","End":"02:32.135","Text":"But then here we can see that there\u0027s a jump between 0 and 7.3,"},{"Start":"02:32.135 ","End":"02:34.460","Text":"3 of just above 7."},{"Start":"02:34.460 ","End":"02:38.180","Text":"Then here we also have a jump of approximately 7."},{"Start":"02:38.180 ","End":"02:39.455","Text":"But as we continue,"},{"Start":"02:39.455 ","End":"02:43.610","Text":"we can see that slowly those jumps of approximately 5,"},{"Start":"02:43.610 ","End":"02:47.010","Text":"and here there\u0027s jumps of less than 3."},{"Start":"02:47.010 ","End":"02:54.365","Text":"Because the difference between each subsequent value isn\u0027t constant,"},{"Start":"02:54.365 ","End":"02:58.850","Text":"we can also see that the graph isn\u0027t going to be linear, and as we\u0027ve seen,"},{"Start":"02:58.850 ","End":"03:04.610","Text":"if we draw it, it will look something like so, like a curve."},{"Start":"03:04.610 ","End":"03:08.360","Text":"All right, so this is the answer to question number 1."},{"Start":"03:08.360 ","End":"03:12.665","Text":"Now let\u0027s take a look at question number 2."},{"Start":"03:12.665 ","End":"03:19.480","Text":"Define the variables or functions needed to achieve a linear graph."},{"Start":"03:20.210 ","End":"03:23.795","Text":"Again, we know from a previous lessons with"},{"Start":"03:23.795 ","End":"03:28.775","Text":"Snell\u0027s Law is that we just use Snell\u0027s Law over here,"},{"Start":"03:28.775 ","End":"03:33.070","Text":"and we just isolate out Sine of Theta 2."},{"Start":"03:33.070 ","End":"03:38.415","Text":"Sine of Theta 2 is our first variable."},{"Start":"03:38.415 ","End":"03:45.660","Text":"This is equal to n1 divided by n2 of Sine of Theta 1,"},{"Start":"03:45.660 ","End":"03:47.400","Text":"which is our second variable."},{"Start":"03:47.400 ","End":"03:50.380","Text":"We can say that this is y,"},{"Start":"03:50.990 ","End":"03:59.890","Text":"n1 divided by n2 is m and Sine of Theta 1 is x."},{"Start":"03:59.890 ","End":"04:03.004","Text":"Then we get a linear graph where we have a variable,"},{"Start":"04:03.004 ","End":"04:05.135","Text":"this is the gradient of the line,"},{"Start":"04:05.135 ","End":"04:06.800","Text":"and this is our second variable."},{"Start":"04:06.800 ","End":"04:10.670","Text":"Then the graph will look something like so where of course we don\u0027t have"},{"Start":"04:10.670 ","End":"04:16.220","Text":"this plus c because we know that in both we\u0027re going through the origin,"},{"Start":"04:16.220 ","End":"04:19.850","Text":"Sine of 0 is equal to 0."},{"Start":"04:19.850 ","End":"04:22.605","Text":"There\u0027s no plus C over here."},{"Start":"04:22.605 ","End":"04:24.840","Text":"That\u0027s the reason."},{"Start":"04:24.840 ","End":"04:29.160","Text":"Now we can answer question 3."},{"Start":"04:29.160 ","End":"04:30.660","Text":"I\u0027ll just square this."},{"Start":"04:30.660 ","End":"04:35.450","Text":"These are variables that will make the graph linear."},{"Start":"04:35.450 ","End":"04:39.290","Text":"Question 3 is to scratch this linear graph."},{"Start":"04:39.290 ","End":"04:44.870","Text":"What we want to do is we want to fill out this table with values for"},{"Start":"04:44.870 ","End":"04:52.980","Text":"Sine of Theta 1 and Sine of Theta 2."},{"Start":"04:52.980 ","End":"04:59.810","Text":"All I\u0027m going to be doing is taking these values and signing them,"},{"Start":"04:59.810 ","End":"05:05.210","Text":"putting them in the calculator with Sine and writing down their corresponding values."},{"Start":"05:05.210 ","End":"05:09.905","Text":"Here we have the graph."},{"Start":"05:09.905 ","End":"05:11.150","Text":"If you want to,"},{"Start":"05:11.150 ","End":"05:14.090","Text":"of course, you can check this yourself."},{"Start":"05:14.090 ","End":"05:18.920","Text":"Or can I just draw these lines to make it clear which values correspond to each."},{"Start":"05:18.920 ","End":"05:23.580","Text":"Now we want to sketch this linear graph."},{"Start":"05:24.280 ","End":"05:32.150","Text":"I\u0027m not going to do it perfectly because I don\u0027t have the ability to over here right now."},{"Start":"05:32.150 ","End":"05:35.585","Text":"But basically what we\u0027re going to be doing,"},{"Start":"05:35.585 ","End":"05:39.455","Text":"you can do this if you want to check this."},{"Start":"05:39.455 ","End":"05:43.320","Text":"We saw Sine of Theta 2 as our y,"},{"Start":"05:43.320 ","End":"05:51.060","Text":"so over here is Sine of Theta 2 and Sine of Theta 1 is our x."},{"Start":"05:51.060 ","End":"05:54.000","Text":"Here we have Sine of Theta 1,"},{"Start":"05:54.000 ","End":"05:57.130","Text":"and then we start at the origin 0,"},{"Start":"05:57.130 ","End":"06:02.880","Text":"0 and then just draw out your spaces over here."},{"Start":"06:02.880 ","End":"06:10.230","Text":"You\u0027ll see that you get a perfectly straight line through the origin, like so."},{"Start":"06:10.230 ","End":"06:13.850","Text":"If you have paper that has squares on it at home,"},{"Start":"06:13.850 ","End":"06:15.410","Text":"feel free to do that."},{"Start":"06:15.410 ","End":"06:19.250","Text":"Or if you know of some computer program that can"},{"Start":"06:19.250 ","End":"06:24.220","Text":"simulate it and you can just plug in these numbers, please do."},{"Start":"06:24.490 ","End":"06:33.405","Text":"All right. Now let\u0027s answer question number 4. let\u0027s do it over here."},{"Start":"06:33.405 ","End":"06:39.110","Text":"Using the graph, calculate the refractive index of the unknown transparent medium."},{"Start":"06:39.110 ","End":"06:46.330","Text":"What we\u0027re trying to find is the refractive index of this in the refracted ray."},{"Start":"06:46.330 ","End":"06:49.480","Text":"We\u0027re trying to calculate n2."},{"Start":"06:50.930 ","End":"06:57.635","Text":"First of all we know that n1 is equal to approximately 1."},{"Start":"06:57.635 ","End":"06:59.750","Text":"Why is this?"},{"Start":"06:59.750 ","End":"07:04.670","Text":"Because we know that the laser beam first goes through air,"},{"Start":"07:04.670 ","End":"07:06.890","Text":"and then through the second medium."},{"Start":"07:06.890 ","End":"07:14.880","Text":"We know that the refractive medium of air is approximately 1. We can just write that."},{"Start":"07:14.880 ","End":"07:19.880","Text":"We also know from what we did previously,"},{"Start":"07:19.880 ","End":"07:22.539","Text":"that given this graph,"},{"Start":"07:22.539 ","End":"07:30.005","Text":"m is the gradient of the graph,"},{"Start":"07:30.005 ","End":"07:34.220","Text":"and m is equal to n1 divided by n2."},{"Start":"07:34.220 ","End":"07:36.110","Text":"We can write it over here,"},{"Start":"07:36.110 ","End":"07:41.490","Text":"m is equal to n1 divided by n2,"},{"Start":"07:41.490 ","End":"07:43.530","Text":"where n1 is equal to 1."},{"Start":"07:43.530 ","End":"07:50.340","Text":"This is what we\u0027re dealing with."},{"Start":"07:50.340 ","End":"07:52.325","Text":"We\u0027re been asked to use the graph."},{"Start":"07:52.325 ","End":"07:55.025","Text":"Now we didn\u0027t draw the graph obviously accurately."},{"Start":"07:55.025 ","End":"07:56.660","Text":"But if we had,"},{"Start":"07:56.660 ","End":"08:02.490","Text":"please make sure that when you choose points on the graph to calculate the gradient,"},{"Start":"08:02.490 ","End":"08:09.155","Text":"we\u0027ll soon go over it, please make sure that you can know the exact value of that point."},{"Start":"08:09.155 ","End":"08:13.200","Text":"All right, so it needs to be as accurate as possible."},{"Start":"08:13.200 ","End":"08:16.265","Text":"It\u0027s best if you\u0027re using square paper where"},{"Start":"08:16.265 ","End":"08:20.120","Text":"the corner of the square lines up with your point,"},{"Start":"08:20.120 ","End":"08:25.905","Text":"Because then you can read the value from the graph in the most accurate way."},{"Start":"08:25.905 ","End":"08:28.410","Text":"But since we haven\u0027t done that,"},{"Start":"08:28.410 ","End":"08:31.325","Text":"what we can just do is use values from the table,"},{"Start":"08:31.325 ","End":"08:33.680","Text":"which is any way the same thing,"},{"Start":"08:33.680 ","End":"08:37.195","Text":"because we drew the graph according to these values."},{"Start":"08:37.195 ","End":"08:39.200","Text":"Or we were meant to, if we were to draw it."},{"Start":"08:39.200 ","End":"08:44.210","Text":"as we know, the gradient m is equal"},{"Start":"08:44.210 ","End":"08:51.730","Text":"to y2 minus y1 divided by x2 minus x1."},{"Start":"08:51.730 ","End":"08:57.615","Text":"In that case, we can just choose 2 values over here."},{"Start":"08:57.615 ","End":"09:01.396","Text":"Let\u0027s take this over here, 0.563,"},{"Start":"09:01.396 ","End":"09:07.980","Text":"0.563 is our y2."},{"Start":"09:07.980 ","End":"09:12.150","Text":"Remember, we\u0027re dividing it also by x2,"},{"Start":"09:12.150 ","End":"09:15.215","Text":"so the corresponding x, 0.563,"},{"Start":"09:15.215 ","End":"09:19.090","Text":"its corresponding x is 0.766,"},{"Start":"09:19.090 ","End":"09:26.430","Text":"so 0.766 minus y1 and minus x1."},{"Start":"09:26.430 ","End":"09:31.605","Text":"Let\u0027s find another value that\u0027s lower down."},{"Start":"09:31.605 ","End":"09:34.405","Text":"Let\u0027s take this over here, 0.368,"},{"Start":"09:34.405 ","End":"09:41.210","Text":"0.368 and its corresponding x 0.5."},{"Start":"09:41.210 ","End":"09:45.829","Text":"Now I\u0027m going to plug this into my calculator."},{"Start":"09:45.829 ","End":"09:52.415","Text":"What we get is 0.733."},{"Start":"09:52.415 ","End":"09:57.600","Text":"Of course we know that m is the inverse of this,"},{"Start":"09:57.600 ","End":"09:59.460","Text":"m is 1 divided by n2,"},{"Start":"09:59.460 ","End":"10:01.720","Text":"and we\u0027re trying to find n2."},{"Start":"10:02.240 ","End":"10:05.370","Text":"In that case, in order to get n2,"},{"Start":"10:05.370 ","End":"10:09.745","Text":"we take 1 divided by m. Again,"},{"Start":"10:09.745 ","End":"10:16.140","Text":"I put this into my calculator and we just get"},{"Start":"10:16.140 ","End":"10:24.640","Text":"an answer of 1.36 to 3 significant figures,"},{"Start":"10:24.640 ","End":"10:27.430","Text":"which is what we\u0027ve been doing up until now,"},{"Start":"10:27.430 ","End":"10:31.210","Text":"pretty much aside from over here where we have 4 significant figures."},{"Start":"10:31.210 ","End":"10:38.495","Text":"This is the answer to question number 4."},{"Start":"10:38.495 ","End":"10:42.590","Text":"This is the refractive index of the unknown transparent medium."},{"Start":"10:42.590 ","End":"10:45.020","Text":"Just so you know, this number corresponds"},{"Start":"10:45.020 ","End":"10:48.500","Text":"approximately to the refractive index of acetone,"},{"Start":"10:48.500 ","End":"10:52.880","Text":"which is also known as nail varnish remover."},{"Start":"10:52.880 ","End":"10:56.160","Text":"That\u0027s the end of this lesson."}],"ID":21463},{"Watched":false,"Name":"Total Internal Reflection and Critical Angle","Duration":"27m 46s","ChapterTopicVideoID":21574,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21574.jpeg","UploadDate":"2020-05-07T18:16:31.6930000","DurationForVideoObject":"PT27M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.235","Text":"Hello. In this lesson,"},{"Start":"00:02.235 ","End":"00:07.800","Text":"we\u0027re going to be speaking about 3 new topics or terms."},{"Start":"00:07.800 ","End":"00:09.450","Text":"We have the grazing angle,"},{"Start":"00:09.450 ","End":"00:13.920","Text":"total internal reflection, and the critical angle."},{"Start":"00:13.920 ","End":"00:16.450","Text":"This is what we\u0027re going to be speaking about."},{"Start":"00:17.900 ","End":"00:21.795","Text":"Let\u0027s begin. Now a quick little note, up until now,"},{"Start":"00:21.795 ","End":"00:24.630","Text":"we\u0027ve used the simplistic term of going"},{"Start":"00:24.630 ","End":"00:29.025","Text":"from when we\u0027re dealing with refraction from 1 medium to another."},{"Start":"00:29.025 ","End":"00:34.530","Text":"Where usually we\u0027ve been speaking about 1 medium that is denser than the other."},{"Start":"00:34.530 ","End":"00:39.345","Text":"Let\u0027s say that here is our interface,"},{"Start":"00:39.345 ","End":"00:41.190","Text":"and here we have air,"},{"Start":"00:41.190 ","End":"00:44.070","Text":"and here we have water,"},{"Start":"00:44.070 ","End":"00:48.380","Text":"and that this is the normal to the interface."},{"Start":"00:48.380 ","End":"00:50.930","Text":"We said that water is denser than air,"},{"Start":"00:50.930 ","End":"00:54.840","Text":"so it will have a higher refractive index."},{"Start":"00:54.940 ","End":"00:57.335","Text":"This is what we\u0027ve been using."},{"Start":"00:57.335 ","End":"01:02.925","Text":"We can also say that water is denser and that air is rarer."},{"Start":"01:02.925 ","End":"01:08.030","Text":"Instead of saying that 1 medium is less dense than the other,"},{"Start":"01:08.030 ","End":"01:12.065","Text":"we can say it\u0027s rarer than the other."},{"Start":"01:12.065 ","End":"01:16.540","Text":"Air, we could call the rarer,"},{"Start":"01:16.540 ","End":"01:19.380","Text":"medium and water,"},{"Start":"01:19.380 ","End":"01:23.595","Text":"we can call the denser medium."},{"Start":"01:23.595 ","End":"01:25.430","Text":"It would be the same if we were going,"},{"Start":"01:25.430 ","End":"01:27.215","Text":"let\u0027s say, from air to glass,"},{"Start":"01:27.215 ","End":"01:31.580","Text":"or from vacuum, which would be rare to glass which would be dense."},{"Start":"01:31.580 ","End":"01:37.290","Text":"These is just 2 extra terms that we can go through."},{"Start":"01:38.560 ","End":"01:42.070","Text":"Let us begin."},{"Start":"01:42.070 ","End":"01:43.625","Text":"First of all, again,"},{"Start":"01:43.625 ","End":"01:45.500","Text":"we\u0027re using Snell\u0027s law."},{"Start":"01:45.500 ","End":"01:47.390","Text":"Let\u0027s remember it, so n_1,"},{"Start":"01:47.390 ","End":"01:49.985","Text":"the refractive index in medium 1,"},{"Start":"01:49.985 ","End":"01:52.860","Text":"multiplied by sine of Theta 1,"},{"Start":"01:52.860 ","End":"01:55.785","Text":"where Theta 1 is the angle of incidence,"},{"Start":"01:55.785 ","End":"02:00.030","Text":"is equal to n_2 sine of Theta 2,"},{"Start":"02:00.030 ","End":"02:02.930","Text":"where n_2 is the refractive index of the second medium,"},{"Start":"02:02.930 ","End":"02:09.060","Text":"and Theta 2 is the angle of the refracted ray."},{"Start":"02:09.170 ","End":"02:12.525","Text":"Let\u0027s look at 2 little examples."},{"Start":"02:12.525 ","End":"02:20.235","Text":"Let\u0027s say that I send down a laser beam like so right along the normal over here."},{"Start":"02:20.235 ","End":"02:24.990","Text":"Its angle is 0 degrees."},{"Start":"02:24.990 ","End":"02:28.225","Text":"In this case, we\u0027ll have our refractive index."},{"Start":"02:28.225 ","End":"02:30.205","Text":"Now it\u0027s for air, but it doesn\u0027t really matter,"},{"Start":"02:30.205 ","End":"02:31.780","Text":"n_1 for any material,"},{"Start":"02:31.780 ","End":"02:36.590","Text":"this is right, multiplied by sine of 0."},{"Start":"02:37.480 ","End":"02:41.150","Text":"We know that sine of 0 is equal to 0,"},{"Start":"02:41.150 ","End":"02:43.565","Text":"so everything over here will equal 0."},{"Start":"02:43.565 ","End":"02:50.190","Text":"Which therefore means that sine of Theta 2 will also be equal to 0,"},{"Start":"02:50.190 ","End":"02:53.675","Text":"so therefore Theta 2 is equal to 0."},{"Start":"02:53.675 ","End":"03:01.150","Text":"In other words, the refracted ray continues down like so at 0 degrees."},{"Start":"03:01.150 ","End":"03:03.635","Text":"When we have this 0 angle,"},{"Start":"03:03.635 ","End":"03:09.090","Text":"we get 0 refraction in this case."},{"Start":"03:10.120 ","End":"03:13.115","Text":"Now, let\u0027s give another example."},{"Start":"03:13.115 ","End":"03:17.850","Text":"Let\u0027s say we send down a ray like so."},{"Start":"03:17.850 ","End":"03:20.175","Text":"This is our incident ray,"},{"Start":"03:20.175 ","End":"03:24.785","Text":"and let\u0027s say that this angle to the normal is 20 degrees."},{"Start":"03:24.785 ","End":"03:28.040","Text":"As we\u0027ve seen, it will be refracted,"},{"Start":"03:28.040 ","End":"03:32.120","Text":"and when we go from a rarer medium into a denser medium,"},{"Start":"03:32.120 ","End":"03:34.985","Text":"or from a less dense medium into a denser medium,"},{"Start":"03:34.985 ","End":"03:37.040","Text":"just a different way of saying it."},{"Start":"03:37.040 ","End":"03:42.615","Text":"We saw that instead of the ray carrying on in a straight line,"},{"Start":"03:42.615 ","End":"03:44.760","Text":"it will be refracted,"},{"Start":"03:44.760 ","End":"03:47.375","Text":"and when we go from rarer to denser,"},{"Start":"03:47.375 ","End":"03:52.910","Text":"the ray will bend slightly towards the normal,"},{"Start":"03:52.910 ","End":"03:57.140","Text":"where the normal is this over here perpendicular to the interface."},{"Start":"03:57.140 ","End":"03:58.970","Text":"This angle over here,"},{"Start":"03:58.970 ","End":"04:01.960","Text":"I don\u0027t know exactly what it will be."},{"Start":"04:01.960 ","End":"04:03.890","Text":"We\u0027re not going to do the calculation now,"},{"Start":"04:03.890 ","End":"04:08.400","Text":"but it\u0027s going to be less than 20 degrees."},{"Start":"04:09.730 ","End":"04:16.235","Text":"Similarly, if we send in now another incident ray,"},{"Start":"04:16.235 ","End":"04:20.480","Text":"and let\u0027s say it\u0027s at 50 degrees."},{"Start":"04:20.480 ","End":"04:24.750","Text":"That means 50 degrees to the normal."},{"Start":"04:24.750 ","End":"04:30.904","Text":"Again, instead of it traveling in a straight line as it did in 0 degrees,"},{"Start":"04:30.904 ","End":"04:38.794","Text":"it\u0027s going to be refracted and again it\u0027s going to bend towards the normal."},{"Start":"04:38.794 ","End":"04:46.220","Text":"It\u0027s going to have a slightly smaller angle than 50 degrees. This is its angle."},{"Start":"04:51.800 ","End":"04:55.440","Text":"Let\u0027s take a look at 80 degrees."},{"Start":"04:55.440 ","End":"04:59.580","Text":"Let\u0027s say 80 degrees is this over here,"},{"Start":"04:59.580 ","End":"05:02.775","Text":"so this is 80 degrees."},{"Start":"05:02.775 ","End":"05:12.150","Text":"Again, this is going to bend in and its angle is going to be less than 80 degrees."},{"Start":"05:12.350 ","End":"05:17.705","Text":"It\u0027s going to look something like so because it\u0027s refracted."},{"Start":"05:17.705 ","End":"05:23.405","Text":"Now, what is going to happen if I take,"},{"Start":"05:23.405 ","End":"05:26.310","Text":"trying to choose a color."},{"Start":"05:26.990 ","End":"05:34.795","Text":"I\u0027ll use gray. If I take my laser beam and I shine the laser,"},{"Start":"05:34.795 ","End":"05:38.660","Text":"obviously I couldn\u0027t get exactly parallel."},{"Start":"05:38.660 ","End":"05:40.130","Text":"Imagine this is a straight line."},{"Start":"05:40.130 ","End":"05:45.110","Text":"I can\u0027t get exactly parallel to the interface, but it\u0027s close."},{"Start":"05:45.110 ","End":"05:52.235","Text":"My angle here is something like 89.9999, recurring,"},{"Start":"05:52.235 ","End":"05:53.885","Text":"or in other words,"},{"Start":"05:53.885 ","End":"05:56.790","Text":"this angle is called Alpha,"},{"Start":"05:57.160 ","End":"06:00.230","Text":"and it\u0027s this over here."},{"Start":"06:00.230 ","End":"06:02.425","Text":"This is Alpha."},{"Start":"06:02.425 ","End":"06:07.125","Text":"Alpha approaches 90 degrees."},{"Start":"06:07.125 ","End":"06:11.540","Text":"It\u0027s never going to be exactly 90 degrees because it\u0027s impossible for us to do that,"},{"Start":"06:11.540 ","End":"06:12.740","Text":"but it\u0027s going to be very,"},{"Start":"06:12.740 ","End":"06:16.080","Text":"very close, 99999 recurring."},{"Start":"06:16.080 ","End":"06:18.555","Text":"This angle is called Alpha,"},{"Start":"06:18.555 ","End":"06:19.910","Text":"or in other words,"},{"Start":"06:19.910 ","End":"06:28.570","Text":"it\u0027s also the grazing angle because it grazes the interface between the 2 media."},{"Start":"06:28.570 ","End":"06:30.655","Text":"Let\u0027s just write this down."},{"Start":"06:30.655 ","End":"06:32.325","Text":"The angle Alpha,"},{"Start":"06:32.325 ","End":"06:36.859","Text":"which is always approaching 90 degrees,"},{"Start":"06:36.859 ","End":"06:38.690","Text":"is called the grazing,"},{"Start":"06:38.690 ","End":"06:42.780","Text":"and another word for it is also glancing."},{"Start":"06:42.780 ","End":"06:45.620","Text":"If you see that in a textbook,"},{"Start":"06:45.620 ","End":"06:48.875","Text":"so grazing or glancing angle."},{"Start":"06:48.875 ","End":"06:50.390","Text":"Why is this?"},{"Start":"06:50.390 ","End":"06:57.560","Text":"Because the incident beam grazes the interface between the 2 media."},{"Start":"06:57.560 ","End":"07:00.845","Text":"This could be a way to remember it."},{"Start":"07:00.845 ","End":"07:05.740","Text":"It just means that it\u0027s approaching 90 degrees."},{"Start":"07:05.740 ","End":"07:16.230","Text":"The refracted ray is again going to bend towards the normal over here."},{"Start":"07:16.300 ","End":"07:20.700","Text":"Here we can see its angle."},{"Start":"07:20.780 ","End":"07:28.715","Text":"Now we want to calculate this angle over here in gray, Theta 2."},{"Start":"07:28.715 ","End":"07:30.970","Text":"Using Snell\u0027s law,"},{"Start":"07:30.970 ","End":"07:33.450","Text":"we\u0027re going to soon substitute in air and water,"},{"Start":"07:33.450 ","End":"07:34.725","Text":"but in the meantime,"},{"Start":"07:34.725 ","End":"07:39.810","Text":"we have n_1 multiplied by sine of Theta 1,"},{"Start":"07:39.810 ","End":"07:42.230","Text":"where Theta 1 is approaching 90."},{"Start":"07:42.230 ","End":"07:45.710","Text":"We can just write that it\u0027s equal to 90 degrees,"},{"Start":"07:45.710 ","End":"07:53.230","Text":"because it\u0027s very close, which is equal to n_2 multiplied by sine of Theta 2."},{"Start":"07:53.230 ","End":"07:57.165","Text":"Sine of 90 is equal to 1."},{"Start":"07:57.165 ","End":"08:06.570","Text":"What we get is that therefore n_1 is equal to n_2 sine of Theta 2."},{"Start":"08:06.570 ","End":"08:08.685","Text":"Or in other words,"},{"Start":"08:08.685 ","End":"08:17.350","Text":"sine of Theta 2 is equal to n_1 divided by n_2."},{"Start":"08:20.030 ","End":"08:29.505","Text":"What this is going to give us is the maximal angle."},{"Start":"08:29.505 ","End":"08:33.140","Text":"The maximal refractive angle in"},{"Start":"08:33.140 ","End":"08:40.160","Text":"a given material is sine to the negative 1 of n_1 divided by n_2."},{"Start":"08:40.160 ","End":"08:43.330","Text":"That\u0027s basically what this gives us."},{"Start":"08:43.330 ","End":"08:46.485","Text":"For our example over here,"},{"Start":"08:46.485 ","End":"08:51.800","Text":"Theta 2 is equal to sine to the negative 1 of n_1,"},{"Start":"08:51.800 ","End":"08:54.350","Text":"the refractive index of air, which is 1,"},{"Start":"08:54.350 ","End":"08:55.955","Text":"divided by n_2,"},{"Start":"08:55.955 ","End":"08:57.590","Text":"the refractive index of water,"},{"Start":"08:57.590 ","End":"09:00.060","Text":"which is approximately 1.33."},{"Start":"09:00.380 ","End":"09:07.140","Text":"Then we arc sign both sides and we get 48.8 degrees."},{"Start":"09:07.140 ","End":"09:12.170","Text":"This is the maximum refractive angle or angle of"},{"Start":"09:12.170 ","End":"09:19.760","Text":"refraction that we can get when we\u0027re going between air and water."},{"Start":"09:20.120 ","End":"09:29.620","Text":"We\u0027re never going to get a larger angle of refraction given air and water as our 2 media."},{"Start":"09:29.990 ","End":"09:34.335","Text":"I want you to remember that Alpha is approaching 90."},{"Start":"09:34.335 ","End":"09:37.040","Text":"When that happens, in other words,"},{"Start":"09:37.040 ","End":"09:40.775","Text":"when the angle is just approaching 90 degrees,"},{"Start":"09:40.775 ","End":"09:43.730","Text":"that\u0027s called the grazing or the glancing angle."},{"Start":"09:43.730 ","End":"09:51.559","Text":"When our incident ray is almost parallel to the interface between 2 medium,"},{"Start":"09:51.559 ","End":"10:00.980","Text":"then what we get is that we can calculate Theta 2 is equal to this."},{"Start":"10:00.980 ","End":"10:05.460","Text":"This is the maximum refractive angle."},{"Start":"10:05.890 ","End":"10:07.910","Text":"In this region,"},{"Start":"10:07.910 ","End":"10:11.500","Text":"we\u0027re never going to get a refracted ray."},{"Start":"10:11.500 ","End":"10:17.900","Text":"These are the 2 things to remember right now."},{"Start":"10:20.850 ","End":"10:24.250","Text":"Here we call this angle Theta 2."},{"Start":"10:24.250 ","End":"10:27.070","Text":"But I want to call it something else."},{"Start":"10:27.070 ","End":"10:36.500","Text":"I want to call it Theta c. Let\u0027s just change this to Theta c. Remember this."},{"Start":"10:39.150 ","End":"10:49.160","Text":"We saw that Theta c when we have air and water is equal to 48.8 degrees."},{"Start":"10:49.680 ","End":"10:54.350","Text":"Let\u0027s imagine the exact same thing."},{"Start":"10:55.080 ","End":"10:57.655","Text":"The same situation."},{"Start":"10:57.655 ","End":"11:00.175","Text":"But this time, we\u0027re going to go from water,"},{"Start":"11:00.175 ","End":"11:04.525","Text":"which is denser to air, which is rarer."},{"Start":"11:04.525 ","End":"11:07.150","Text":"Let\u0027s begin again like we did before."},{"Start":"11:07.150 ","End":"11:09.805","Text":"Imagine I have my incident ray,"},{"Start":"11:09.805 ","End":"11:13.165","Text":"like so at an angle of,"},{"Start":"11:13.165 ","End":"11:15.370","Text":"sorry, I keep writing Theta."},{"Start":"11:15.370 ","End":"11:18.190","Text":"At an angle of 0 degrees to the normal."},{"Start":"11:18.190 ","End":"11:23.860","Text":"Just like before, same calculation as this over here."},{"Start":"11:23.860 ","End":"11:29.905","Text":"We\u0027re going to get our refracted ray over here also at 0 degrees to the normal."},{"Start":"11:29.905 ","End":"11:33.235","Text":"In other words, there\u0027s going to be no refraction."},{"Start":"11:33.235 ","End":"11:36.910","Text":"Our laser beam just goes straight through."},{"Start":"11:36.910 ","End":"11:44.485","Text":"Now, let\u0027s imagine that we again have from water,"},{"Start":"11:44.485 ","End":"11:49.520","Text":"our incident ray is at 20 degrees."},{"Start":"11:50.040 ","End":"11:52.360","Text":"It reaches the interface."},{"Start":"11:52.360 ","End":"11:55.495","Text":"Just as we\u0027ve learned, when a ray goes from"},{"Start":"11:55.495 ","End":"12:00.430","Text":"a denser medium to a less dense medium or to a rarer medium,"},{"Start":"12:00.430 ","End":"12:05.170","Text":"the light ray is going to bend away from the normal."},{"Start":"12:05.170 ","End":"12:09.010","Text":"It\u0027s going to go with something like so where"},{"Start":"12:09.010 ","End":"12:13.790","Text":"this angle over here is bigger than 20 degrees."},{"Start":"12:14.130 ","End":"12:23.245","Text":"Similarly, if we take another incident ray,"},{"Start":"12:23.245 ","End":"12:27.309","Text":"let\u0027s say that this is 30 degrees."},{"Start":"12:27.309 ","End":"12:33.400","Text":"It\u0027s going to bend further away from the normal, like so."},{"Start":"12:33.400 ","End":"12:39.699","Text":"Its angle from the normal will be greater than 30 degrees."},{"Start":"12:39.699 ","End":"12:41.740","Text":"Then we, of course,"},{"Start":"12:41.740 ","End":"12:48.190","Text":"have our Theta c over here,"},{"Start":"12:48.190 ","End":"12:54.080","Text":"which we saw is at about 48.8 degrees."},{"Start":"12:54.300 ","End":"12:59.605","Text":"Let\u0027s write this over here 48.8 degrees,"},{"Start":"12:59.605 ","End":"13:05.635","Text":"which is our Theta c. Just like"},{"Start":"13:05.635 ","End":"13:09.625","Text":"before in previous lessons where we read"},{"Start":"13:09.625 ","End":"13:14.635","Text":"that whether we\u0027re going from air to water or water to air,"},{"Start":"13:14.635 ","End":"13:17.935","Text":"the pattern is going to look the same."},{"Start":"13:17.935 ","End":"13:20.905","Text":"We saw that if we\u0027re at a critical angle,"},{"Start":"13:20.905 ","End":"13:25.540","Text":"it corresponds to our grazing or a glancing angle,"},{"Start":"13:25.540 ","End":"13:30.595","Text":"where Alpha is at 90 degrees or approaching 90 degrees."},{"Start":"13:30.595 ","End":"13:34.015","Text":"Our refracted ray is going to be this,"},{"Start":"13:34.015 ","End":"13:36.040","Text":"which we called Alpha."},{"Start":"13:36.040 ","End":"13:42.520","Text":"Remember the critical angle also works backwards."},{"Start":"13:42.520 ","End":"13:45.895","Text":"Just like if Alpha was the incident ray,"},{"Start":"13:45.895 ","End":"13:47.920","Text":"we would get the critical angle."},{"Start":"13:47.920 ","End":"13:51.226","Text":"If our critical angle is the incident ray,"},{"Start":"13:51.226 ","End":"13:55.040","Text":"we\u0027ll get Alpha, just like in a previous lesson."},{"Start":"13:55.080 ","End":"14:02.170","Text":"Now, what happens if let\u0027s take gray again."},{"Start":"14:02.170 ","End":"14:09.340","Text":"If I have an incident ray going from my denser medium to my less dense medium,"},{"Start":"14:09.340 ","End":"14:13.570","Text":"which is bigger than 48.8 degrees."},{"Start":"14:13.570 ","End":"14:18.520","Text":"Let\u0027s say I have an incident ray, like so."},{"Start":"14:18.520 ","End":"14:25.070","Text":"Let\u0027s imagine that my incident ray is at 60 degrees."},{"Start":"14:25.170 ","End":"14:28.540","Text":"We\u0027ve already seen that when we\u0027re looking at my Theta c,"},{"Start":"14:28.540 ","End":"14:30.235","Text":"my critical angle,"},{"Start":"14:30.235 ","End":"14:33.610","Text":"which soon we\u0027re going to speak about a little bit more."},{"Start":"14:33.610 ","End":"14:36.020","Text":"We\u0027ll define it properly."},{"Start":"14:36.780 ","End":"14:44.230","Text":"We\u0027ve already seen that my refracted ray is going to be at approximately 90 degrees."},{"Start":"14:44.230 ","End":"14:49.030","Text":"It\u0027s going to be parallel to the interface."},{"Start":"14:49.030 ","End":"14:54.280","Text":"There\u0027s no more that it can bend away from the normal."},{"Start":"14:54.280 ","End":"14:59.230","Text":"Because that\u0027s what we\u0027ve seen when going from a denser to a less dense medium,"},{"Start":"14:59.230 ","End":"15:07.455","Text":"the angle Theta 2 is always going to be greater than the incident angle."},{"Start":"15:07.455 ","End":"15:12.670","Text":"But now we can\u0027t go more because we\u0027re already at 90 degrees."},{"Start":"15:12.860 ","End":"15:16.575","Text":"What happens is, well,"},{"Start":"15:16.575 ","End":"15:17.640","Text":"we can see what happens."},{"Start":"15:17.640 ","End":"15:19.970","Text":"Let\u0027s do a calculation."},{"Start":"15:19.970 ","End":"15:24.025","Text":"We\u0027re going from water, that\u0027s our n1."},{"Start":"15:24.025 ","End":"15:32.410","Text":"Our refractive index is 1.33 multiplied by sine of Theta 1."},{"Start":"15:32.410 ","End":"15:33.826","Text":"That\u0027s our incident ray,"},{"Start":"15:33.826 ","End":"15:39.070","Text":"so over here that\u0027s 60 degrees is equal to n2,"},{"Start":"15:39.070 ","End":"15:40.810","Text":"the refractive index of air,"},{"Start":"15:40.810 ","End":"15:45.325","Text":"which is 1 multiplied by sine of Theta 2."},{"Start":"15:45.325 ","End":"15:49.735","Text":"This is the angle that we\u0027re trying to calculate."},{"Start":"15:49.735 ","End":"15:55.030","Text":"What we will get is that sine of Theta 2,"},{"Start":"15:55.030 ","End":"15:57.235","Text":"once we plug it into the calculator,"},{"Start":"15:57.235 ","End":"16:00.140","Text":"is equal to 1.15."},{"Start":"16:01.020 ","End":"16:03.940","Text":"Up until now, great."},{"Start":"16:03.940 ","End":"16:08.170","Text":"Now, if we arc sine all of this in order to get Theta 2,"},{"Start":"16:08.170 ","End":"16:13.460","Text":"our calculator is going to say, \"Math error\"."},{"Start":"16:13.560 ","End":"16:20.125","Text":"Pretty shocking. The calculator cannot calculate this."},{"Start":"16:20.125 ","End":"16:26.300","Text":"What is actually going to happen in this case?"},{"Start":"16:27.870 ","End":"16:35.604","Text":"What happens is that when we go from a denser medium to a less dense medium,"},{"Start":"16:35.604 ","End":"16:42.610","Text":"and we shine a laser beam at an angle greater than the critical angle."},{"Start":"16:42.610 ","End":"16:46.855","Text":"What happens is that we do not get refraction."},{"Start":"16:46.855 ","End":"16:51.070","Text":"We get total internal reflection."},{"Start":"16:51.070 ","End":"16:54.490","Text":"Total internal reflection. In other words,"},{"Start":"16:54.490 ","End":"16:55.757","Text":"if this ray,"},{"Start":"16:55.757 ","End":"16:57.430","Text":"our incident ray,"},{"Start":"16:57.430 ","End":"16:58.480","Text":"is at 60 degrees,"},{"Start":"16:58.480 ","End":"17:02.050","Text":"it\u0027s going to be totally internally reflected."},{"Start":"17:02.050 ","End":"17:06.550","Text":"The whole ray is going to be reflected."},{"Start":"17:06.550 ","End":"17:11.275","Text":"I\u0027ll draw this in gray to correspond to the incident ray."},{"Start":"17:11.275 ","End":"17:20.935","Text":"It\u0027s going to be completely reflected at a 60-degree angle to the normal."},{"Start":"17:20.935 ","End":"17:26.390","Text":"All the rules of reflection apply."},{"Start":"17:26.430 ","End":"17:34.360","Text":"Similarly, if we had an incident ray like so from a dense medium"},{"Start":"17:34.360 ","End":"17:42.715","Text":"to a rarer medium and let\u0027s say that its angle was 70 degrees."},{"Start":"17:42.715 ","End":"17:46.090","Text":"When we\u0027re going from a denser medium to"},{"Start":"17:46.090 ","End":"17:50.890","Text":"a less dense medium we\u0027re going to get no refraction."},{"Start":"17:50.890 ","End":"17:53.350","Text":"This is total internal reflection."},{"Start":"17:53.350 ","End":"17:58.010","Text":"Everything total, everything is reflected."},{"Start":"17:58.260 ","End":"18:03.445","Text":"Our ray is going to come out like so."},{"Start":"18:03.445 ","End":"18:09.025","Text":"Also, at an angle of 70 degrees to the normal,"},{"Start":"18:09.025 ","End":"18:14.446","Text":"just like in reflection."},{"Start":"18:14.446 ","End":"18:18.820","Text":"Total internal reflection. Here,"},{"Start":"18:18.820 ","End":"18:21.310","Text":"we have the rule."},{"Start":"18:21.310 ","End":"18:24.280","Text":"When the refractive index of"},{"Start":"18:24.280 ","End":"18:29.130","Text":"the first medium is greater than the refractive index of the second medium,"},{"Start":"18:29.130 ","End":"18:32.920","Text":"in other words, we\u0027re going from"},{"Start":"18:32.920 ","End":"18:40.610","Text":"a denser medium to a less dense or a rare medium."},{"Start":"18:42.930 ","End":"18:46.165","Text":"In addition, Theta 1,"},{"Start":"18:46.165 ","End":"18:53.964","Text":"so the incident ray is greater than Theta c, the critical angle."},{"Start":"18:53.964 ","End":"18:57.190","Text":"There is no refraction."},{"Start":"18:57.190 ","End":"18:59.950","Text":"The ray is just reflected."},{"Start":"18:59.950 ","End":"19:09.115","Text":"As we saw, we went from water and supposedly into air where water is denser."},{"Start":"19:09.115 ","End":"19:14.530","Text":"That means that n1 is greater than n2."},{"Start":"19:14.530 ","End":"19:23.170","Text":"Also, we set our incident angle Theta 1 at greater than the critical ray,"},{"Start":"19:23.170 ","End":"19:29.689","Text":"so the gray and black incident rays we can see they\u0027re 60 and 70 degrees."},{"Start":"19:30.870 ","End":"19:33.895","Text":"We got no refraction."},{"Start":"19:33.895 ","End":"19:40.820","Text":"We just got reflection as we saw according to the laws of reflection."},{"Start":"19:43.010 ","End":"19:45.255","Text":"This is this equation."},{"Start":"19:45.255 ","End":"19:47.130","Text":"Now, Theta c, as we said,"},{"Start":"19:47.130 ","End":"19:49.080","Text":"is the critical angle."},{"Start":"19:49.080 ","End":"19:52.410","Text":"Here we have the critical angle."},{"Start":"19:52.410 ","End":"19:57.240","Text":"This is measured when n_1 is less than n_2."},{"Start":"19:57.240 ","End":"20:06.609","Text":"We\u0027re going from a rarer medium to a denser medium."},{"Start":"20:07.940 ","End":"20:13.480","Text":"I.e., we\u0027re going from air to glass."},{"Start":"20:13.610 ","End":"20:18.975","Text":"When we\u0027re going from a less dense medium to a more dense medium,"},{"Start":"20:18.975 ","End":"20:22.680","Text":"such as from air to water, the maximum angle of refraction,"},{"Start":"20:22.680 ","End":"20:28.395","Text":"the maximum value for Theta 2,"},{"Start":"20:28.395 ","End":"20:33.645","Text":"that is Theta c, the critical angle."},{"Start":"20:33.645 ","End":"20:39.510","Text":"This is achieved when the incident ray is at grazing angle."},{"Start":"20:39.510 ","End":"20:45.880","Text":"As in when our incident ray is at its maximal angle."},{"Start":"20:47.390 ","End":"20:51.360","Text":"Here, imagine the arrow was like so,"},{"Start":"20:51.360 ","End":"20:55.785","Text":"i.e., it is at approximately 90 degrees."},{"Start":"20:55.785 ","End":"20:57.660","Text":"Or in other words,"},{"Start":"20:57.660 ","End":"21:00.615","Text":"it\u0027s at grazing or glancing angle."},{"Start":"21:00.615 ","End":"21:03.240","Text":"Then, this incident ray,"},{"Start":"21:03.240 ","End":"21:12.525","Text":"then the refracted ray is going to be at its maximal angle of refraction,"},{"Start":"21:12.525 ","End":"21:20.110","Text":"which corresponds or is just called Theta c, the critical angle."},{"Start":"21:20.210 ","End":"21:27.450","Text":"When we have an incident ray which is at 90 degrees to the interface"},{"Start":"21:27.450 ","End":"21:34.740","Text":"between the two media and we\u0027re going from a less dense medium to a more dense medium,"},{"Start":"21:34.740 ","End":"21:38.700","Text":"then we will achieve critical angle."},{"Start":"21:38.700 ","End":"21:42.400","Text":"The maximum angle of refraction."},{"Start":"21:43.070 ","End":"21:45.420","Text":"That is what this means."},{"Start":"21:45.420 ","End":"21:50.820","Text":"Then, if we do this in the other direction,"},{"Start":"21:50.820 ","End":"21:55.020","Text":"now we\u0027re looking at this way."},{"Start":"21:55.020 ","End":"22:00.060","Text":"Now when we\u0027re going from n_1,"},{"Start":"22:00.060 ","End":"22:02.910","Text":"which is denser than n_2,"},{"Start":"22:02.910 ","End":"22:05.520","Text":"if n_1 is greater than n_2,"},{"Start":"22:05.520 ","End":"22:10.425","Text":"that means we\u0027re going from a more dense medium to a less dense medium."},{"Start":"22:10.425 ","End":"22:13.695","Text":"Let\u0027s say from water to air."},{"Start":"22:13.695 ","End":"22:19.950","Text":"If the incident ray is at critical angle to the normal,"},{"Start":"22:19.950 ","End":"22:29.790","Text":"then the refracted ray will be at grazing or glancing angle."},{"Start":"22:29.790 ","End":"22:35.190","Text":"It will be at 90 degrees or approaching 90 degrees to the normal."},{"Start":"22:35.190 ","End":"22:45.255","Text":"This will be the refracted ray corresponding to an incident ray at critical angle."},{"Start":"22:45.255 ","End":"22:48.975","Text":"If we\u0027re going from, again,"},{"Start":"22:48.975 ","End":"22:52.260","Text":"the more dense medium to the less dense medium"},{"Start":"22:52.260 ","End":"22:57.720","Text":"at any angle greater than the critical angle,"},{"Start":"22:57.720 ","End":"23:01.575","Text":"further away from the normal."},{"Start":"23:01.575 ","End":"23:07.185","Text":"Then what we get is that there will be no refraction,"},{"Start":"23:07.185 ","End":"23:09.180","Text":"the ray is reflected."},{"Start":"23:09.180 ","End":"23:18.600","Text":"If I were in a denser medium to a less dense medium and our critical angle is x,"},{"Start":"23:18.600 ","End":"23:24.165","Text":"if we shine in a beam at x plus 1 degree away from the normal,"},{"Start":"23:24.165 ","End":"23:29.490","Text":"what we\u0027re going to get is reflection where the reflected angle"},{"Start":"23:29.490 ","End":"23:36.070","Text":"is also going to be x plus 1 degree away from the normal."},{"Start":"23:37.310 ","End":"23:44.050","Text":"This reflection that we get is called total internal reflection."},{"Start":"23:44.930 ","End":"23:48.150","Text":"Of course, the grazing angle,"},{"Start":"23:48.150 ","End":"23:50.057","Text":"we\u0027re just going to go over it again,"},{"Start":"23:50.057 ","End":"23:52.290","Text":"we can denote it by Alpha."},{"Start":"23:52.290 ","End":"23:56.880","Text":"Where Alpha is just an angle that is approaching 90 degrees and"},{"Start":"23:56.880 ","End":"24:01.860","Text":"we can also call it Theta 1 if you want to."},{"Start":"24:01.860 ","End":"24:06.060","Text":"When it\u0027s going like so this is Alpha,"},{"Start":"24:06.060 ","End":"24:08.860","Text":"which is also equal to Theta 1."},{"Start":"24:10.430 ","End":"24:16.260","Text":"If Theta 1 is equal to Alpha,"},{"Start":"24:16.260 ","End":"24:19.680","Text":"which is approaching 90 degrees,"},{"Start":"24:19.680 ","End":"24:27.710","Text":"then we know that Theta 2 is going to be equal to Theta c, the critical angle."},{"Start":"24:27.710 ","End":"24:32.255","Text":"This is also something that you can write."},{"Start":"24:32.255 ","End":"24:38.430","Text":"This just corresponds to this definition over here."},{"Start":"24:40.130 ","End":"24:42.885","Text":"Sorry if this is getting repetitive,"},{"Start":"24:42.885 ","End":"24:49.530","Text":"it\u0027s just important that you remember or understand this."},{"Start":"24:49.530 ","End":"24:52.335","Text":"Just like we learned a few lessons ago,"},{"Start":"24:52.335 ","End":"24:57.585","Text":"where if this is our incident ray,"},{"Start":"24:57.585 ","End":"25:01.565","Text":"then this will be the angle of our refracted ray,"},{"Start":"25:01.565 ","End":"25:06.510","Text":"and if this same angle is now suddenly our incident ray,"},{"Start":"25:06.510 ","End":"25:10.170","Text":"we\u0027re going to get the same angle in the other side is the refracted ray."},{"Start":"25:10.170 ","End":"25:12.000","Text":"If a line is in this shape,"},{"Start":"25:12.000 ","End":"25:15.540","Text":"it doesn\u0027t matter if the arrow is pointing in this direction"},{"Start":"25:15.540 ","End":"25:20.490","Text":"or if it\u0027s pointing in this direction,"},{"Start":"25:20.490 ","End":"25:26.205","Text":"the shape of the line is always going to be the same relative to the normal,"},{"Start":"25:26.205 ","End":"25:27.615","Text":"if we remember that."},{"Start":"25:27.615 ","End":"25:34.635","Text":"Similarly here, if we have that Theta 1 is equal to Theta c,"},{"Start":"25:34.635 ","End":"25:40.230","Text":"now we\u0027re in the other medium therefore,"},{"Start":"25:40.230 ","End":"25:45.585","Text":"we can say that Theta 2 is going to be equal to Alpha."},{"Start":"25:45.585 ","End":"25:55.140","Text":"Which of course, we know Alpha is approaching 90 degrees."},{"Start":"25:55.140 ","End":"25:59.535","Text":"That means that here we\u0027re going from the second medium,"},{"Start":"25:59.535 ","End":"26:04.590","Text":"and we\u0027re shining our beam at the critical angle so we know that"},{"Start":"26:04.590 ","End":"26:10.455","Text":"the refracted angle is going to be Alpha or approximately 90 degrees."},{"Start":"26:10.455 ","End":"26:14.680","Text":"Just from this rule over here."},{"Start":"26:14.690 ","End":"26:17.985","Text":"I\u0027m going to scroll down."},{"Start":"26:17.985 ","End":"26:21.150","Text":"Now, in some textbooks,"},{"Start":"26:21.150 ","End":"26:26.775","Text":"this definition for the critical angle will be slightly different."},{"Start":"26:26.775 ","End":"26:29.280","Text":"It will just give you similar to over here."},{"Start":"26:29.280 ","End":"26:31.260","Text":"It will just give you the inverse."},{"Start":"26:31.260 ","End":"26:39.165","Text":"It will say that when n_1 is greater than n_2,"},{"Start":"26:39.165 ","End":"26:46.330","Text":"when the first medium is denser than the second medium,"},{"Start":"26:48.080 ","End":"26:52.570","Text":"it will define the critical angle as"},{"Start":"26:52.940 ","End":"27:02.650","Text":"the angle at which any angle greater than it is going to be internally reflected."},{"Start":"27:04.490 ","End":"27:07.605","Text":"What we have here for the critical angle,"},{"Start":"27:07.605 ","End":"27:09.030","Text":"or in other words,"},{"Start":"27:09.030 ","End":"27:12.405","Text":"when n_1 is greater than n_2,"},{"Start":"27:12.405 ","End":"27:19.230","Text":"we\u0027re going from a denser medium to a rarer or a less dense medium."},{"Start":"27:19.230 ","End":"27:25.200","Text":"Then any incident angle greater than Theta c,"},{"Start":"27:25.200 ","End":"27:32.920","Text":"greater than the critical angle will experience TIR or total internal reflection."},{"Start":"27:33.320 ","End":"27:37.650","Text":"These two definitions basically mean the same thing"},{"Start":"27:37.650 ","End":"27:42.975","Text":"but you may see it written like this so that you won\u0027t get confused."},{"Start":"27:42.975 ","End":"27:46.720","Text":"That is the end of this lesson."}],"ID":22427},{"Watched":false,"Name":"Optical Fiber","Duration":"14m 26s","ChapterTopicVideoID":21575,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21575.jpeg","UploadDate":"2020-05-07T18:20:43.4670000","DurationForVideoObject":"PT14M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"Hello. In this lesson,"},{"Start":"00:02.295 ","End":"00:08.130","Text":"we\u0027re going to be speaking about one of the uses of total internal reflection,"},{"Start":"00:08.130 ","End":"00:10.715","Text":"which is what we learned in the previous lesson."},{"Start":"00:10.715 ","End":"00:15.360","Text":"This is the use of optical fibers."},{"Start":"00:15.360 ","End":"00:20.160","Text":"Optical fibers are just flexible, transparent fibers,"},{"Start":"00:20.160 ","End":"00:26.680","Text":"which means just very thin hoses."},{"Start":"00:27.560 ","End":"00:30.005","Text":"When I say very thin,"},{"Start":"00:30.005 ","End":"00:35.490","Text":"I mean they are narrower than a strand of hair"},{"Start":"00:35.490 ","End":"00:42.185","Text":"mostly and they are made out of some transparent material."},{"Start":"00:42.185 ","End":"00:51.335","Text":"What we want is that the refractive index of the material will be as high as possible."},{"Start":"00:51.335 ","End":"00:59.050","Text":"Often the optical fibers are made out of a material called silica."},{"Start":"00:59.050 ","End":"01:05.060","Text":"Let\u0027s speak about the uses first of all of optical fiber and then we\u0027re going to speak"},{"Start":"01:05.060 ","End":"01:11.430","Text":"about why we need this high refractive index and how they actually work."},{"Start":"01:13.280 ","End":"01:17.380","Text":"Another thing about the optical fiber is that it is used"},{"Start":"01:17.380 ","End":"01:21.205","Text":"to transmit light. What do we do?"},{"Start":"01:21.205 ","End":"01:25.300","Text":"Let\u0027s say you are a doctor and a patient"},{"Start":"01:25.300 ","End":"01:29.934","Text":"comes to you and says that they have a problem something in their esophagus."},{"Start":"01:29.934 ","End":"01:36.080","Text":"A section of their throat or in their sinuses up their nose."},{"Start":"01:36.080 ","End":"01:41.290","Text":"If we imagine the face of a person."},{"Start":"01:41.290 ","End":"01:46.435","Text":"Please, excuse my drawing."},{"Start":"01:46.435 ","End":"01:52.275","Text":"Here we have a nice big nose."},{"Start":"01:52.275 ","End":"01:55.530","Text":"This is what the person looks like."},{"Start":"01:55.530 ","End":"02:01.740","Text":"This is their ear and this is their eye."},{"Start":"02:01.740 ","End":"02:06.570","Text":"If they have a problem in their sinuses or in their throat, in their esophagus."},{"Start":"02:06.570 ","End":"02:13.210","Text":"If you want to look at a section over here or somewhere in their sinuses up here."},{"Start":"02:13.210 ","End":"02:17.300","Text":"We don\u0027t have a direct line that we"},{"Start":"02:17.300 ","End":"02:21.650","Text":"can look down in order to get to these distant places."},{"Start":"02:21.650 ","End":"02:26.000","Text":"Instead, what we can do is we can take our optical fiber and"},{"Start":"02:26.000 ","End":"02:31.310","Text":"just push it through the nose, like so."},{"Start":"02:31.310 ","End":"02:36.380","Text":"I\u0027m reminding you it\u0027s approximately the width of a hair or down the throat."},{"Start":"02:36.380 ","End":"02:39.153","Text":"Of course, this isn\u0027t very comfortable,"},{"Start":"02:39.153 ","End":"02:40.895","Text":"but it does the job."},{"Start":"02:40.895 ","End":"02:44.854","Text":"Using these optical fibers,"},{"Start":"02:44.854 ","End":"02:48.500","Text":"which use total internal reflection,"},{"Start":"02:48.500 ","End":"02:53.450","Text":"we can actually see what\u0027s going on in these hard to reach places."},{"Start":"02:53.450 ","End":"02:58.650","Text":"What I\u0027m going to do is I\u0027m going to draw this now in Bic."},{"Start":"02:58.650 ","End":"03:02.390","Text":"Right now we have this optical fiber that is"},{"Start":"03:02.390 ","End":"03:06.720","Text":"going through the nose and the sinuses or the throat."},{"Start":"03:07.250 ","End":"03:11.160","Text":"Here is what we\u0027re looking at."},{"Start":"03:11.160 ","End":"03:17.700","Text":"Over here there\u0027s something that emits light, some torch."},{"Start":"03:17.700 ","End":"03:21.580","Text":"The light that is reflected from the point that we\u0027re looking at"},{"Start":"03:21.580 ","End":"03:27.350","Text":"will come into the optical fiber."},{"Start":"03:28.580 ","End":"03:35.025","Text":"A ray of light will enter the fiber."},{"Start":"03:35.025 ","End":"03:39.030","Text":"What we want is total internal reflection,"},{"Start":"03:39.030 ","End":"03:42.490","Text":"because if the ray is refracted we\u0027re going"},{"Start":"03:42.490 ","End":"03:46.060","Text":"to lose it out into the throat or into the sinuses,"},{"Start":"03:46.060 ","End":"03:53.825","Text":"in which case, whoever is over here won\u0027t be able to see what is going on inside."},{"Start":"03:53.825 ","End":"03:56.270","Text":"What we want is total internal reflection."},{"Start":"03:56.270 ","End":"03:59.900","Text":"We want the ray to reflect all the way until it"},{"Start":"03:59.900 ","End":"04:05.310","Text":"reaches outside so that the person looking can see exactly what\u0027s going on."},{"Start":"04:06.400 ","End":"04:15.460","Text":"I\u0027m reminding you, if this angle over here, let\u0027s say Theta."},{"Start":"04:16.420 ","End":"04:22.355","Text":"If Theta is greater than Theta c,"},{"Start":"04:22.355 ","End":"04:30.720","Text":"the critical angle, then we\u0027re going to get total internal reflection."},{"Start":"04:33.440 ","End":"04:37.840","Text":"In which case, we\u0027ll be able to see what\u0027s happening."},{"Start":"04:37.840 ","End":"04:42.950","Text":"If this Theta over here,"},{"Start":"04:42.950 ","End":"04:47.165","Text":"the angle of incidence is greater than the critical angle,"},{"Start":"04:47.165 ","End":"04:55.240","Text":"then it\u0027s going to be reflected in the normal at the exact same angle,"},{"Start":"04:55.240 ","End":"04:58.625","Text":"then it will be reflected something like so."},{"Start":"04:58.625 ","End":"05:01.925","Text":"Where this angle is of course still Theta,"},{"Start":"05:01.925 ","End":"05:06.545","Text":"or rather, this isn\u0027t exactly correct."},{"Start":"05:06.545 ","End":"05:10.155","Text":"Technically, this angle over here is Theta,"},{"Start":"05:10.155 ","End":"05:14.650","Text":"because it\u0027s the angle to the normal, remember that."},{"Start":"05:14.650 ","End":"05:19.265","Text":"Again, if this is the normal."},{"Start":"05:19.265 ","End":"05:24.745","Text":"Again, if this angle Theta over here is greater than the critical angle, so again,"},{"Start":"05:24.745 ","End":"05:29.769","Text":"we\u0027re going to get total internal reflection"},{"Start":"05:29.769 ","End":"05:34.285","Text":"to something like so at the same angle Theta."},{"Start":"05:34.285 ","End":"05:39.475","Text":"Again, if this angle Theta is greater than the critical angle,"},{"Start":"05:39.475 ","End":"05:45.050","Text":"we\u0027re going to get total internal reflection at the same angle Theta."},{"Start":"05:45.290 ","End":"05:48.450","Text":"Again, you\u0027ll get the point."},{"Start":"05:48.450 ","End":"05:53.685","Text":"It will continue reflecting like"},{"Start":"05:53.685 ","End":"06:01.415","Text":"so until it reflects out and reaches the eye."},{"Start":"06:01.415 ","End":"06:07.325","Text":"Now we can just draw the arrow so we can see the direction of the light ray."},{"Start":"06:07.325 ","End":"06:09.005","Text":"This is why we want"},{"Start":"06:09.005 ","End":"06:15.750","Text":"total internal reflection in the optical fiber so that we can see what\u0027s going on."},{"Start":"06:16.940 ","End":"06:21.120","Text":"Now we understand how it works and its use."},{"Start":"06:21.120 ","End":"06:24.965","Text":"Why do we want a very high refractive index?"},{"Start":"06:24.965 ","End":"06:28.380","Text":"I\u0027m reminding you of Snell\u0027s Law."},{"Start":"06:30.980 ","End":"06:35.200","Text":"We have this over here."},{"Start":"06:36.530 ","End":"06:40.030","Text":"In order to find the critical angle,"},{"Start":"06:40.030 ","End":"06:43.900","Text":"what we do is we know"},{"Start":"06:43.900 ","End":"06:52.555","Text":"that when we\u0027re going from this more dense medium into a less dense medium."},{"Start":"06:52.555 ","End":"06:56.230","Text":"Because that from the previous lesson was what we know we"},{"Start":"06:56.230 ","End":"07:00.125","Text":"need in order to get total internal reflection."},{"Start":"07:00.125 ","End":"07:06.220","Text":"We\u0027re going from a denser medium in to a rarer medium or a less dense medium."},{"Start":"07:06.680 ","End":"07:09.970","Text":"In order to calculate the critical angle,"},{"Start":"07:09.970 ","End":"07:19.000","Text":"we know that Theta 2 is going to be at 90 degrees."},{"Start":"07:19.640 ","End":"07:22.305","Text":"I\u0027ll just give a little recap."},{"Start":"07:22.305 ","End":"07:26.880","Text":"Let\u0027s say this is our interface and this is our normal."},{"Start":"07:26.880 ","End":"07:29.550","Text":"This is the dense medium,"},{"Start":"07:29.550 ","End":"07:31.380","Text":"and this is the rarer medium,"},{"Start":"07:31.380 ","End":"07:33.000","Text":"the less dense medium."},{"Start":"07:33.000 ","End":"07:36.960","Text":"In order to calculate the critical angle,"},{"Start":"07:36.960 ","End":"07:41.970","Text":"either we could look at a ray coming from this side,"},{"Start":"07:41.970 ","End":"07:44.940","Text":"this direction, from the rarer medium,"},{"Start":"07:44.940 ","End":"07:47.865","Text":"where the angle over here,"},{"Start":"07:47.865 ","End":"07:52.125","Text":"Alpha is approaching 90 degrees."},{"Start":"07:52.125 ","End":"07:57.480","Text":"Then we\u0027ll find our critical angle over here,"},{"Start":"07:57.480 ","End":"08:01.634","Text":"Theta c. If we do that,"},{"Start":"08:01.634 ","End":"08:08.340","Text":"then what we can say is that n_1 sine of 90,"},{"Start":"08:08.340 ","End":"08:13.500","Text":"which is equal to 1."},{"Start":"08:13.500 ","End":"08:16.350","Text":"Then we have just n_1 multiplied by 1,"},{"Start":"08:16.350 ","End":"08:25.080","Text":"which is equal to n_2 multiplied by sine of our Theta 2,"},{"Start":"08:25.080 ","End":"08:27.105","Text":"which over here is our critical angle."},{"Start":"08:27.105 ","End":"08:30.180","Text":"Because we know that if Theta 1 is 90,"},{"Start":"08:30.180 ","End":"08:33.870","Text":"then this is going to be our critical angle."},{"Start":"08:33.870 ","End":"08:36.420","Text":"Then specifically if we\u0027re going,"},{"Start":"08:36.420 ","End":"08:40.500","Text":"let\u0027s say here we know that the ray is going to be coming out in air,"},{"Start":"08:40.500 ","End":"08:43.890","Text":"we can just say that here we have air,"},{"Start":"08:43.890 ","End":"08:46.380","Text":"where the n_1,"},{"Start":"08:46.380 ","End":"08:48.915","Text":"its refractive index is 1."},{"Start":"08:48.915 ","End":"08:57.735","Text":"That means therefore that 1 divided by n_2 is equal to sine of"},{"Start":"08:57.735 ","End":"09:03.450","Text":"Theta c. Now what we can do is we can try"},{"Start":"09:03.450 ","End":"09:10.110","Text":"and find a material that has the highest refractive index as possible."},{"Start":"09:10.110 ","End":"09:14.070","Text":"The higher the greater the value for n_2,"},{"Start":"09:14.070 ","End":"09:16.200","Text":"the smaller the fraction."},{"Start":"09:16.200 ","End":"09:26.040","Text":"Then when we do sine of minus 1 of 1 divided by n_2 will get our Theta c,"},{"Start":"09:26.040 ","End":"09:29.235","Text":"our critical angle,"},{"Start":"09:29.235 ","End":"09:36.825","Text":"and sine of the smallest number gives us a small degree."},{"Start":"09:36.825 ","End":"09:42.735","Text":"Let\u0027s say if we choose a refractive index n_2 to be equal to 3."},{"Start":"09:42.735 ","End":"09:46.620","Text":"Let\u0027s say that n_2 is equal to 3."},{"Start":"09:46.620 ","End":"09:53.010","Text":"Sine to the minus 1 of 1/3 gives us"},{"Start":"09:53.010 ","End":"10:02.565","Text":"a critical angle that is equal to approximately 19.5 degrees."},{"Start":"10:02.565 ","End":"10:07.185","Text":"I\u0027m reminding you that this is 19.5 degrees away from the normal."},{"Start":"10:07.185 ","End":"10:11.505","Text":"That means that our critical angle over here,"},{"Start":"10:11.505 ","End":"10:15.330","Text":"we could draw it something like so."},{"Start":"10:15.330 ","End":"10:17.085","Text":"Where this over here,"},{"Start":"10:17.085 ","End":"10:22.330","Text":"Theta c is equal to 19.5 degrees."},{"Start":"10:22.850 ","End":"10:26.310","Text":"This is a very small angle."},{"Start":"10:26.310 ","End":"10:30.030","Text":"That means that any angle greater than this is"},{"Start":"10:30.030 ","End":"10:33.480","Text":"going to experience total internal reflection."},{"Start":"10:33.480 ","End":"10:37.110","Text":"Which means that the whole ray is going to be reflected"},{"Start":"10:37.110 ","End":"10:41.595","Text":"through the optical fiber to the doctor,"},{"Start":"10:41.595 ","End":"10:46.300","Text":"physician who is looking into your sinus."},{"Start":"10:46.790 ","End":"10:52.770","Text":"Why is it critical for us to have a small angle for are critical angle as possible?"},{"Start":"10:52.770 ","End":"10:55.845","Text":"Because when we put the hose in,"},{"Start":"10:55.845 ","End":"11:00.075","Text":"it\u0027s not going in a straight line like so."},{"Start":"11:00.075 ","End":"11:03.390","Text":"There\u0027s going to be significant bending"},{"Start":"11:03.390 ","End":"11:07.335","Text":"because you have to let say in the nose go all the way up like this."},{"Start":"11:07.335 ","End":"11:10.455","Text":"If you\u0027re a critical angle is bigger,"},{"Start":"11:10.455 ","End":"11:15.825","Text":"then the more you bend the optical fiber,"},{"Start":"11:15.825 ","End":"11:18.960","Text":"the more of your ray is going to be lost"},{"Start":"11:18.960 ","End":"11:23.385","Text":"and the less clearly you\u0027ll see what\u0027s going on inside the body."},{"Start":"11:23.385 ","End":"11:25.875","Text":"Whereas the smaller the critical angle,"},{"Start":"11:25.875 ","End":"11:30.630","Text":"that means that there\u0027s a higher chance or the more you can bend"},{"Start":"11:30.630 ","End":"11:36.310","Text":"the optical fiber and none of the light rays will be lost."},{"Start":"11:36.440 ","End":"11:39.390","Text":"If we draw a diagram of this,"},{"Start":"11:39.390 ","End":"11:47.490","Text":"let\u0027s say over here where there\u0027s extreme curvature like so,"},{"Start":"11:47.490 ","End":"11:50.730","Text":"and this is where our light is coming out,"},{"Start":"11:50.730 ","End":"11:53.655","Text":"and this is where our problem is inside our sinus,"},{"Start":"11:53.655 ","End":"11:56.620","Text":"and here\u0027s the person that\u0027s looking."},{"Start":"11:57.080 ","End":"12:08.220","Text":"What will happen if we have a large critical angle is that if this ray comes like this,"},{"Start":"12:08.220 ","End":"12:17.280","Text":"to the normal, let\u0027s say we need a critical angle of 30 degrees."},{"Start":"12:17.280 ","End":"12:23.490","Text":"If this angle here is smaller than 30 degrees,"},{"Start":"12:23.490 ","End":"12:25.785","Text":"then we\u0027re going to get refraction."},{"Start":"12:25.785 ","End":"12:27.900","Text":"However, if it\u0027s greater than 30 degrees,"},{"Start":"12:27.900 ","End":"12:29.325","Text":"let\u0027s say 40 degrees,"},{"Start":"12:29.325 ","End":"12:33.824","Text":"then here it will be reflected or so at 40 degrees."},{"Start":"12:33.824 ","End":"12:38.069","Text":"Similarly here it will be reflected at 40 degrees."},{"Start":"12:38.069 ","End":"12:42.060","Text":"But then because of this curvature to the normal over"},{"Start":"12:42.060 ","End":"12:47.690","Text":"here it will also"},{"Start":"12:47.690 ","End":"12:52.445","Text":"be reflected because it\u0027s greater than the critical angle."},{"Start":"12:52.445 ","End":"12:54.960","Text":"But then over here,"},{"Start":"12:58.820 ","End":"13:01.830","Text":"we might get diffraction."},{"Start":"13:01.830 ","End":"13:06.300","Text":"Because once it gets to this position over here,"},{"Start":"13:06.300 ","End":"13:10.140","Text":"this could be potentially 27 degrees."},{"Start":"13:10.140 ","End":"13:13.230","Text":"Because notice the angle is constantly changing."},{"Start":"13:13.230 ","End":"13:19.020","Text":"Sometimes, it\u0027s most clear here we have a very opaque angle,"},{"Start":"13:19.020 ","End":"13:22.480","Text":"whereas here the angle is more acute."},{"Start":"13:25.670 ","End":"13:30.630","Text":"If our critical angle is as small as possible,"},{"Start":"13:30.630 ","End":"13:34.725","Text":"the chance of us getting to this position is much lower."},{"Start":"13:34.725 ","End":"13:41.160","Text":"Let\u0027s say if our critical angle over here was indeed 19.5 degrees,"},{"Start":"13:41.160 ","End":"13:48.600","Text":"then we would still experience this total internal refraction."},{"Start":"13:48.600 ","End":"13:54.930","Text":"Whereas if it was 30, then our problem would end already over here."},{"Start":"13:54.930 ","End":"13:57.600","Text":"Or rather our solution."},{"Start":"13:57.600 ","End":"14:00.330","Text":"We wouldn\u0027t be able to see what\u0027s going on."},{"Start":"14:00.330 ","End":"14:05.385","Text":"That\u0027s why we need a very small critical angle."},{"Start":"14:05.385 ","End":"14:08.445","Text":"In order to get a small critical angle,"},{"Start":"14:08.445 ","End":"14:14.670","Text":"we need as large as possible of a refractive index"},{"Start":"14:14.670 ","End":"14:22.180","Text":"as we have demonstrated in this equation by using Snell\u0027s law."},{"Start":"14:23.420 ","End":"14:26.740","Text":"That\u0027s the end of this lesson."}],"ID":22428},{"Watched":false,"Name":"Exercise - Total Internal Reflection","Duration":"12m 21s","ChapterTopicVideoID":21576,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21576.jpeg","UploadDate":"2020-05-07T18:25:40.5900000","DurationForVideoObject":"PT12M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this lesson,"},{"Start":"00:01.830 ","End":"00:04.770","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.770 ","End":"00:08.655","Text":"A ray of light propagates through glass where the refractive index of"},{"Start":"00:08.655 ","End":"00:14.730","Text":"1.5 and reaches the interface with water or its interface with water,"},{"Start":"00:14.730 ","End":"00:18.640","Text":"where water has a refractive index of 1.33."},{"Start":"00:18.640 ","End":"00:22.400","Text":"Sketch the ray\u0027s path as it crosses the interface."},{"Start":"00:22.400 ","End":"00:24.985","Text":"It\u0027s angles of incident are,"},{"Start":"00:24.985 ","End":"00:27.840","Text":"Theta_1, 0 degrees,"},{"Start":"00:27.840 ","End":"00:30.465","Text":"30 degrees, and 70 degrees."},{"Start":"00:30.465 ","End":"00:35.435","Text":"For each case, we\u0027re going to just sketch what is going on."},{"Start":"00:35.435 ","End":"00:37.835","Text":"This will be case number 1,"},{"Start":"00:37.835 ","End":"00:39.050","Text":"this will be case number 2,"},{"Start":"00:39.050 ","End":"00:40.795","Text":"and this is case number 3."},{"Start":"00:40.795 ","End":"00:43.140","Text":"Let\u0027s look at number 1."},{"Start":"00:43.140 ","End":"00:52.259","Text":"First of all, here\u0027s our interface and we\u0027re going through glass into water."},{"Start":"00:52.760 ","End":"00:56.210","Text":"From the refractive indexes,"},{"Start":"00:56.210 ","End":"01:01.370","Text":"we can see that glass has a higher refractive index than water,"},{"Start":"01:01.370 ","End":"01:11.460","Text":"which means that glass is dense and water in this case is rare or in other words,"},{"Start":"01:11.460 ","End":"01:18.210","Text":"we\u0027re going from a more dense medium to a less dense medium."},{"Start":"01:18.880 ","End":"01:22.985","Text":"I really advise you to write this down"},{"Start":"01:22.985 ","End":"01:27.720","Text":"every time you get to one of these types of questions."},{"Start":"01:27.720 ","End":"01:35.995","Text":"Now, this is our normal to the interface."},{"Start":"01:35.995 ","End":"01:39.630","Text":"Let\u0027s deal with case number 1 when our angles of"},{"Start":"01:39.630 ","End":"01:49.745","Text":"incidence is 0 degrees."},{"Start":"01:49.745 ","End":"01:57.200","Text":"Let\u0027s draw our incoming ray in blue. Here it is."},{"Start":"01:57.200 ","End":"02:06.640","Text":"This is our incident ray and it is at 0 degrees to the normal."},{"Start":"02:06.640 ","End":"02:11.250","Text":"Now from what we know of Snell\u0027s Law,"},{"Start":"02:11.250 ","End":"02:20.520","Text":"so n_1 sine of Theta_1 is equal to n_2 sine(Theta 2)."},{"Start":"02:20.520 ","End":"02:23.990","Text":"N_1 is the refractive index in glass."},{"Start":"02:23.990 ","End":"02:28.700","Text":"We have 1.5 multiplied by sine(Theta_1)."},{"Start":"02:28.700 ","End":"02:36.445","Text":"Theta_1 is 0 degrees is equal to n_2,"},{"Start":"02:36.445 ","End":"02:43.985","Text":"which we have is the refractive index of water 1.33 multiplied by sine(Theta_2),"},{"Start":"02:43.985 ","End":"02:47.540","Text":"which is what we\u0027re trying to calculate right now."},{"Start":"02:47.540 ","End":"02:51.750","Text":"Sine(0) is 0 as we know."},{"Start":"02:51.750 ","End":"02:54.495","Text":"This whole side is equal to 0."},{"Start":"02:54.495 ","End":"03:00.135","Text":"Therefore, we know that Theta_2 is also equal to 0."},{"Start":"03:00.135 ","End":"03:04.250","Text":"Of course, this is what we also saw in the previous videos."},{"Start":"03:04.250 ","End":"03:08.360","Text":"If our incident ray is at 0 degrees,"},{"Start":"03:08.360 ","End":"03:10.415","Text":"so we have 0 refraction,"},{"Start":"03:10.415 ","End":"03:16.380","Text":"it just continues in a straight line, just through."},{"Start":"03:16.460 ","End":"03:19.050","Text":"For case number 1,"},{"Start":"03:19.050 ","End":"03:25.205","Text":"we can see that Theta_2 is equal to 0 or so from what we know and also from Snell\u0027s Law."},{"Start":"03:25.205 ","End":"03:28.955","Text":"Now, let\u0027s look at our second case."},{"Start":"03:28.955 ","End":"03:33.000","Text":"Now our Theta_1 is 30 degrees."},{"Start":"03:33.000 ","End":"03:36.495","Text":"Again, let\u0027s draw the interface."},{"Start":"03:36.495 ","End":"03:44.090","Text":"Again, we\u0027re going from glass, which is dense,"},{"Start":"03:44.090 ","End":"03:50.780","Text":"to water, which is less dense or we can just call it rare,"},{"Start":"03:50.780 ","End":"03:52.160","Text":"and here again,"},{"Start":"03:52.160 ","End":"03:53.930","Text":"we have the normal."},{"Start":"03:53.930 ","End":"03:57.775","Text":"Now our incident ray is at 30 degrees."},{"Start":"03:57.775 ","End":"03:59.640","Text":"We can draw it like so."},{"Start":"03:59.640 ","End":"04:04.445","Text":"Of course, this is 30 degrees to the normal."},{"Start":"04:04.445 ","End":"04:11.300","Text":"When we\u0027re going from a more dense medium to a less dense medium."},{"Start":"04:11.300 ","End":"04:16.010","Text":"We know that the ray is going to refract,"},{"Start":"04:16.010 ","End":"04:20.915","Text":"and it\u0027s going to move away from the normal."},{"Start":"04:20.915 ","End":"04:27.365","Text":"It\u0027s going to come out somewhere like so."},{"Start":"04:27.365 ","End":"04:31.375","Text":"We want to know what this angle is."},{"Start":"04:31.375 ","End":"04:35.180","Text":"Again, we\u0027re going to do the same equation."},{"Start":"04:35.180 ","End":"04:37.865","Text":"Our n_1 is for glass."},{"Start":"04:37.865 ","End":"04:45.310","Text":"1.5 multiplied by sine(30) degrees is equal to n_2."},{"Start":"04:45.310 ","End":"04:52.835","Text":"The refractive index of water 1.33 multiplied by sine(Theta_2)."},{"Start":"04:52.835 ","End":"04:56.100","Text":"This is what we\u0027re trying to calculate."},{"Start":"04:57.490 ","End":"05:01.815","Text":"Sine(30) is equal to 0.5."},{"Start":"05:01.815 ","End":"05:11.039","Text":"We have 1.5 times 0.5 or let\u0027s just calculate Theta_2 straightaway."},{"Start":"05:11.039 ","End":"05:19.780","Text":"It\u0027s just going to be sine to the minus 1 or arc sine(1.5) multiplied by sine(30),"},{"Start":"05:19.780 ","End":"05:22.635","Text":"which is, as we said,"},{"Start":"05:22.635 ","End":"05:28.725","Text":"0.5 divided by 1.33."},{"Start":"05:28.725 ","End":"05:33.200","Text":"We just plug this into our calculator and we\u0027ll get that Theta_2"},{"Start":"05:33.200 ","End":"05:39.425","Text":"is equal to 34.3 degrees."},{"Start":"05:39.425 ","End":"05:45.285","Text":"Over here, we\u0027ll get 34.3 degrees."},{"Start":"05:45.285 ","End":"05:53.249","Text":"As we saw, our ray diffracted away from the normal by 4.3 degrees."},{"Start":"05:54.460 ","End":"05:59.535","Text":"Now let\u0027s take a look at case number 3,"},{"Start":"05:59.535 ","End":"06:02.655","Text":"where Theta_1 is 70 degrees."},{"Start":"06:02.655 ","End":"06:05.850","Text":"Again, let\u0027s draw this."},{"Start":"06:05.850 ","End":"06:12.135","Text":"We\u0027re going again, I\u0027m just going to write glass to water,"},{"Start":"06:12.135 ","End":"06:14.955","Text":"from dense to less dense."},{"Start":"06:14.955 ","End":"06:19.475","Text":"Here is our normal and what we have is"},{"Start":"06:19.475 ","End":"06:26.705","Text":"our incident ray coming in at 70 degrees to the normal."},{"Start":"06:26.705 ","End":"06:34.240","Text":"Now we want to know what our refracted ray with the angle is going to be."},{"Start":"06:34.760 ","End":"06:37.350","Text":"Let\u0027s do this."},{"Start":"06:37.350 ","End":"06:40.095","Text":"We have, once again,"},{"Start":"06:40.095 ","End":"06:45.475","Text":"n_1 is 1.5 multiplied by sine(70)"},{"Start":"06:45.475 ","End":"06:52.955","Text":"degrees is equal to 1.33 multiplied by sine of our Theta_2."},{"Start":"06:52.955 ","End":"06:55.265","Text":"In order to find Theta_2,"},{"Start":"06:55.265 ","End":"06:59.340","Text":"we\u0027re going to arc sine all of this."},{"Start":"06:59.590 ","End":"07:02.975","Text":"After doing the calculation,"},{"Start":"07:02.975 ","End":"07:08.045","Text":"we\u0027re going to get that Theta_2 is equal to"},{"Start":"07:08.045 ","End":"07:16.810","Text":"the arc sine of 1.06, approximately."},{"Start":"07:16.810 ","End":"07:25.685","Text":"What we\u0027re going to get is that this leads to our glorious math error on the calculator,"},{"Start":"07:25.685 ","End":"07:28.790","Text":"which means when you get this,"},{"Start":"07:28.790 ","End":"07:36.090","Text":"that 70 degrees is greater than the critical angle."},{"Start":"07:36.470 ","End":"07:42.875","Text":"This is great because now we know that we\u0027re going to get total internal reflection,"},{"Start":"07:42.875 ","End":"07:47.570","Text":"which also makes sense because we need to notice that every time we"},{"Start":"07:47.570 ","End":"07:53.120","Text":"go from a dense medium to a rare medium,"},{"Start":"07:53.120 ","End":"07:57.830","Text":"know that you\u0027re going to have to deal with the critical angle."},{"Start":"07:57.830 ","End":"08:00.725","Text":"That is something that you should really notice"},{"Start":"08:00.725 ","End":"08:04.100","Text":"anytime you have a question dealing with Snell\u0027s Law,"},{"Start":"08:04.100 ","End":"08:08.800","Text":"to look if you\u0027re going from a less dense to a more dense."},{"Start":"08:08.800 ","End":"08:13.640","Text":"Then there isn\u0027t this problem you don\u0027t really need to focus"},{"Start":"08:13.640 ","End":"08:18.050","Text":"on the critical angle and total internal reflection."},{"Start":"08:18.050 ","End":"08:23.675","Text":"However, if you see that you\u0027re going from a denser medium to a less dense medium,"},{"Start":"08:23.675 ","End":"08:25.970","Text":"you know that chances are,"},{"Start":"08:25.970 ","End":"08:31.510","Text":"they\u0027re going to introduce this issue of total internal reflection."},{"Start":"08:31.510 ","End":"08:35.280","Text":"Straightaway, we see that\u0027s what\u0027s going to happen."},{"Start":"08:35.280 ","End":"08:37.350","Text":"We\u0027re going to get the reflection."},{"Start":"08:37.350 ","End":"08:41.630","Text":"Array is just going to bounce back according to the laws of reflection,"},{"Start":"08:41.630 ","End":"08:45.470","Text":"where this angle is also 70 degrees,"},{"Start":"08:45.470 ","End":"08:48.600","Text":"again to the normal."},{"Start":"08:48.670 ","End":"08:51.680","Text":"There we have it. Now just as a side note,"},{"Start":"08:51.680 ","End":"08:54.770","Text":"what is the critical angle?"},{"Start":"08:56.650 ","End":"08:59.840","Text":"The critical angle, as we know,"},{"Start":"08:59.840 ","End":"09:05.959","Text":"we said that we can calculate it when we go from the less dense medium."},{"Start":"09:05.959 ","End":"09:10.025","Text":"That would be, in this case from water to glass."},{"Start":"09:10.025 ","End":"09:13.710","Text":"Let\u0027s just draw this out,"},{"Start":"09:13.710 ","End":"09:20.880","Text":"and our normal and this is our rare medium and this is our dense medium."},{"Start":"09:20.950 ","End":"09:29.160","Text":"We know that we\u0027re going to go in at this grazing angle like so,"},{"Start":"09:29.160 ","End":"09:32.130","Text":"where it\u0027s approaching 90 degrees."},{"Start":"09:32.130 ","End":"09:35.510","Text":"Then we want to calculate this over here,"},{"Start":"09:35.510 ","End":"09:38.160","Text":"which is our Theta_C."},{"Start":"09:39.110 ","End":"09:47.985","Text":"What we have is we have n_1 multiplied by sine(Theta_1),"},{"Start":"09:47.985 ","End":"09:52.920","Text":"where Theta_1 over here is 90 degrees."},{"Start":"09:52.920 ","End":"09:58.875","Text":"This is equal to n_2 multiplied by sine(Theta_2),"},{"Start":"09:58.875 ","End":"10:01.675","Text":"which in this case is Theta_C,"},{"Start":"10:01.675 ","End":"10:05.310","Text":"which is what we\u0027re trying to calculate the critical angle."},{"Start":"10:06.060 ","End":"10:09.295","Text":"Sine(90) degrees is 1."},{"Start":"10:09.295 ","End":"10:16.600","Text":"What we have is n_1 divided by n_2."},{"Start":"10:16.600 ","End":"10:24.435","Text":"Then we can arc sine this is equal to the critical angle."},{"Start":"10:24.435 ","End":"10:28.155","Text":"What we can do in this case,"},{"Start":"10:28.155 ","End":"10:32.805","Text":"our n_1, we said was the less."},{"Start":"10:32.805 ","End":"10:36.580","Text":"In this case, because we\u0027re going from the rare to the denser."},{"Start":"10:36.580 ","End":"10:40.305","Text":"We\u0027re going from water to glass."},{"Start":"10:40.305 ","End":"10:41.880","Text":"N_1, in this case,"},{"Start":"10:41.880 ","End":"10:45.270","Text":"is 1.332 and n_2 is 1.5."},{"Start":"10:45.270 ","End":"10:48.570","Text":"Because we were going the opposite direction."},{"Start":"10:48.570 ","End":"10:56.685","Text":"We have sine to the minus 1(1.33 divided by 1.5)."},{"Start":"10:56.685 ","End":"11:00.365","Text":"If we plug this into our calculator,"},{"Start":"11:00.365 ","End":"11:11.829","Text":"we get that the critical angle is equal to 62.5 degrees."},{"Start":"11:12.410 ","End":"11:17.420","Text":"In other words, any angle greater than 62.5"},{"Start":"11:17.420 ","End":"11:22.980","Text":"degrees when we\u0027re going from the dense to the less dense."},{"Start":"11:23.290 ","End":"11:25.430","Text":"Not what we did here."},{"Start":"11:25.430 ","End":"11:30.590","Text":"This was just a calculation to calculate what Theta_C is."},{"Start":"11:30.590 ","End":"11:35.105","Text":"But anytime we\u0027re going from dense to less dense,"},{"Start":"11:35.105 ","End":"11:40.250","Text":"if our angle of incidence is greater than the critical angle,"},{"Start":"11:40.250 ","End":"11:43.870","Text":"we\u0027re going to get total internal reflection."},{"Start":"11:43.870 ","End":"11:46.860","Text":"We\u0027re not going to have refraction."},{"Start":"11:46.860 ","End":"11:51.365","Text":"As we can see in this case over here from glass to water,"},{"Start":"11:51.365 ","End":"11:56.180","Text":"our critical angle is 62.5 degrees and in the question,"},{"Start":"11:56.180 ","End":"11:59.735","Text":"we were told that our angle of incidence is 70 degrees,"},{"Start":"11:59.735 ","End":"12:04.045","Text":"70 degrees is obviously greater than 62.5."},{"Start":"12:04.045 ","End":"12:12.070","Text":"We know that when we\u0027re going here from the denser to the less dense,"},{"Start":"12:12.070 ","End":"12:14.630","Text":"where exceeding our critical angle,"},{"Start":"12:14.630 ","End":"12:18.995","Text":"in which case we have total internal reflection."},{"Start":"12:18.995 ","End":"12:22.170","Text":"That\u0027s the end of this lesson."}],"ID":22429},{"Watched":false,"Name":"Wave Equations Solutions and Equations","Duration":"36m 26s","ChapterTopicVideoID":21577,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21577.jpeg","UploadDate":"2020-05-07T18:36:50.2130000","DurationForVideoObject":"PT36M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.350","Text":"Hello. This lesson is a continuation of what we learned in the previous lesson."},{"Start":"00:06.350 ","End":"00:07.829","Text":"In the previous lesson,"},{"Start":"00:07.829 ","End":"00:14.399","Text":"we looked at Maxwell\u0027s equations and we saw how to derive the wave equations,"},{"Start":"00:14.399 ","End":"00:21.870","Text":"eventually arriving at this solution to this differential equation over here."},{"Start":"00:21.870 ","End":"00:30.490","Text":"In this lesson, we\u0027re going to speak about this solution in slightly more detail."},{"Start":"00:31.460 ","End":"00:35.410","Text":"Let\u0027s take a look at this over here."},{"Start":"00:35.410 ","End":"00:36.680","Text":"First of all, again,"},{"Start":"00:36.680 ","End":"00:40.625","Text":"we\u0027re looking at the x component of the electric field."},{"Start":"00:40.625 ","End":"00:43.355","Text":"As we know, the electric field is a vector field."},{"Start":"00:43.355 ","End":"00:46.024","Text":"If we\u0027re looking at Cartesian coordinates,"},{"Start":"00:46.024 ","End":"00:47.824","Text":"it will have an x component,"},{"Start":"00:47.824 ","End":"00:50.720","Text":"y component, and a z component."},{"Start":"00:50.720 ","End":"00:54.664","Text":"Here at our example right now we\u0027re just looking at the x component,"},{"Start":"00:54.664 ","End":"00:57.760","Text":"but it doesn\u0027t really matter."},{"Start":"00:58.220 ","End":"01:05.164","Text":"This equation over here gives us the magnitude of the electric field in"},{"Start":"01:05.164 ","End":"01:11.705","Text":"the x direction at some given time as a function of time."},{"Start":"01:11.705 ","End":"01:15.529","Text":"What we can see is that the electric field over here is dependent on"},{"Start":"01:15.529 ","End":"01:19.445","Text":"time and on this r over here,"},{"Start":"01:19.445 ","End":"01:23.405","Text":"where r is a vector representing its position,"},{"Start":"01:23.405 ","End":"01:25.175","Text":"the position of the wave,"},{"Start":"01:25.175 ","End":"01:28.770","Text":"or of the electric field in space."},{"Start":"01:28.880 ","End":"01:39.139","Text":"We can rewrite this as showing that it\u0027s dependent on r and t. What is our r vector?"},{"Start":"01:39.139 ","End":"01:43.052","Text":"Our r vector is equal to in Cartesian coordinates,"},{"Start":"01:43.052 ","End":"01:47.235","Text":"some x, y, and z values."},{"Start":"01:47.235 ","End":"01:50.060","Text":"If we\u0027re looking at some axes,"},{"Start":"01:50.060 ","End":"01:52.513","Text":"so this is the x-direction,"},{"Start":"01:52.513 ","End":"01:57.990","Text":"this is the z direction,"},{"Start":"01:57.990 ","End":"02:01.710","Text":"and that this is the y-direction."},{"Start":"02:01.710 ","End":"02:08.660","Text":"Then we\u0027re looking over here from the origin at some point."},{"Start":"02:08.660 ","End":"02:15.349","Text":"The electric field in the x direction at this point is,"},{"Start":"02:15.349 ","End":"02:19.460","Text":"this is going to be our r point, some x,"},{"Start":"02:19.460 ","End":"02:24.550","Text":"y, z value giving us this position over here."},{"Start":"02:24.550 ","End":"02:26.850","Text":"That\u0027s the first thing."},{"Start":"02:26.850 ","End":"02:29.344","Text":"Then we also want to look at this k vector."},{"Start":"02:29.344 ","End":"02:33.709","Text":"Now, the k vector we\u0027re going to speak about in a little bit more detail later."},{"Start":"02:33.709 ","End":"02:41.585","Text":"But right now, just think of it as just a vector with some constant values."},{"Start":"02:41.585 ","End":"02:47.809","Text":"It also represents something which we\u0027ll speak about what it means soon."},{"Start":"02:47.809 ","End":"02:49.640","Text":"But what it does is just constants."},{"Start":"02:49.640 ","End":"02:57.289","Text":"We have k in the x direction k in the y direction and k in the z direction."},{"Start":"02:57.289 ","End":"02:58.610","Text":"Then if we look at here,"},{"Start":"02:58.610 ","End":"03:02.105","Text":"we have the dot-product between k and r,"},{"Start":"03:02.105 ","End":"03:04.940","Text":"which means that we take the x, y,"},{"Start":"03:04.940 ","End":"03:06.709","Text":"and z component of each,"},{"Start":"03:06.709 ","End":"03:09.980","Text":"multiply them together, and then sum this up."},{"Start":"03:09.980 ","End":"03:13.354","Text":"Given the r vector and the k vector,"},{"Start":"03:13.354 ","End":"03:23.255","Text":"we can rewrite this as being equal to A cosine of k_x multiplied by x plus"},{"Start":"03:23.255 ","End":"03:33.390","Text":"k y multiplied by y plus k_z multiplied by z and"},{"Start":"03:33.390 ","End":"03:40.940","Text":"then minus Omega t. This equation over"},{"Start":"03:40.940 ","End":"03:45.079","Text":"here is correct exactly the same equation"},{"Start":"03:45.079 ","End":"03:50.735","Text":"for all of the other components of the E field also y and z."},{"Start":"03:50.735 ","End":"03:54.560","Text":"The only difference between each component x,"},{"Start":"03:54.560 ","End":"03:57.860","Text":"y and z will be this constant a over here,"},{"Start":"03:57.860 ","End":"04:02.960","Text":"which represents the amplitude of the wave in that direction."},{"Start":"04:02.960 ","End":"04:09.160","Text":"The amplitude will be different for x, y, and z."},{"Start":"04:09.530 ","End":"04:15.170","Text":"Obviously that means that sometimes it will be equal in each direction,"},{"Start":"04:15.170 ","End":"04:19.369","Text":"sometimes the amplitude could also be 0 in one direction."},{"Start":"04:19.369 ","End":"04:23.929","Text":"But in general, this whole equation is correct for each component,"},{"Start":"04:23.929 ","End":"04:27.650","Text":"just the amplitude is the thing that changes."},{"Start":"04:27.650 ","End":"04:29.450","Text":"The k, the x, y,"},{"Start":"04:29.450 ","End":"04:32.360","Text":"z, and the Omega is different."},{"Start":"04:32.360 ","End":"04:34.969","Text":"The t of course just represents the time,"},{"Start":"04:34.969 ","End":"04:38.340","Text":"so we can just plug in t is equal to 0,"},{"Start":"04:38.340 ","End":"04:42.550","Text":"t is equal to 2 seconds and calculate it."},{"Start":"04:43.790 ","End":"04:48.739","Text":"Equally, this exact same equation and the ideas"},{"Start":"04:48.739 ","End":"04:53.359","Text":"with the different components is also correct for the B field,"},{"Start":"04:53.359 ","End":"04:54.904","Text":"so the magnetic field,"},{"Start":"04:54.904 ","End":"05:00.829","Text":"whether we\u0027re looking at the electric field in the x direction,"},{"Start":"05:00.829 ","End":"05:04.282","Text":"or the magnetic field in the z direction,"},{"Start":"05:04.282 ","End":"05:06.740","Text":"this equation will be exactly the same,"},{"Start":"05:06.740 ","End":"05:10.645","Text":"just the amplitude will be different."},{"Start":"05:10.645 ","End":"05:16.069","Text":"What am I going to do now is we\u0027re going to give a very easy example."},{"Start":"05:16.069 ","End":"05:20.140","Text":"Just to make this a little bit clearer."},{"Start":"05:20.140 ","End":"05:25.185","Text":"Let\u0027s just draw out a border."},{"Start":"05:25.185 ","End":"05:27.780","Text":"Now we\u0027re doing the example."},{"Start":"05:27.780 ","End":"05:31.910","Text":"Let\u0027s say that we have an electric field that is"},{"Start":"05:31.910 ","End":"05:36.260","Text":"given as a function of its possession of time."},{"Start":"05:36.260 ","End":"05:38.719","Text":"It\u0027s equal to A,"},{"Start":"05:38.719 ","End":"05:48.510","Text":"the amplitude multiplied by cosine of 2z minus Omega t x in the x-direction."},{"Start":"05:48.510 ","End":"05:58.493","Text":"What we can see over here is that our electric field given is only in the x direction."},{"Start":"05:58.493 ","End":"06:02.270","Text":"It only has an amplitude in the x direction, or in other words,"},{"Start":"06:02.270 ","End":"06:09.485","Text":"its amplitude in the y and in the z directions is equal to 0."},{"Start":"06:09.485 ","End":"06:12.300","Text":"It means the same thing."},{"Start":"06:13.070 ","End":"06:16.050","Text":"That\u0027s with respect to the direction."},{"Start":"06:16.050 ","End":"06:17.790","Text":"Now what we want to do is,"},{"Start":"06:17.790 ","End":"06:20.190","Text":"we want to find our k vector."},{"Start":"06:20.190 ","End":"06:24.784","Text":"Our k vector we saw that we\u0027d have the x component of it"},{"Start":"06:24.784 ","End":"06:30.320","Text":"multiplied by the x component of our r vector,"},{"Start":"06:30.320 ","End":"06:31.733","Text":"of our position vector,"},{"Start":"06:31.733 ","End":"06:36.260","Text":"the y component multiplied by the y component of"},{"Start":"06:36.260 ","End":"06:41.735","Text":"the vector and the z component multiplied by the z component of r vector."},{"Start":"06:41.735 ","End":"06:47.970","Text":"Here we can see that we have a 0 multiplied by x."},{"Start":"06:47.970 ","End":"06:52.669","Text":"We can see that our k vector is going to be equal to 0."},{"Start":"06:52.669 ","End":"06:55.594","Text":"We also don\u0027t have any y variable either,"},{"Start":"06:55.594 ","End":"06:58.880","Text":"so it\u0027s going to be 0 multiplied by y."},{"Start":"06:58.880 ","End":"07:04.895","Text":"Then we can see that we have a z variable and its coefficient is 2."},{"Start":"07:04.895 ","End":"07:08.900","Text":"The z component of the k vector is 2."},{"Start":"07:08.900 ","End":"07:13.350","Text":"This is how you would find the k vector."},{"Start":"07:14.390 ","End":"07:21.259","Text":"What we can see is that we have this electric field which is only in the x direction."},{"Start":"07:21.259 ","End":"07:24.950","Text":"However, it is dependent on z."},{"Start":"07:24.950 ","End":"07:27.095","Text":"Don\u0027t get confused between this."},{"Start":"07:27.095 ","End":"07:28.249","Text":"It\u0027s in the x direction,"},{"Start":"07:28.249 ","End":"07:29.960","Text":"but it\u0027s dependent on z."},{"Start":"07:29.960 ","End":"07:33.710","Text":"Let\u0027s draw out an axis."},{"Start":"07:33.710 ","End":"07:36.454","Text":"Here we have our x-axis,"},{"Start":"07:36.454 ","End":"07:40.550","Text":"here we have our z-axis,"},{"Start":"07:40.550 ","End":"07:44.030","Text":"and here we have our y-axis."},{"Start":"07:44.030 ","End":"07:46.760","Text":"Now let\u0027s look at a certain time."},{"Start":"07:46.760 ","End":"07:53.445","Text":"Let\u0027s take a look at the initial moment so where t is equal to 0."},{"Start":"07:53.445 ","End":"07:55.175","Text":"Here, if t is equal to 0,"},{"Start":"07:55.175 ","End":"08:03.600","Text":"this all cancels out because of the t. What we\u0027re left with is just A cosine of 2z."},{"Start":"08:03.610 ","End":"08:07.375","Text":"Let\u0027s look over here at the origin."},{"Start":"08:07.375 ","End":"08:10.010","Text":"We can see at the origin,"},{"Start":"08:10.010 ","End":"08:13.250","Text":"we\u0027re at the origin, so z is equal to 0."},{"Start":"08:13.250 ","End":"08:16.535","Text":"Then all will be left with is A cosine of 0."},{"Start":"08:16.535 ","End":"08:18.574","Text":"Cosine of 0 is just 1,"},{"Start":"08:18.574 ","End":"08:23.989","Text":"so we have just the maximum amplitude in the x direction."},{"Start":"08:23.989 ","End":"08:25.985","Text":"Remember, it\u0027s in the x direction,"},{"Start":"08:25.985 ","End":"08:30.230","Text":"the maximum amplitude of our electric field at this point."},{"Start":"08:30.230 ","End":"08:33.080","Text":"Then as z increases,"},{"Start":"08:33.080 ","End":"08:38.960","Text":"we can see just as the cosine function behaves, after 0,"},{"Start":"08:38.960 ","End":"08:43.595","Text":"the cosine function begins decreasing,"},{"Start":"08:43.595 ","End":"08:46.835","Text":"like so, as a type of wave,"},{"Start":"08:46.835 ","End":"08:49.430","Text":"so we get something like this."},{"Start":"08:49.430 ","End":"08:54.625","Text":"Then it moves over to this side,"},{"Start":"08:54.625 ","End":"08:59.370","Text":"until it reaches its maximum amplitude."},{"Start":"08:59.370 ","End":"09:06.255","Text":"Then it begins decreasing again and again on the other side."},{"Start":"09:06.255 ","End":"09:09.880","Text":"This of course continues to infinity."},{"Start":"09:10.130 ","End":"09:14.695","Text":"Then what we can do is we can draw the outline."},{"Start":"09:14.695 ","End":"09:18.440","Text":"I haven\u0027t drawn the arrows in the best way,"},{"Start":"09:18.440 ","End":"09:22.715","Text":"but as you know, it\u0027s meant to look like a cosine wave like so."},{"Start":"09:22.715 ","End":"09:26.640","Text":"Imagine that that is what is drawn here."},{"Start":"09:27.230 ","End":"09:31.130","Text":"Now let\u0027s take a look at a different moment."},{"Start":"09:31.130 ","End":"09:34.050","Text":"Let\u0027s look 1 second later,"},{"Start":"09:34.050 ","End":"09:39.760","Text":"at t is equal to 1 and let\u0027s see what the wave looks like."},{"Start":"09:39.760 ","End":"09:42.380","Text":"We\u0027ll just draw it in pink."},{"Start":"09:42.380 ","End":"09:43.889","Text":"1 second later,"},{"Start":"09:43.889 ","End":"09:47.615","Text":"we can see that here our t will be equal to 1."},{"Start":"09:47.615 ","End":"09:53.105","Text":"We have negative Omega multiplied by 1 or just negative Omega."},{"Start":"09:53.105 ","End":"09:55.393","Text":"Because there\u0027s a negative over here,"},{"Start":"09:55.393 ","End":"10:01.140","Text":"that means that this peak will move forward some distance,"},{"Start":"10:01.140 ","End":"10:04.510","Text":"which is dependent on Omega."},{"Start":"10:05.470 ","End":"10:08.960","Text":"Let\u0027s say that it moves up until here."},{"Start":"10:08.960 ","End":"10:13.130","Text":"That means that now our peak is going to be here"},{"Start":"10:13.130 ","End":"10:18.490","Text":"and then slowly it will decrease like so."},{"Start":"10:19.400 ","End":"10:24.960","Text":"Then it just looks like this."},{"Start":"10:24.960 ","End":"10:27.015","Text":"Exactly the same thing."},{"Start":"10:27.015 ","End":"10:31.290","Text":"What we see is that this peak is moving."},{"Start":"10:31.290 ","End":"10:36.625","Text":"This gives us the direction of the propagation of the wave,"},{"Start":"10:36.625 ","End":"10:41.255","Text":"this Omega, and the minus tells us that it\u0027s moving in the rightwards direction."},{"Start":"10:41.255 ","End":"10:42.500","Text":"If there was a plus,"},{"Start":"10:42.500 ","End":"10:46.379","Text":"the peak would move in the leftwards direction."},{"Start":"10:46.430 ","End":"10:50.794","Text":"Then of course, after a certain period of time,"},{"Start":"10:50.794 ","End":"10:53.779","Text":"our peak or 1 of the other peaks and the wave will"},{"Start":"10:53.779 ","End":"10:57.289","Text":"reach this exact same point again because"},{"Start":"10:57.289 ","End":"11:03.690","Text":"right now we have a situation like so."},{"Start":"11:03.690 ","End":"11:08.795","Text":"Now let\u0027s speak about all the components in this equation."},{"Start":"11:08.795 ","End":"11:14.900","Text":"As we can see, Omega is exactly the same as what we saw in circular motion."},{"Start":"11:14.900 ","End":"11:17.794","Text":"It\u0027s equal to 2Pif,"},{"Start":"11:17.794 ","End":"11:19.069","Text":"where f is of course,"},{"Start":"11:19.069 ","End":"11:27.915","Text":"the frequency of the wave and of course this is also equal to 2Pi divided by T,"},{"Start":"11:27.915 ","End":"11:31.080","Text":"where T is the period of the wave,"},{"Start":"11:31.080 ","End":"11:38.635","Text":"the time taken for 1 oscillation."},{"Start":"11:38.635 ","End":"11:40.359","Text":"As we were saying before,"},{"Start":"11:40.359 ","End":"11:45.910","Text":"if we started with our peak at the origin."},{"Start":"11:45.910 ","End":"11:52.524","Text":"The period or capital T is the time taken for us to,"},{"Start":"11:52.524 ","End":"11:56.600","Text":"again get to a point where there\u0027s a peak at the origin,"},{"Start":"11:56.600 ","End":"12:01.780","Text":"to return to the same image that we had at the beginning."},{"Start":"12:02.550 ","End":"12:06.085","Text":"That is what Omega tells us."},{"Start":"12:06.085 ","End":"12:12.475","Text":"It gives us information on the frequency and on the time period of our wave,"},{"Start":"12:12.475 ","End":"12:20.500","Text":"so that in turn gives us information regarding the propagation of the wave."},{"Start":"12:20.500 ","End":"12:26.350","Text":"How long it takes for our peak to move from some arbitrary point,"},{"Start":"12:26.350 ","End":"12:33.355","Text":"let\u0027s say from being over here at the origin to being a certain distance away."},{"Start":"12:33.355 ","End":"12:36.295","Text":"That\u0027s what our Omega gives us."},{"Start":"12:36.295 ","End":"12:41.380","Text":"A larger Omega will mean that in the same time frame between t is 0,"},{"Start":"12:41.380 ","End":"12:48.115","Text":"and t is 1 our peak could move either less distance or more."},{"Start":"12:48.115 ","End":"12:51.350","Text":"That\u0027s what our Omega gives us."},{"Start":"12:52.020 ","End":"12:56.680","Text":"As said our Omega tells us the distance moved,"},{"Start":"12:56.680 ","End":"12:59.290","Text":"what does our k tell us?"},{"Start":"12:59.290 ","End":"13:04.615","Text":"Our k tells us the direction of this movement."},{"Start":"13:04.615 ","End":"13:08.860","Text":"The distance moved, it\u0027s I\u0027m walking distance and then the k tells"},{"Start":"13:08.860 ","End":"13:13.119","Text":"us the direction it gives it a size."},{"Start":"13:13.119 ","End":"13:14.230","Text":"This is, if you will,"},{"Start":"13:14.230 ","End":"13:20.870","Text":"the size and this is the direction of the wave."},{"Start":"13:21.930 ","End":"13:26.019","Text":"Omega tells us how far the peak will move in"},{"Start":"13:26.019 ","End":"13:31.869","Text":"every time increments and k will tell us in which direction or in which axis."},{"Start":"13:31.869 ","End":"13:36.205","Text":"Here specifically, our wave as we can see,"},{"Start":"13:36.205 ","End":"13:41.770","Text":"it\u0027s not moving in either the x or y direction because we can see 0 in both."},{"Start":"13:41.770 ","End":"13:44.515","Text":"We can see that it\u0027s only moving in"},{"Start":"13:44.515 ","End":"13:49.930","Text":"the z direction and this also has a magnitude in the z direction which ties in."},{"Start":"13:49.930 ","End":"13:52.809","Text":"But we can see that it\u0027s only moving in the z direction,"},{"Start":"13:52.809 ","End":"13:55.104","Text":"which is exactly how we drew it."},{"Start":"13:55.104 ","End":"13:57.339","Text":"Now, this is of course an easy example."},{"Start":"13:57.339 ","End":"14:00.609","Text":"A lot of the times we\u0027ll have"},{"Start":"14:00.609 ","End":"14:05.335","Text":"a k vector that has components also in the x and y direction."},{"Start":"14:05.335 ","End":"14:10.230","Text":"That just means that it moves a certain magnitude in the x,"},{"Start":"14:10.230 ","End":"14:12.569","Text":"a certain in the y, and a certain in the z."},{"Start":"14:12.569 ","End":"14:14.730","Text":"But here just in this easy example,"},{"Start":"14:14.730 ","End":"14:22.189","Text":"we can see that we are moving with a magnitude of 2 in the z direction each time."},{"Start":"14:22.189 ","End":"14:31.225","Text":"Now another important quality of the k vector is that it\u0027s connected to the wavelength."},{"Start":"14:31.225 ","End":"14:33.190","Text":"When we\u0027re speaking about wavelength,"},{"Start":"14:33.190 ","End":"14:38.154","Text":"we\u0027re speaking about the distance from peak to peak."},{"Start":"14:38.154 ","End":"14:43.660","Text":"This distance over here from peak to peak is Lambda,"},{"Start":"14:43.660 ","End":"14:45.820","Text":"which is the wavelength."},{"Start":"14:45.820 ","End":"14:51.325","Text":"If we take the magnitude of our k vector,"},{"Start":"14:51.325 ","End":"14:59.330","Text":"it is equal to 2 Pi divided by Lambda, the wavelength."},{"Start":"15:01.020 ","End":"15:07.480","Text":"These are 2 very important equations to remember."},{"Start":"15:07.480 ","End":"15:09.295","Text":"Now there\u0027s another one."},{"Start":"15:09.295 ","End":"15:14.004","Text":"A moment ago, I related the Omega"},{"Start":"15:14.004 ","End":"15:19.885","Text":"to how much the wave moves in a certain direction,"},{"Start":"15:19.885 ","End":"15:21.984","Text":"the distance, if you will,"},{"Start":"15:21.984 ","End":"15:25.839","Text":"and the k to the direction moved."},{"Start":"15:25.839 ","End":"15:29.695","Text":"But as we can see, the k also has a magnitude."},{"Start":"15:29.695 ","End":"15:31.690","Text":"It\u0027s not a unit vector."},{"Start":"15:31.690 ","End":"15:33.760","Text":"For instance, here we can see it\u0027s 2."},{"Start":"15:33.760 ","End":"15:35.199","Text":"If it was a unit vector,"},{"Start":"15:35.199 ","End":"15:38.090","Text":"this value here would just be 1,"},{"Start":"15:38.100 ","End":"15:41.934","Text":"so k, in actual fact,"},{"Start":"15:41.934 ","End":"15:48.369","Text":"isn\u0027t just the direction that the wave is propagating in,"},{"Start":"15:48.369 ","End":"15:51.070","Text":"but it\u0027s also connected to the distance."},{"Start":"15:51.070 ","End":"15:56.155","Text":"That must mean that k and Omega are somehow related."},{"Start":"15:56.155 ","End":"16:00.819","Text":"Because Omega also gives us something to do with the distance"},{"Start":"16:00.819 ","End":"16:03.130","Text":"that a peak will propagate or any points"},{"Start":"16:03.130 ","End":"16:06.490","Text":"along the wave will propagate in a certain direction."},{"Start":"16:06.490 ","End":"16:13.330","Text":"We can relate Omega via this equation to k. Omega is equal to c,"},{"Start":"16:13.330 ","End":"16:14.904","Text":"which is the speed of light,"},{"Start":"16:14.904 ","End":"16:21.430","Text":"multiplied by the magnitude of the k vector."},{"Start":"16:21.430 ","End":"16:29.570","Text":"This equation over here is called the dispersion relationship."},{"Start":"16:29.820 ","End":"16:32.950","Text":"These equations are super important."},{"Start":"16:32.950 ","End":"16:36.175","Text":"Please write them in your equation sheets."},{"Start":"16:36.175 ","End":"16:39.265","Text":"I\u0027ve just labeled that f over here is frequency,"},{"Start":"16:39.265 ","End":"16:41.559","Text":"capital T over here is period,"},{"Start":"16:41.559 ","End":"16:45.280","Text":"Lambda over here is the wavelength."},{"Start":"16:45.280 ","End":"16:47.335","Text":"That this entire equation,"},{"Start":"16:47.335 ","End":"16:51.040","Text":"Omega is equal to c multiplied by the magnitude of"},{"Start":"16:51.040 ","End":"16:55.855","Text":"k. This whole equation is called the dispersion relationship."},{"Start":"16:55.855 ","End":"17:02.725","Text":"As we\u0027ve already spoken about in the past few lessons,"},{"Start":"17:02.725 ","End":"17:05.995","Text":"we\u0027ve said that whenever we have an electric field,"},{"Start":"17:05.995 ","End":"17:08.320","Text":"we also have a magnetic field."},{"Start":"17:08.320 ","End":"17:13.885","Text":"We even said that they are identical just from different reference frames."},{"Start":"17:13.885 ","End":"17:17.320","Text":"We can see that here we have an electric field,"},{"Start":"17:17.320 ","End":"17:20.305","Text":"which in turn is going to create a magnetic field."},{"Start":"17:20.305 ","End":"17:22.929","Text":"If we had a magnetic field,"},{"Start":"17:22.929 ","End":"17:25.525","Text":"it would in turn creates an electric field."},{"Start":"17:25.525 ","End":"17:29.785","Text":"Possible question that they could ask is given this electric field,"},{"Start":"17:29.785 ","End":"17:31.840","Text":"find the magnetic field."},{"Start":"17:31.840 ","End":"17:34.750","Text":"Now we\u0027re going to look again at a simple example of how"},{"Start":"17:34.750 ","End":"17:38.140","Text":"to get from the electric field to the magnetic field."},{"Start":"17:38.140 ","End":"17:40.029","Text":"In order to do this,"},{"Start":"17:40.029 ","End":"17:44.454","Text":"we\u0027re going to again use Maxwell\u0027s equations."},{"Start":"17:44.454 ","End":"17:46.765","Text":"Let\u0027s just go up."},{"Start":"17:46.765 ","End":"17:49.150","Text":"In order to convert between fields,"},{"Start":"17:49.150 ","End":"17:53.560","Text":"we use Maxwell\u0027s third and fourth equations."},{"Start":"17:53.560 ","End":"17:58.929","Text":"This is the third equation, all of this,"},{"Start":"17:58.929 ","End":"18:04.480","Text":"and this is the fourth equation, all of this."},{"Start":"18:04.480 ","End":"18:07.419","Text":"Remember that we\u0027re in a vacuum,"},{"Start":"18:07.419 ","End":"18:09.355","Text":"so this is equal to 0."},{"Start":"18:09.355 ","End":"18:15.445","Text":"We just take into account this section over here."},{"Start":"18:15.445 ","End":"18:20.320","Text":"If we\u0027re converting, if we have an electric field as we have here,"},{"Start":"18:20.320 ","End":"18:25.225","Text":"and we want to find the magnetic field we\u0027ll use equation number 3,"},{"Start":"18:25.225 ","End":"18:28.900","Text":"where we\u0027re taking the rotor of the electric field."},{"Start":"18:28.900 ","End":"18:34.450","Text":"Whereas if we were given a magnetic field and told to calculate the electric field,"},{"Start":"18:34.450 ","End":"18:36.969","Text":"we would use Maxwell\u0027s fourth equation,"},{"Start":"18:36.969 ","End":"18:39.910","Text":"where we take the rotor of the magnetic field and"},{"Start":"18:39.910 ","End":"18:43.690","Text":"then we will calculate the electric field."},{"Start":"18:43.690 ","End":"18:49.360","Text":"In other words, you take the rotor of the field that you\u0027re given."},{"Start":"18:49.360 ","End":"18:51.399","Text":"If you\u0027re given the electric field,"},{"Start":"18:51.399 ","End":"18:53.275","Text":"you take the rotor of the electric field."},{"Start":"18:53.275 ","End":"18:54.819","Text":"If you\u0027re given the magnetic field,"},{"Start":"18:54.819 ","End":"18:58.089","Text":"you take the rotor of the magnetic field and you use that"},{"Start":"18:58.089 ","End":"19:02.020","Text":"in order to calculate the other field."},{"Start":"19:02.020 ","End":"19:04.615","Text":"In our question over here,"},{"Start":"19:04.615 ","End":"19:06.640","Text":"we\u0027re given the electric field,"},{"Start":"19:06.640 ","End":"19:08.980","Text":"so we\u0027re going to use Maxwell\u0027s third equation."},{"Start":"19:08.980 ","End":"19:13.299","Text":"We take the rotor of the electric field and we see that it is equal"},{"Start":"19:13.299 ","End":"19:18.694","Text":"to the negative time derivative of the magnetic field."},{"Start":"19:18.694 ","End":"19:21.660","Text":"Let\u0027s solve this."},{"Start":"19:21.660 ","End":"19:29.685","Text":"Let\u0027s give us some more space and let\u0027s rewrite out our equation."},{"Start":"19:29.685 ","End":"19:37.649","Text":"We\u0027re using Maxwell\u0027s third equation that says that the rotor of"},{"Start":"19:37.649 ","End":"19:46.440","Text":"the electric field is equal to the negative time derivative of the magnetic field."},{"Start":"19:46.440 ","End":"19:55.995","Text":"Let\u0027s first of all do the rotor of the magnetic of the electric field."},{"Start":"19:55.995 ","End":"19:59.534","Text":"This is going to give us,"},{"Start":"19:59.534 ","End":"20:02.250","Text":"we\u0027ll just go over how to solve this."},{"Start":"20:02.250 ","End":"20:06.794","Text":"We have our partial derivatives, d by dx,"},{"Start":"20:06.794 ","End":"20:14.055","Text":"d by dy, d by dz cross multiplied with our E field."},{"Start":"20:14.055 ","End":"20:17.190","Text":"So our E field is only in the x-direction,"},{"Start":"20:17.190 ","End":"20:20.730","Text":"so it\u0027s equal to A cosine of"},{"Start":"20:20.730 ","End":"20:28.620","Text":"2z minus Omega t. Then we have 0 and 0 in the x and y-directions."},{"Start":"20:28.620 ","End":"20:31.470","Text":"Now let\u0027s calculate this."},{"Start":"20:31.470 ","End":"20:35.400","Text":"Remember from the rotor we get again a vector."},{"Start":"20:35.400 ","End":"20:40.679","Text":"First what we do is we cross out over here and then"},{"Start":"20:40.679 ","End":"20:46.064","Text":"we do d by dy of 0 minus d by dz of 0,"},{"Start":"20:46.064 ","End":"20:50.519","Text":"which is equal to 0 in the x-direction."},{"Start":"20:50.519 ","End":"20:58.560","Text":"Then we cross out the middle row and then we flip it over so we have d by"},{"Start":"20:58.560 ","End":"21:06.960","Text":"dz of A cosine of 2z minus Omega t minus d by dx of 0, which is 0."},{"Start":"21:06.960 ","End":"21:13.380","Text":"What we have is d by dz of A cosine"},{"Start":"21:13.380 ","End":"21:23.589","Text":"2z minus Omega t. Then we cross out the bottom row."},{"Start":"21:23.990 ","End":"21:29.400","Text":"We have d by d y of this,"},{"Start":"21:29.400 ","End":"21:34.619","Text":"so d by dy of A cosine of"},{"Start":"21:34.619 ","End":"21:40.169","Text":"2z minus Omega t and then minus d by dx of 0,"},{"Start":"21:40.169 ","End":"21:42.765","Text":"which is just 0."},{"Start":"21:42.765 ","End":"21:45.285","Text":"This is the vector that we have."},{"Start":"21:45.285 ","End":"21:52.200","Text":"Then we can say that this is then equal to 0."},{"Start":"21:52.200 ","End":"22:02.430","Text":"Then we have d by dz of A cosine of 2z minus Omega t. This is going to give us negative."},{"Start":"22:02.430 ","End":"22:11.160","Text":"Then we\u0027ll have, cosine becomes negative sine so that\u0027s"},{"Start":"22:11.160 ","End":"22:13.800","Text":"why negative sine of"},{"Start":"22:13.800 ","End":"22:20.850","Text":"2z minus Omega t. Then this is going to be multiplied by the inner derivative."},{"Start":"22:20.850 ","End":"22:22.920","Text":"Here only this term has a z,"},{"Start":"22:22.920 ","End":"22:26.620","Text":"so the derivative of 2z is just 2."},{"Start":"22:27.110 ","End":"22:33.015","Text":"Then here we have d by dy of the same function."},{"Start":"22:33.015 ","End":"22:34.215","Text":"As we can see,"},{"Start":"22:34.215 ","End":"22:38.805","Text":"we have no y variables so when we take the derivative with respect to y,"},{"Start":"22:38.805 ","End":"22:42.190","Text":"we\u0027re just left with 0."},{"Start":"22:42.770 ","End":"22:46.379","Text":"This is the rotor. In other words,"},{"Start":"22:46.379 ","End":"22:53.084","Text":"we can say that our rotor is equal to"},{"Start":"22:53.084 ","End":"23:03.390","Text":"negative 2A sine of 2z minus Omega t in the y-direction."},{"Start":"23:03.390 ","End":"23:07.335","Text":"We can see it\u0027s the y component of this vector over here."},{"Start":"23:07.335 ","End":"23:08.684","Text":"This is equal to,"},{"Start":"23:08.684 ","End":"23:11.054","Text":"according to Maxwell\u0027s third equation,"},{"Start":"23:11.054 ","End":"23:16.320","Text":"to negative dB by dt."},{"Start":"23:16.320 ","End":"23:21.390","Text":"First of all, we can see that the magnetic field is in the y-direction."},{"Start":"23:21.390 ","End":"23:25.980","Text":"Sometimes we\u0027ll also have other components like x and z."},{"Start":"23:25.980 ","End":"23:31.810","Text":"But specifically here, we can see that the magnetic field is only in the y-direction."},{"Start":"23:31.880 ","End":"23:37.905","Text":"Let\u0027s just erase this so that we can play around with both sides."},{"Start":"23:37.905 ","End":"23:43.470","Text":"Both sides have negative signs so we can cancel them out."},{"Start":"23:43.470 ","End":"23:46.065","Text":"They just become pluses."},{"Start":"23:46.065 ","End":"23:51.870","Text":"Now what we\u0027re going to do is we\u0027re going to integrate both sides."},{"Start":"23:51.870 ","End":"24:00.450","Text":"In that case, we\u0027re going to get that the magnetic field is equal to the time"},{"Start":"24:00.450 ","End":"24:06.064","Text":"integral of 2A sine"},{"Start":"24:06.064 ","End":"24:12.200","Text":"of 2z minus Omega t dt,"},{"Start":"24:12.200 ","End":"24:16.360","Text":"and of course this is in the y-direction."},{"Start":"24:16.360 ","End":"24:22.889","Text":"We\u0027ve just integrated both sides with respect to t. Of course,"},{"Start":"24:22.889 ","End":"24:26.430","Text":"what we\u0027re going to get over here in the y-direction,"},{"Start":"24:26.430 ","End":"24:31.020","Text":"the integral with respect 2t will give us"},{"Start":"24:31.020 ","End":"24:36.450","Text":"negative because the integral of sine is negative cosine so"},{"Start":"24:36.450 ","End":"24:40.095","Text":"we have negative cosine of"},{"Start":"24:40.095 ","End":"24:47.235","Text":"2z minus Omega t and then multiplied by our constants over here 2A."},{"Start":"24:47.235 ","End":"24:51.389","Text":"Then we divide by the inner derivative with respect to"},{"Start":"24:51.389 ","End":"24:55.439","Text":"t. Only this term over here has a t"},{"Start":"24:55.439 ","End":"24:59.550","Text":"so the derivative of negative Omega t is just negative Omega."},{"Start":"24:59.550 ","End":"25:04.090","Text":"We divide all of this by negative Omega."},{"Start":"25:04.160 ","End":"25:08.280","Text":"Then again, we can see that the minuses here cancel out,"},{"Start":"25:08.280 ","End":"25:11.115","Text":"so they just become positives."},{"Start":"25:11.115 ","End":"25:12.870","Text":"Then, in other words,"},{"Start":"25:12.870 ","End":"25:20.775","Text":"we can see that our magnetic field is equal to 2A divided by Omega"},{"Start":"25:20.775 ","End":"25:30.760","Text":"multiplied by cosine of 2z minus Omega t in the y-direction."},{"Start":"25:31.130 ","End":"25:33.795","Text":"Now of course sometime,"},{"Start":"25:33.795 ","End":"25:36.300","Text":"we can add a constant over here,"},{"Start":"25:36.300 ","End":"25:38.564","Text":"because we had an indefinite integral."},{"Start":"25:38.564 ","End":"25:41.550","Text":"But usually we say that this constant is equal to 0,"},{"Start":"25:41.550 ","End":"25:44.129","Text":"so you can just erase it,"},{"Start":"25:44.129 ","End":"25:45.945","Text":"and this is the magnetic field."},{"Start":"25:45.945 ","End":"25:50.205","Text":"What\u0027s important to note is that the argument,"},{"Start":"25:50.205 ","End":"25:51.780","Text":"or in other words,"},{"Start":"25:51.780 ","End":"25:54.490","Text":"this over here,"},{"Start":"25:54.500 ","End":"26:00.299","Text":"cosine 2z minus Omega t for the magnetic field"},{"Start":"26:00.299 ","End":"26:05.925","Text":"is the exact same thing as we saw for the electric field."},{"Start":"26:05.925 ","End":"26:15.240","Text":"Also, cosine 2z minus Omega t. The only difference that there is is the axes."},{"Start":"26:15.240 ","End":"26:18.960","Text":"Here the electric field is in the x-direction,"},{"Start":"26:18.960 ","End":"26:20.714","Text":"whereas as we can see here,"},{"Start":"26:20.714 ","End":"26:23.595","Text":"the magnetic field is in the y-direction."},{"Start":"26:23.595 ","End":"26:25.499","Text":"The other difference is,"},{"Start":"26:25.499 ","End":"26:27.645","Text":"as we\u0027ve mentioned before, the amplitude."},{"Start":"26:27.645 ","End":"26:31.290","Text":"Here, the amplitude of the electric field is just A,"},{"Start":"26:31.290 ","End":"26:36.510","Text":"whereas the amplitude of the magnetic field is 2A divided by Omega."},{"Start":"26:36.510 ","End":"26:40.320","Text":"However, the direction of the propagation and"},{"Start":"26:40.320 ","End":"26:44.190","Text":"the wavelength of the wave is exactly the same."},{"Start":"26:44.190 ","End":"26:50.204","Text":"If we draw this over here back on our axis,"},{"Start":"26:50.204 ","End":"26:53.925","Text":"so let\u0027s draw it in pink."},{"Start":"26:53.925 ","End":"26:59.670","Text":"What we can see is that our wave is in the y-direction,"},{"Start":"26:59.670 ","End":"27:01.830","Text":"but it\u0027s also dependent on z."},{"Start":"27:01.830 ","End":"27:04.035","Text":"If we start at the origin,"},{"Start":"27:04.035 ","End":"27:09.810","Text":"we\u0027ll again have our peak over here at the origin just it\u0027s a different peak because"},{"Start":"27:09.810 ","End":"27:15.480","Text":"it\u0027s a different amplitude and it\u0027s in the y-direction,"},{"Start":"27:15.480 ","End":"27:16.770","Text":"as we can see, however,"},{"Start":"27:16.770 ","End":"27:18.270","Text":"it\u0027s dependent on z."},{"Start":"27:18.270 ","End":"27:23.970","Text":"We\u0027re going along the z-axis, like so."},{"Start":"27:23.970 ","End":"27:28.570","Text":"I haven\u0027t actually drawn this very well."},{"Start":"27:28.610 ","End":"27:33.584","Text":"We have the peak over here and then here,"},{"Start":"27:33.584 ","End":"27:36.840","Text":"so the peaks correspond."},{"Start":"27:36.840 ","End":"27:39.840","Text":"Then we get to this minimum point,"},{"Start":"27:39.840 ","End":"27:42.930","Text":"and then it flips at the exact same point."},{"Start":"27:42.930 ","End":"27:47.835","Text":"The only difference is that the amplitude is different and"},{"Start":"27:47.835 ","End":"27:54.890","Text":"it\u0027s in the y-direction."},{"Start":"27:54.890 ","End":"28:03.040","Text":"We can just see that this is the wave representing the magnetic field."},{"Start":"28:03.040 ","End":"28:06.100","Text":"Both of them, the electric field and"},{"Start":"28:06.100 ","End":"28:10.164","Text":"the magnetic field are propagating in the same direction."},{"Start":"28:10.164 ","End":"28:13.360","Text":"They\u0027re both propagating in this direction."},{"Start":"28:13.360 ","End":"28:16.360","Text":"They have the exact same wavelength."},{"Start":"28:16.360 ","End":"28:20.350","Text":"The same lambda, as we can see,"},{"Start":"28:20.350 ","End":"28:22.900","Text":"it\u0027s the exact same lambda from peak to peak,"},{"Start":"28:22.900 ","End":"28:24.939","Text":"the same period and everything,"},{"Start":"28:24.939 ","End":"28:30.894","Text":"just their amplitude is different and they\u0027re in the different direction."},{"Start":"28:30.894 ","End":"28:35.995","Text":"The electric field is in the x direction and the magnetic field is in the y direction."},{"Start":"28:35.995 ","End":"28:40.370","Text":"Notice that they\u0027re perpendicular to one another."},{"Start":"28:40.620 ","End":"28:43.179","Text":"The fact that they are perpendicular,"},{"Start":"28:43.179 ","End":"28:46.885","Text":"we can actually write this over here."},{"Start":"28:46.885 ","End":"28:52.449","Text":"The electric field is always going to be perpendicular to"},{"Start":"28:52.449 ","End":"28:58.164","Text":"the magnetic field and perpendicular to the direction of propagation."},{"Start":"28:58.164 ","End":"29:03.295","Text":"For instance here the direction of propagation is z."},{"Start":"29:03.295 ","End":"29:08.425","Text":"The direction that the wave is moving in is z."},{"Start":"29:08.425 ","End":"29:11.649","Text":"As we can see, the electric field is in the x-axis,"},{"Start":"29:11.649 ","End":"29:13.315","Text":"which is perpendicular to z,"},{"Start":"29:13.315 ","End":"29:17.425","Text":"and the magnetic field is in the y-axis,"},{"Start":"29:17.425 ","End":"29:20.110","Text":"which is perpendicular to z as well."},{"Start":"29:20.110 ","End":"29:25.909","Text":"Of course, the x and y are also perpendicular to one another."},{"Start":"29:26.550 ","End":"29:32.530","Text":"This is also something important to include in your equation sheets."},{"Start":"29:32.530 ","End":"29:36.325","Text":"Of course, the direction of propagation is given to us by"},{"Start":"29:36.325 ","End":"29:41.590","Text":"k. The amplitude of the electric field will"},{"Start":"29:41.590 ","End":"29:45.054","Text":"be perpendicular to k. The amplitude of"},{"Start":"29:45.054 ","End":"29:50.829","Text":"the magnetic field will also be perpendicular to k. K it\u0027s in the y direction,"},{"Start":"29:50.829 ","End":"29:55.854","Text":"and k represents the direction of propagation."},{"Start":"29:55.854 ","End":"30:00.440","Text":"We can just write this as the k vector."},{"Start":"30:00.510 ","End":"30:03.115","Text":"Now, just a little note,"},{"Start":"30:03.115 ","End":"30:04.390","Text":"we\u0027ve already said this,"},{"Start":"30:04.390 ","End":"30:09.670","Text":"here because the coefficient of omega was negative 1, or in other words,"},{"Start":"30:09.670 ","End":"30:11.560","Text":"because there\u0027s a negative over here,"},{"Start":"30:11.560 ","End":"30:18.045","Text":"the direction of propagation is as given in our vector k. However,"},{"Start":"30:18.045 ","End":"30:20.715","Text":"if there was a plus here."},{"Start":"30:20.715 ","End":"30:25.830","Text":"Then we said that the direction of propagation would be leftwards."},{"Start":"30:25.830 ","End":"30:33.189","Text":"Or in other words, in the opposite direction to k. If there\u0027s a minus here,"},{"Start":"30:33.189 ","End":"30:34.495","Text":"it might sound a bit weird."},{"Start":"30:34.495 ","End":"30:36.474","Text":"If there\u0027s a minus over here,"},{"Start":"30:36.474 ","End":"30:40.885","Text":"It\u0027s in the direction of k. If there\u0027s a plus,"},{"Start":"30:40.885 ","End":"30:48.144","Text":"then it\u0027s the opposite direction to K. Now"},{"Start":"30:48.144 ","End":"30:55.974","Text":"let\u0027s do a little recap of everything we\u0027ve done in this lesson and the previous one."},{"Start":"30:55.974 ","End":"30:57.849","Text":"In the previous lesson,"},{"Start":"30:57.849 ","End":"31:04.270","Text":"we looked at Maxwell\u0027s 4 equations from which we can derive the wave equation."},{"Start":"31:04.270 ","End":"31:07.464","Text":"First of all, we said that the wave equations are"},{"Start":"31:07.464 ","End":"31:10.974","Text":"correct when we\u0027re looking at waves in a vacuum."},{"Start":"31:10.974 ","End":"31:12.519","Text":"If we\u0027re in a vacuum,"},{"Start":"31:12.519 ","End":"31:15.579","Text":"that means that there\u0027s no free particles or charges,"},{"Start":"31:15.579 ","End":"31:18.129","Text":"which means that rho is equal to 0."},{"Start":"31:18.129 ","End":"31:22.809","Text":"Also that current can pass because there\u0027s no charges around,"},{"Start":"31:22.809 ","End":"31:25.224","Text":"so j is equal to 0."},{"Start":"31:25.224 ","End":"31:28.404","Text":"Then using these equations,"},{"Start":"31:28.404 ","End":"31:31.580","Text":"we got to the wave equations."},{"Start":"31:32.250 ","End":"31:37.764","Text":"It\u0027s not that important to remember how we got to these."},{"Start":"31:37.764 ","End":"31:43.480","Text":"Then we also explained what this nabla squared means."},{"Start":"31:43.480 ","End":"31:47.889","Text":"We looked at that and we got this differential equation over here."},{"Start":"31:47.889 ","End":"31:53.799","Text":"Nabla squared is just each partial derivative taken twice and these are some of"},{"Start":"31:53.799 ","End":"32:00.580","Text":"them on each component of the field be it electric or magnetic."},{"Start":"32:00.580 ","End":"32:04.510","Text":"We got this differential equation where from this differential equation we"},{"Start":"32:04.510 ","End":"32:08.484","Text":"got this solution which is where it becomes interesting."},{"Start":"32:08.484 ","End":"32:11.469","Text":"We saw that we have the I,"},{"Start":"32:11.469 ","End":"32:14.905","Text":"which gives us the position in space."},{"Start":"32:14.905 ","End":"32:18.520","Text":"We have the k vector, which represents,"},{"Start":"32:18.520 ","End":"32:21.504","Text":"as we spoke about a little bit later,"},{"Start":"32:21.504 ","End":"32:29.919","Text":"the direction of propagation and also the magnitude of this propagation."},{"Start":"32:29.919 ","End":"32:36.524","Text":"We saw that when we do the dot product between k and I we get this over here."},{"Start":"32:36.524 ","End":"32:38.654","Text":"This type of equation,"},{"Start":"32:38.654 ","End":"32:40.950","Text":"and that we have a over here,"},{"Start":"32:40.950 ","End":"32:43.800","Text":"which gives us the amplitude and omega,"},{"Start":"32:43.800 ","End":"32:47.129","Text":"which gives us also the direction of propagation."},{"Start":"32:47.129 ","End":"32:54.924","Text":"It also gives us the frequency of the wave and the period of the wave and"},{"Start":"32:54.924 ","End":"32:58.420","Text":"therefore from that also get"},{"Start":"32:58.420 ","End":"33:04.239","Text":"the wavelength of the wave by playing around with certain things."},{"Start":"33:04.239 ","End":"33:07.660","Text":"Sorry, from k, we get the wavelength of the wave."},{"Start":"33:07.660 ","End":"33:10.599","Text":"Then we looked at an easy example."},{"Start":"33:10.599 ","End":"33:15.629","Text":"We saw how to find k. We just look for any variables x,"},{"Start":"33:15.629 ","End":"33:19.890","Text":"y, z, and then place their coefficients."},{"Start":"33:19.890 ","End":"33:23.280","Text":"Here we had 0 times x plus 0 times y,"},{"Start":"33:23.280 ","End":"33:26.620","Text":"so it\u0027s 00 plus 2 times z."},{"Start":"33:27.450 ","End":"33:30.310","Text":"This is the direction of propagation."},{"Start":"33:30.310 ","End":"33:34.600","Text":"We can see that the wave is propagating in the z direction and"},{"Start":"33:34.600 ","End":"33:39.040","Text":"the amplitude of the electric field was in the direction given here."},{"Start":"33:39.040 ","End":"33:46.760","Text":"Specifically, it\u0027s in the x direction as we had 0 in the y and 0 in the z-direction."},{"Start":"33:46.920 ","End":"33:50.710","Text":"Then we looked at some other important equations."},{"Start":"33:50.710 ","End":"33:53.080","Text":"We have our omega, which as we said,"},{"Start":"33:53.080 ","End":"33:57.219","Text":"is connected to the frequency of the wave and to the time period."},{"Start":"33:57.219 ","End":"34:03.174","Text":"We saw that the magnitude of k can also give us the wavelength."},{"Start":"34:03.174 ","End":"34:08.034","Text":"We saw that omega is also connected to k,"},{"Start":"34:08.034 ","End":"34:13.060","Text":"where omega is equal to the speed of light multiplied by the magnitude of"},{"Start":"34:13.060 ","End":"34:20.694","Text":"k. Then we also saw how to convert between the electric field and the magnetic field."},{"Start":"34:20.694 ","End":"34:23.080","Text":"We said that if we\u0027re given the magnetic fields,"},{"Start":"34:23.080 ","End":"34:25.210","Text":"we use Maxwell\u0027s third equation."},{"Start":"34:25.210 ","End":"34:27.399","Text":"If we have the magnetic fields,"},{"Start":"34:27.399 ","End":"34:30.175","Text":"then we use Maxwell\u0027s fourth equation."},{"Start":"34:30.175 ","End":"34:32.035","Text":"Whichever field we have,"},{"Start":"34:32.035 ","End":"34:37.659","Text":"we do the root of x to calculate the corresponding fields."},{"Start":"34:37.659 ","End":"34:41.649","Text":"Remember equations 3 and 4 are not the same."},{"Start":"34:41.649 ","End":"34:47.889","Text":"In Equation 4 we also have to take into account mu naught and epsilon naught."},{"Start":"34:47.889 ","End":"34:58.075","Text":"Here we showed an example of how we do the rotor of a vector field."},{"Start":"34:58.075 ","End":"35:01.105","Text":"Then we got the differential equation."},{"Start":"35:01.105 ","End":"35:03.594","Text":"We integrated both sides,"},{"Start":"35:03.594 ","End":"35:08.125","Text":"and therefore, we got to our magnetic field."},{"Start":"35:08.125 ","End":"35:11.139","Text":"We also saw that the argument is going to be the"},{"Start":"35:11.139 ","End":"35:15.010","Text":"same for the magnetic and the electric field."},{"Start":"35:15.010 ","End":"35:16.749","Text":"What changes is that"},{"Start":"35:16.749 ","End":"35:22.239","Text":"the electric and magnetic field will be perpendicular to one another."},{"Start":"35:22.239 ","End":"35:26.140","Text":"Here our electric field was in the x-direction,"},{"Start":"35:26.140 ","End":"35:28.704","Text":"and here we can see it\u0027s in the y-direction."},{"Start":"35:28.704 ","End":"35:32.019","Text":"Of course, both fields are going to also be"},{"Start":"35:32.019 ","End":"35:35.259","Text":"perpendicular to the direction of propagation."},{"Start":"35:35.259 ","End":"35:37.480","Text":"Here the direction of propagation was z."},{"Start":"35:37.480 ","End":"35:41.995","Text":"The x and the y-axis are of course, perpendicular to z."},{"Start":"35:41.995 ","End":"35:45.340","Text":"The direction of propagation is of course, as we said,"},{"Start":"35:45.340 ","End":"35:49.285","Text":"given by the vector k. In other words, E,"},{"Start":"35:49.285 ","End":"35:55.495","Text":"B, and k are all going to be perpendicular to one another."},{"Start":"35:55.495 ","End":"36:00.219","Text":"Then the only other difference is just the amplitude of the wave,"},{"Start":"36:00.219 ","End":"36:02.485","Text":"but everything else is the same."},{"Start":"36:02.485 ","End":"36:05.065","Text":"Their peaks are at the same points."},{"Start":"36:05.065 ","End":"36:08.155","Text":"Their minimums are at the same points."},{"Start":"36:08.155 ","End":"36:12.249","Text":"They cut the x or the axes of their propagation here it\u0027s"},{"Start":"36:12.249 ","End":"36:17.260","Text":"the z-axis at the same points also."},{"Start":"36:17.260 ","End":"36:21.429","Text":"That is what we saw in the past 2 lessons."},{"Start":"36:21.429 ","End":"36:26.479","Text":"That\u0027s a recap, and that is the end of this lesson."}],"ID":22430},{"Watched":false,"Name":"Deriving the Wave Equations Using Maxwell","Duration":"19m 58s","ChapterTopicVideoID":21578,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21578.jpeg","UploadDate":"2020-05-07T18:44:10.4870000","DurationForVideoObject":"PT19M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.935","Text":"Hello. In this lesson,"},{"Start":"00:01.935 ","End":"00:07.170","Text":"we\u0027re going to be speaking about the wave equation and its derivation."},{"Start":"00:07.170 ","End":"00:12.570","Text":"Now, the derivation isn\u0027t that useful for solving questions, however,"},{"Start":"00:12.570 ","End":"00:15.405","Text":"the wave equation is imperative,"},{"Start":"00:15.405 ","End":"00:18.555","Text":"but we\u0027re going to look at how we derive it."},{"Start":"00:18.555 ","End":"00:21.540","Text":"Now, we\u0027ve seen these 4 equations before."},{"Start":"00:21.540 ","End":"00:28.780","Text":"These are 4 of Maxwell\u0027s equations that we\u0027ve seen in the previous chapters."},{"Start":"00:28.820 ","End":"00:34.275","Text":"The wave equation comes from these 4 equations."},{"Start":"00:34.275 ","End":"00:42.730","Text":"Now, what is important is that the wave equation speaks about waves that are in a vacuum."},{"Start":"00:42.730 ","End":"00:46.053","Text":"This is very important."},{"Start":"00:46.053 ","End":"00:47.345","Text":"What does this mean?"},{"Start":"00:47.345 ","End":"00:48.710","Text":"If it\u0027s in a vacuum,"},{"Start":"00:48.710 ","End":"00:51.650","Text":"that means that the space is completely empty."},{"Start":"00:51.650 ","End":"00:53.630","Text":"There is no other particles."},{"Start":"00:53.630 ","End":"00:58.475","Text":"What does that mean for us with respect to the equation?"},{"Start":"00:58.475 ","End":"01:00.865","Text":"If there\u0027s no other particles,"},{"Start":"01:00.865 ","End":"01:04.395","Text":"then that means that there\u0027s no Rho,"},{"Start":"01:04.395 ","End":"01:07.635","Text":"because Rho is the charge density."},{"Start":"01:07.635 ","End":"01:10.970","Text":"If there\u0027s no particles,"},{"Start":"01:10.970 ","End":"01:13.490","Text":"that means there\u0027s no charge density,"},{"Start":"01:13.490 ","End":"01:19.850","Text":"so this becomes 0 and in that case this whole equation is equal to 0."},{"Start":"01:19.850 ","End":"01:22.834","Text":"Additionally, if there\u0027s no particles,"},{"Start":"01:22.834 ","End":"01:26.360","Text":"then there can be no current passing through."},{"Start":"01:26.360 ","End":"01:31.169","Text":"We can also see that by the fact that there\u0027s no charge density."},{"Start":"01:31.310 ","End":"01:35.355","Text":"There\u0027s no charges causing the current."},{"Start":"01:35.355 ","End":"01:37.565","Text":"That means that J,"},{"Start":"01:37.565 ","End":"01:39.800","Text":"which is the current density,"},{"Start":"01:39.800 ","End":"01:41.700","Text":"is also equal to 0."},{"Start":"01:41.700 ","End":"01:47.674","Text":"Then we\u0027re just left with this section over here for this equation."},{"Start":"01:47.674 ","End":"01:52.430","Text":"What we can see is that we\u0027re left with 2 variables,"},{"Start":"01:52.430 ","End":"01:56.510","Text":"E for the electric field and B for the magnetic field."},{"Start":"01:56.510 ","End":"01:58.940","Text":"What we want to do is we want to play around with"},{"Start":"01:58.940 ","End":"02:03.420","Text":"these equations in order to get to 2 different equations;"},{"Start":"02:03.420 ","End":"02:08.555","Text":"1 dependent solely on E and 1 dependent solely on B,"},{"Start":"02:08.555 ","End":"02:11.195","Text":"and those are called the wave equations."},{"Start":"02:11.195 ","End":"02:14.510","Text":"I\u0027ll write them out over here."},{"Start":"02:14.630 ","End":"02:17.819","Text":"These are the 2 equations,"},{"Start":"02:17.819 ","End":"02:20.925","Text":"the 2 wave equations."},{"Start":"02:20.925 ","End":"02:23.030","Text":"We can see that they\u0027re identical."},{"Start":"02:23.030 ","End":"02:28.640","Text":"Just 1 is with a variable of E and the other 1 is with a variable of B."},{"Start":"02:28.640 ","End":"02:33.510","Text":"What I\u0027m going to do is I\u0027m going to show how we get to the first equation."},{"Start":"02:33.730 ","End":"02:35.990","Text":"To get to the second equation,"},{"Start":"02:35.990 ","End":"02:41.670","Text":"we do the exact same thing just with the magnetic field."},{"Start":"02:42.140 ","End":"02:44.771","Text":"Let\u0027s number these equations."},{"Start":"02:44.771 ","End":"02:47.260","Text":"This is 1,"},{"Start":"02:47.260 ","End":"02:49.505","Text":"2, 3, and 4."},{"Start":"02:49.505 ","End":"02:54.530","Text":"The first thing we\u0027re going to do is we\u0027re going to take Equation 3,"},{"Start":"02:54.530 ","End":"02:56.270","Text":"so I\u0027ll write it like this."},{"Start":"02:56.270 ","End":"02:57.748","Text":"We\u0027re taking Equation 3,"},{"Start":"02:57.748 ","End":"03:00.260","Text":"and we\u0027re doing a rotor on both sides."},{"Start":"03:00.260 ","End":"03:05.475","Text":"That means taking the Nabla symbol cross all of this to both sides."},{"Start":"03:05.475 ","End":"03:10.980","Text":"What we have is Nabla or Del cross and then Equation 3."},{"Start":"03:10.980 ","End":"03:17.930","Text":"Then we have Nabla cross E. This is equal to,"},{"Start":"03:17.930 ","End":"03:20.810","Text":"again, we\u0027re doing the same thing to both sides,"},{"Start":"03:20.810 ","End":"03:23.375","Text":"Nabla cross and then"},{"Start":"03:23.375 ","End":"03:31.350","Text":"negative dB by dt."},{"Start":"03:31.350 ","End":"03:35.840","Text":"What we\u0027re going to do is we\u0027re going to look at the left side of"},{"Start":"03:35.840 ","End":"03:41.990","Text":"the equation now to show how we get to the left side of the final equation."},{"Start":"03:41.990 ","End":"03:45.830","Text":"Then we\u0027re going to deal with the right side of the equation to see how we"},{"Start":"03:45.830 ","End":"03:50.100","Text":"get to the right side of the final equation."},{"Start":"03:50.100 ","End":"03:53.645","Text":"Using mathematical laws and operators,"},{"Start":"03:53.645 ","End":"03:56.960","Text":"which you may have learned in 1 of your maths courses,"},{"Start":"03:56.960 ","End":"04:06.815","Text":"we have seen or you may know that the rotor of the rotor of a vector field"},{"Start":"04:06.815 ","End":"04:14.625","Text":"is equal to Nabla dot open brackets Nabla dot"},{"Start":"04:14.625 ","End":"04:24.850","Text":"the vector field minus Nabla squared of the vector field."},{"Start":"04:25.000 ","End":"04:28.040","Text":"From the rule of operators,"},{"Start":"04:28.040 ","End":"04:32.570","Text":"what we have here is the gradient of the divergence of the E field,"},{"Start":"04:32.570 ","End":"04:38.250","Text":"and here we have the Laplacian if you\u0027ve heard of this before."},{"Start":"04:38.300 ","End":"04:42.180","Text":"Now notice these 2 terms are different."},{"Start":"04:42.180 ","End":"04:45.590","Text":"Here first we find the divergence of E,"},{"Start":"04:45.590 ","End":"04:49.378","Text":"and then we do Nabla."},{"Start":"04:49.378 ","End":"04:54.130","Text":"Here we take the Nabla, the Dell,"},{"Start":"04:54.130 ","End":"04:55.990","Text":"and we square it,"},{"Start":"04:55.990 ","End":"05:00.182","Text":"and only then do we apply it to our vector field."},{"Start":"05:00.182 ","End":"05:02.980","Text":"These 2 are different."},{"Start":"05:02.980 ","End":"05:06.130","Text":"If you haven\u0027t learned this or you don\u0027t remember this,"},{"Start":"05:06.130 ","End":"05:12.810","Text":"you can either research it online and look at the mathematical introduction,"},{"Start":"05:12.810 ","End":"05:16.545","Text":"but otherwise, it doesn\u0027t really matter,"},{"Start":"05:16.545 ","End":"05:18.540","Text":"as we said at the beginning of the lesson,"},{"Start":"05:18.540 ","End":"05:22.555","Text":"we really just want to know how to use these equations,"},{"Start":"05:22.555 ","End":"05:25.540","Text":"and getting there is just or so interesting,"},{"Start":"05:25.540 ","End":"05:29.200","Text":"and it\u0027s quite important to know but for solving questions,"},{"Start":"05:29.200 ","End":"05:31.695","Text":"you don\u0027t really need to know this."},{"Start":"05:31.695 ","End":"05:35.075","Text":"Don\u0027t panic if you don\u0027t know what\u0027s going on here."},{"Start":"05:35.075 ","End":"05:42.480","Text":"We can see this side is equal to just this."},{"Start":"05:42.480 ","End":"05:47.780","Text":"Now, notice from Equation 1 here we have the divergence of E,"},{"Start":"05:47.780 ","End":"05:49.870","Text":"which from Equation 1,"},{"Start":"05:49.870 ","End":"05:52.245","Text":"we see is equal to 0."},{"Start":"05:52.245 ","End":"05:56.360","Text":"This then cancels out. This is equal to 0."},{"Start":"05:56.360 ","End":"06:00.410","Text":"What we\u0027re left with is that the whole left side of"},{"Start":"06:00.410 ","End":"06:05.405","Text":"the equation is equal to the negative Laplacian."},{"Start":"06:05.405 ","End":"06:13.679","Text":"We can already see this Laplacian over here on this left side of the final equation."},{"Start":"06:13.690 ","End":"06:17.360","Text":"We\u0027ve gotten to this side,"},{"Start":"06:17.360 ","End":"06:23.070","Text":"so now we\u0027re going to look at this section over here."},{"Start":"06:24.200 ","End":"06:26.580","Text":"Let\u0027s begin it over here."},{"Start":"06:26.580 ","End":"06:34.785","Text":"What we have is the rotor of negative dB by dt."},{"Start":"06:34.785 ","End":"06:38.750","Text":"First of all, the negative is like a constant,"},{"Start":"06:38.750 ","End":"06:41.410","Text":"it\u0027s equivalent to negative 1."},{"Start":"06:41.410 ","End":"06:43.110","Text":"Because it\u0027s a constant,"},{"Start":"06:43.110 ","End":"06:47.240","Text":"we can just take it out and put it to the side over here."},{"Start":"06:47.240 ","End":"06:49.895","Text":"Also when we\u0027re looking at the rotor,"},{"Start":"06:49.895 ","End":"06:57.890","Text":"this Nabla or this Del is taking partial differentials with respect to space."},{"Start":"06:57.890 ","End":"07:02.270","Text":"Coordinates in space, for instance, the x, y,"},{"Start":"07:02.270 ","End":"07:07.415","Text":"z axes or R Theta z or R Theta Phi,"},{"Start":"07:07.415 ","End":"07:10.340","Text":"depending on the coordinate system that we\u0027re using."},{"Start":"07:10.340 ","End":"07:15.140","Text":"Whereas this differential is with respect to time."},{"Start":"07:15.140 ","End":"07:21.755","Text":"Of course, over here we can see that a differential with respect to time,"},{"Start":"07:21.755 ","End":"07:25.595","Text":"and with respect to space are independent of 1 another."},{"Start":"07:25.595 ","End":"07:29.845","Text":"In other words, we can look at this d by"},{"Start":"07:29.845 ","End":"07:35.900","Text":"dt as also being a constant which is independent of x,"},{"Start":"07:35.900 ","End":"07:40.655","Text":"y, z, or coordinate systems relating to space."},{"Start":"07:40.655 ","End":"07:49.160","Text":"Therefore, we can just write this as being equal to negative d by dt"},{"Start":"07:49.160 ","End":"07:55.755","Text":"of the rotor of"},{"Start":"07:55.755 ","End":"08:00.515","Text":"the magnetic field like so."},{"Start":"08:00.515 ","End":"08:09.250","Text":"Then we can rewrite this as being equal to negative d by dt."},{"Start":"08:09.250 ","End":"08:13.473","Text":"Now the rotor of the magnetic field,"},{"Start":"08:13.473 ","End":"08:17.870","Text":"so this we can see over here in Maxwell\u0027s equation number 4,"},{"Start":"08:17.870 ","End":"08:20.480","Text":"we can see the rotor of the magnetic field where"},{"Start":"08:20.480 ","End":"08:24.485","Text":"this term canceled out and we\u0027re left with this term over here,"},{"Start":"08:24.485 ","End":"08:26.830","Text":"which is equal to"},{"Start":"08:26.830 ","End":"08:36.515","Text":"Mu_naught Epsilon_naught dE by dt."},{"Start":"08:36.515 ","End":"08:41.950","Text":"Then of course we can see that Mu Naught and Epsilon Naught are like constants."},{"Start":"08:41.950 ","End":"08:45.220","Text":"What we\u0027re left with is negative Mu Naught,"},{"Start":"08:45.220 ","End":"08:49.090","Text":"Epsilon Naught k because we can move them out because they\u0027re like constants."},{"Start":"08:49.090 ","End":"08:57.685","Text":"D by dt of d by dt of the electric field."},{"Start":"08:57.685 ","End":"08:59.290","Text":"Or in other words,"},{"Start":"08:59.290 ","End":"09:03.955","Text":"this is equal to negative Mu Naught Epsilon Naught over"},{"Start":"09:03.955 ","End":"09:13.090","Text":"d^2E by dt^2,"},{"Start":"09:13.090 ","End":"09:16.315","Text":"which is exactly the same as over here."},{"Start":"09:16.315 ","End":"09:25.135","Text":"Now of course, then we can say that here we have that this is equal to this over here."},{"Start":"09:25.135 ","End":"09:26.665","Text":"We have negative,"},{"Start":"09:26.665 ","End":"09:31.555","Text":"so here we have this negative over here,"},{"Start":"09:31.555 ","End":"09:36.340","Text":"and here we have this negative over here so we can multiply just both sides."},{"Start":"09:36.340 ","End":"09:39.355","Text":"Remember this is equal to the left side,"},{"Start":"09:39.355 ","End":"09:43.340","Text":"and this is equal to the right side of the equation."},{"Start":"09:43.370 ","End":"09:47.670","Text":"We can just multiply both sides of the equation by negative 1 to"},{"Start":"09:47.670 ","End":"09:52.420","Text":"get rid of the negative sign and we\u0027re left with this over here."},{"Start":"09:52.560 ","End":"09:59.200","Text":"As we\u0027ve seen, the equations are exactly the same just 1 has the variable E and"},{"Start":"09:59.200 ","End":"10:05.140","Text":"1 has the variable B and we derive both of them,"},{"Start":"10:05.140 ","End":"10:08.680","Text":"so also the 2nd equation in the exact same way."},{"Start":"10:08.680 ","End":"10:13.855","Text":"We do wrote a 2 both sides and then we substitute"},{"Start":"10:13.855 ","End":"10:19.990","Text":"in this equation over here and we just solve,"},{"Start":"10:19.990 ","End":"10:25.120","Text":"and another explanation which we may speak about later is to show,"},{"Start":"10:25.120 ","End":"10:27.235","Text":"or you might learn it in a different course,"},{"Start":"10:27.235 ","End":"10:30.160","Text":"is that the electric field and the magnetic field are actually the"},{"Start":"10:30.160 ","End":"10:34.015","Text":"same just from different frames of reference,"},{"Start":"10:34.015 ","End":"10:38.800","Text":"it\u0027s not important to know now just notes and then that\u0027s another explanation of"},{"Start":"10:38.800 ","End":"10:44.395","Text":"why this is a very similar equation."},{"Start":"10:44.395 ","End":"10:46.855","Text":"Now, another thing to note,"},{"Start":"10:46.855 ","End":"10:48.670","Text":"we have this constant over here,"},{"Start":"10:48.670 ","End":"10:50.980","Text":"Mu Naught Epsilon Naught."},{"Start":"10:50.980 ","End":"11:00.720","Text":"Mu naught epsilon naught multiplied together is also equal to 1 divided by C^2,"},{"Start":"11:00.720 ","End":"11:04.330","Text":"where C is the speed of light."},{"Start":"11:05.730 ","End":"11:09.850","Text":"Now let\u0027s speak a little bit about the equations,"},{"Start":"11:09.850 ","End":"11:15.850","Text":"so I\u0027m just going to speak about the equation with the variable E for the electric field."},{"Start":"11:15.850 ","End":"11:19.510","Text":"But everything I say about this equation is also"},{"Start":"11:19.510 ","End":"11:24.860","Text":"correct for the equation relating to the magnetic field."},{"Start":"11:24.960 ","End":"11:27.985","Text":"Let\u0027s just look at the electric field."},{"Start":"11:27.985 ","End":"11:34.690","Text":"The equation for the electric field over here is actually made of 3 equations."},{"Start":"11:34.690 ","End":"11:38.620","Text":"Because we have over here this nabla."},{"Start":"11:38.620 ","End":"11:40.540","Text":"Whenever we see the nabla,"},{"Start":"11:40.540 ","End":"11:45.955","Text":"we know that it separates out the vector field into each of its components in space."},{"Start":"11:45.955 ","End":"11:49.210","Text":"Let\u0027s say we\u0027re using Cartesian coordinates it\u0027s with"},{"Start":"11:49.210 ","End":"11:52.540","Text":"respect to the components in the x direction,"},{"Start":"11:52.540 ","End":"11:55.850","Text":"y direction, and z direction."},{"Start":"11:56.370 ","End":"12:02.860","Text":"We can see that this would be nabla squared E_x,"},{"Start":"12:02.860 ","End":"12:08.890","Text":"which is equal to Mu Naught Epsilon Naught d^2E,"},{"Start":"12:08.890 ","End":"12:16.480","Text":"x by dt squared or nabla squared e_y,"},{"Start":"12:16.480 ","End":"12:22.540","Text":"which is equal to Mu Naught Epsilon Naught d^2E_y"},{"Start":"12:22.540 ","End":"12:29.785","Text":"by dt^2 and nabla squared E_z,"},{"Start":"12:29.785 ","End":"12:39.010","Text":"which is equal to Mu Naught Epsilon Naught d^2E_z by dt^2."},{"Start":"12:39.010 ","End":"12:41.320","Text":"This is the exact same thing,"},{"Start":"12:41.320 ","End":"12:47.050","Text":"like when we saw Sigma F is equal to ma."},{"Start":"12:47.050 ","End":"12:51.850","Text":"That means that we would have Sigma, sorry,"},{"Start":"12:51.850 ","End":"12:57.865","Text":"this is sigma Fx is equal to ma in the x direction."},{"Start":"12:57.865 ","End":"13:02.380","Text":"The sum of all the forces in the y-direction would equal to the mass multiplied"},{"Start":"13:02.380 ","End":"13:08.180","Text":"by the acceleration in the y-direction and the same for the z-axis."},{"Start":"13:09.000 ","End":"13:13.480","Text":"Now let\u0027s look at each one of these, what it means."},{"Start":"13:13.480 ","End":"13:18.415","Text":"Over here we understand what\u0027s going on the right side of the equation."},{"Start":"13:18.415 ","End":"13:23.950","Text":"These are constants multiplied by the E_x component,"},{"Start":"13:23.950 ","End":"13:29.730","Text":"which has been derived twice with respect to t,"},{"Start":"13:29.730 ","End":"13:34.470","Text":"or its derivative has been taken twice with respect to t. If"},{"Start":"13:34.470 ","End":"13:39.990","Text":"our electric field was some constant t squared,"},{"Start":"13:39.990 ","End":"13:42.615","Text":"so we would take its derivative once,"},{"Start":"13:42.615 ","End":"13:45.480","Text":"which would give 2At,"},{"Start":"13:45.480 ","End":"13:51.250","Text":"and then the 2nd time derivative would just leave us with 2A."},{"Start":"13:52.290 ","End":"13:55.405","Text":"That\u0027s what we do here that\u0027s not very new."},{"Start":"13:55.405 ","End":"13:57.280","Text":"But now let\u0027s look at the left side,"},{"Start":"13:57.280 ","End":"14:01.105","Text":"which is something that is new to us."},{"Start":"14:01.105 ","End":"14:05.665","Text":"We have this nabla squared."},{"Start":"14:05.665 ","End":"14:07.405","Text":"What is nabla squared?"},{"Start":"14:07.405 ","End":"14:10.555","Text":"Nabla we know are the partial derivatives."},{"Start":"14:10.555 ","End":"14:14.260","Text":"If it\u0027s squared, it\u0027s just the partial derivative squared."},{"Start":"14:14.260 ","End":"14:22.915","Text":"We have d^2 by dx^2 plus d^2 by"},{"Start":"14:22.915 ","End":"14:33.350","Text":"dy^2 plus d^2 by dz^2 in Cartesian coordinates."},{"Start":"14:33.780 ","End":"14:36.805","Text":"In actual fact what we\u0027re doing,"},{"Start":"14:36.805 ","End":"14:40.370","Text":"when we have this nabla squared E,"},{"Start":"14:41.190 ","End":"14:44.350","Text":"we split it up into 3 equations."},{"Start":"14:44.350 ","End":"14:45.910","Text":"In this section,"},{"Start":"14:45.910 ","End":"14:50.710","Text":"we\u0027re taking the x component of the magnetic field and"},{"Start":"14:50.710 ","End":"14:55.525","Text":"we are taking its partial derivatives with respect to x twice,"},{"Start":"14:55.525 ","End":"14:56.980","Text":"with respect to y twice,"},{"Start":"14:56.980 ","End":"14:58.705","Text":"and with respect to z twice,"},{"Start":"14:58.705 ","End":"15:00.910","Text":"and then adding them up."},{"Start":"15:00.910 ","End":"15:11.440","Text":"Over here we would have dE_x with respect to x, like so."},{"Start":"15:11.440 ","End":"15:13.495","Text":"Then we just add them up."},{"Start":"15:13.495 ","End":"15:16.520","Text":"That is all that this means."},{"Start":"15:17.640 ","End":"15:20.290","Text":"Let\u0027s just go over this again,"},{"Start":"15:20.290 ","End":"15:23.200","Text":"let\u0027s imagine or not imagine our E field is"},{"Start":"15:23.200 ","End":"15:26.365","Text":"a vector field so it could be something like this,"},{"Start":"15:26.365 ","End":"15:30.190","Text":"x y, y^2, z."},{"Start":"15:30.190 ","End":"15:37.510","Text":"Let\u0027s just say that this is our vector field so let\u0027s say I\u0027m doing this equation."},{"Start":"15:37.510 ","End":"15:41.140","Text":"This side of the equation just for the x-component."},{"Start":"15:41.140 ","End":"15:45.715","Text":"I\u0027m just looking at this over here, let\u0027s underline it."},{"Start":"15:45.715 ","End":"15:49.310","Text":"That is what I\u0027m doing."},{"Start":"15:49.950 ","End":"16:00.550","Text":"That means that all I\u0027m doing over here from this nabla squared E_x is I\u0027m taking"},{"Start":"16:00.550 ","End":"16:10.600","Text":"d^2 of the x component by dx^2 plus d^2 of"},{"Start":"16:10.600 ","End":"16:16.465","Text":"the x-component of the E field by y^2 plus"},{"Start":"16:16.465 ","End":"16:24.295","Text":"d of the x-component by dz^2."},{"Start":"16:24.295 ","End":"16:28.600","Text":"This is our E_x,"},{"Start":"16:28.600 ","End":"16:31.255","Text":"this is the x-component of our E field."},{"Start":"16:31.255 ","End":"16:35.965","Text":"Over here, I\u0027ll just change it to x^2, y^2."},{"Start":"16:35.965 ","End":"16:40.533","Text":"Let\u0027s take the x component of the electric field,"},{"Start":"16:40.533 ","End":"16:44.110","Text":"and take its derivative with respect to x twice."},{"Start":"16:44.110 ","End":"16:53.260","Text":"The first time we\u0027ll be left with 2xy^2 and the 2nd time we\u0027ll just be left with 2y^2."},{"Start":"16:53.260 ","End":"16:56.005","Text":"Then again the same component,"},{"Start":"16:56.005 ","End":"16:59.530","Text":"but we\u0027re taking the derivative twice with respect to y."},{"Start":"16:59.530 ","End":"17:02.635","Text":"The first time we\u0027ll have 2x^2y,"},{"Start":"17:02.635 ","End":"17:06.895","Text":"and the second derivative will just be 2x^2."},{"Start":"17:06.895 ","End":"17:11.665","Text":"Then we can take it again with respect to z."},{"Start":"17:11.665 ","End":"17:15.040","Text":"We can see that there\u0027s no z variable over here"},{"Start":"17:15.040 ","End":"17:20.325","Text":"so both the 1st and 2nd derivatives will give us 0."},{"Start":"17:20.325 ","End":"17:25.580","Text":"Then when we\u0027re just calculating the nabla squared,"},{"Start":"17:25.580 ","End":"17:33.130","Text":"so we\u0027re taking the sum of this so what we\u0027re going to have is 2y^2 plus 2x^2."},{"Start":"17:33.130 ","End":"17:35.600","Text":"That\u0027s all that this means and of course,"},{"Start":"17:35.600 ","End":"17:37.580","Text":"you do it for each component,"},{"Start":"17:37.580 ","End":"17:41.480","Text":"so also for the y component and also for the z component,"},{"Start":"17:41.480 ","End":"17:43.950","Text":"the exact same thing."},{"Start":"17:44.880 ","End":"17:48.695","Text":"Then of course this is equal to the right side,"},{"Start":"17:48.695 ","End":"17:50.255","Text":"which is this Mu Naught,"},{"Start":"17:50.255 ","End":"17:56.360","Text":"Epsilon Naught and then we take the 2nd derivative of the same component,"},{"Start":"17:56.360 ","End":"18:00.155","Text":"the x-component of the E field in this example over here."},{"Start":"18:00.155 ","End":"18:04.135","Text":"But this time with respect to time."},{"Start":"18:04.135 ","End":"18:10.805","Text":"What we get is some equation in just the x-direction,"},{"Start":"18:10.805 ","End":"18:16.850","Text":"or here just using 1 of the components of the E field you can also do it for y and for z"},{"Start":"18:16.850 ","End":"18:24.140","Text":"that links the 2nd derivative of it with respect to its position in space,"},{"Start":"18:24.140 ","End":"18:35.390","Text":"to the 2nd derivative with respect to its change with respect to time."},{"Start":"18:36.270 ","End":"18:41.990","Text":"As we can see, we\u0027re left with this differential equation that we need to solve somehow."},{"Start":"18:41.990 ","End":"18:45.305","Text":"I\u0027m not going to go over right now how we solve it."},{"Start":"18:45.305 ","End":"18:47.015","Text":"But in the end,"},{"Start":"18:47.015 ","End":"18:51.380","Text":"if we\u0027re still looking at this x component of our E field,"},{"Start":"18:51.380 ","End":"18:58.825","Text":"so we\u0027re meant to get a solution that is E_x is equal to A cosine"},{"Start":"18:58.825 ","End":"19:07.415","Text":"of k.r minus Omega t. Now,"},{"Start":"19:07.415 ","End":"19:09.170","Text":"often added to this,"},{"Start":"19:09.170 ","End":"19:13.595","Text":"we\u0027ll have another component over here of Phi,"},{"Start":"19:13.595 ","End":"19:17.540","Text":"where Phi refers to the phase."},{"Start":"19:17.540 ","End":"19:21.105","Text":"But often this is left out."},{"Start":"19:21.105 ","End":"19:25.550","Text":"For simplicity, let\u0027s just leave it at this,"},{"Start":"19:25.550 ","End":"19:27.079","Text":"and in later lessons,"},{"Start":"19:27.079 ","End":"19:32.165","Text":"we\u0027ll explain what is going on over here what all of these terms are."},{"Start":"19:32.165 ","End":"19:35.930","Text":"This over here is what is really important to us."},{"Start":"19:35.930 ","End":"19:40.895","Text":"Now of course, there are versions with cosine or using sine you can use."},{"Start":"19:40.895 ","End":"19:43.845","Text":"Either right now we\u0027re just looking at the cosine version."},{"Start":"19:43.845 ","End":"19:47.855","Text":"This is what is important and in the next lessons,"},{"Start":"19:47.855 ","End":"19:50.465","Text":"we\u0027re going to really be looking at this,"},{"Start":"19:50.465 ","End":"19:51.860","Text":"how we use it,"},{"Start":"19:51.860 ","End":"19:55.055","Text":"and what is going on inside here."},{"Start":"19:55.055 ","End":"19:58.470","Text":"That is the end of this lesson."}],"ID":22431},{"Watched":false,"Name":"Deriving the Equation for the Dispersion Relationship","Duration":"11m 47s","ChapterTopicVideoID":21579,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21579.jpeg","UploadDate":"2020-05-07T18:48:19.9970000","DurationForVideoObject":"PT11M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.385","Text":"Hello. In the previous lesson,"},{"Start":"00:02.385 ","End":"00:06.750","Text":"we learned about the dispersion relationship which was the equation"},{"Start":"00:06.750 ","End":"00:11.520","Text":"linking Omega to k in the wave equation."},{"Start":"00:11.520 ","End":"00:14.730","Text":"We saw that Omega is equal to c multiplied by"},{"Start":"00:14.730 ","End":"00:18.990","Text":"the magnitude of k where c is of course the speed of light."},{"Start":"00:18.990 ","End":"00:24.730","Text":"In this lesson we\u0027re going to see how we arrive at this relationship."},{"Start":"00:25.460 ","End":"00:28.390","Text":"Let\u0027s show how we do this."},{"Start":"00:28.390 ","End":"00:33.605","Text":"In the previous lesson we saw Maxwell\u0027s 4 equations which"},{"Start":"00:33.605 ","End":"00:40.940","Text":"by using them we get to the wave equations which describe waves in a vacuum."},{"Start":"00:40.940 ","End":"00:47.095","Text":"Then we saw what this Nabla^2 was equal to."},{"Start":"00:47.095 ","End":"00:50.360","Text":"It\u0027s the partial derivative or"},{"Start":"00:50.360 ","End":"00:52.715","Text":"the second partial derivative of"},{"Start":"00:52.715 ","End":"00:56.525","Text":"a certain component of the electric field according to x,"},{"Start":"00:56.525 ","End":"01:00.260","Text":"y, and z and then their summation."},{"Start":"01:00.260 ","End":"01:02.225","Text":"Also we saw that Mu_0,"},{"Start":"01:02.225 ","End":"01:07.160","Text":"Epsilon_0 was equal to 1 divided by c^2 where,"},{"Start":"01:07.160 ","End":"01:11.495","Text":"of course c is the speed of light."},{"Start":"01:11.495 ","End":"01:16.320","Text":"Now, let\u0027s just write this out again and"},{"Start":"01:16.320 ","End":"01:18.950","Text":"of course we can do this exact same thing as we"},{"Start":"01:18.950 ","End":"01:22.130","Text":"saw here for the other components of the E field."},{"Start":"01:22.130 ","End":"01:25.340","Text":"The y component and also the z component,"},{"Start":"01:25.340 ","End":"01:26.735","Text":"it\u0027s the exact same thing."},{"Start":"01:26.735 ","End":"01:31.910","Text":"We\u0027re just showing the example over here with the x component."},{"Start":"01:31.910 ","End":"01:40.195","Text":"We can just rewrite this as being equal to 1 divided by c^2 for the Mu_0,"},{"Start":"01:40.195 ","End":"01:45.140","Text":"Epsilon naught of d_2E_x"},{"Start":"01:45.140 ","End":"01:50.620","Text":"by dt^2."},{"Start":"01:50.620 ","End":"01:54.215","Text":"Now what we\u0027re going to do is we\u0027re going to plug"},{"Start":"01:54.215 ","End":"01:58.940","Text":"in the equation for the x component of the E field."},{"Start":"01:58.940 ","End":"02:00.695","Text":"As we\u0027ve seen before,"},{"Start":"02:00.695 ","End":"02:05.540","Text":"we can write it as E_x which is as a function of r,"},{"Start":"02:05.540 ","End":"02:07.700","Text":"its position and t,"},{"Start":"02:07.700 ","End":"02:17.060","Text":"time is equal to A cosine of our k vector dot product with our r vector"},{"Start":"02:17.060 ","End":"02:23.015","Text":"minus Omega t. This is"},{"Start":"02:23.015 ","End":"02:29.688","Text":"our x component and of course we can write the same thing for the y and z components,"},{"Start":"02:29.688 ","End":"02:32.960","Text":"just they might have different amplitudes."},{"Start":"02:32.960 ","End":"02:39.500","Text":"We can, again, write this as A cosine of and then we remember"},{"Start":"02:39.500 ","End":"02:46.760","Text":"that the r vector is equal to some position x,"},{"Start":"02:46.760 ","End":"02:51.740","Text":"y, z, so when we do the dot product between k and r we get"},{"Start":"02:51.740 ","End":"02:57.515","Text":"k_xX plus k_yY plus"},{"Start":"02:57.515 ","End":"03:04.170","Text":"k_zZ minus Omega t."},{"Start":"03:04.220 ","End":"03:08.160","Text":"This is our x component of the E field."},{"Start":"03:08.160 ","End":"03:17.550","Text":"Now what we\u0027re going to do is we\u0027re going to plug this in to this equation over here."},{"Start":"03:18.290 ","End":"03:27.000","Text":"Let\u0027s first of all look at this d_2E_x by dx^2."},{"Start":"03:27.000 ","End":"03:28.830","Text":"We\u0027re just plugging this in."},{"Start":"03:28.830 ","End":"03:34.502","Text":"Let\u0027s just write this d_2E_x by dx^2,"},{"Start":"03:34.502 ","End":"03:40.560","Text":"so that will be d_2 by dx^2 of this,"},{"Start":"03:40.560 ","End":"03:45.555","Text":"so of A cosine of"},{"Start":"03:45.555 ","End":"03:56.260","Text":"k_xX plus k_yY plus k_zZ minus Omega t."},{"Start":"03:57.110 ","End":"04:03.355","Text":"Now we\u0027re taking the derivative twice with respect to x."},{"Start":"04:03.355 ","End":"04:06.485","Text":"If we take the first derivative,"},{"Start":"04:06.485 ","End":"04:11.186","Text":"so let\u0027s write the first derivative in blue at the bottom,"},{"Start":"04:11.186 ","End":"04:14.675","Text":"so the derivative of cosine is negative sine so we\u0027ll have"},{"Start":"04:14.675 ","End":"04:19.250","Text":"negative A sine of whatever is in the brackets,"},{"Start":"04:19.250 ","End":"04:29.175","Text":"k_xX plus k_yY plus k_zZ minus Omega t"},{"Start":"04:29.175 ","End":"04:36.240","Text":"and then we\u0027re going to multiply by the inner derivative with respect to x."},{"Start":"04:36.240 ","End":"04:42.000","Text":"This is over here, it will just be multiplied by k_x."},{"Start":"04:42.000 ","End":"04:45.465","Text":"Remember k_x is the x component of the k vector,"},{"Start":"04:45.465 ","End":"04:48.230","Text":"k_y is the y component of the k vector,"},{"Start":"04:48.230 ","End":"04:52.190","Text":"and z is the z component of the k vector just in case you forgot that."},{"Start":"04:52.190 ","End":"04:57.645","Text":"Now we\u0027ll write out the second derivative over here,"},{"Start":"04:57.645 ","End":"05:00.535","Text":"so we\u0027re taking this derivative again."},{"Start":"05:00.535 ","End":"05:03.980","Text":"The derivative of sine is just cosine,"},{"Start":"05:03.980 ","End":"05:13.525","Text":"so we still have negative Ak_x and then cosine"},{"Start":"05:13.525 ","End":"05:24.335","Text":"of k_xX plus k_yY plus k_zZ minus Omega t."},{"Start":"05:24.335 ","End":"05:30.560","Text":"Then we multiply this again by the derivative of our x value."},{"Start":"05:30.560 ","End":"05:34.865","Text":"Again, only here inside the brackets do we have this."},{"Start":"05:34.865 ","End":"05:38.150","Text":"Again, we\u0027re multiplying by just k_x,"},{"Start":"05:38.150 ","End":"05:42.330","Text":"so we can write k_x^2 over here."},{"Start":"05:43.040 ","End":"05:50.175","Text":"As we\u0027ll notice, here we have some coefficients, these are constants."},{"Start":"05:50.175 ","End":"05:58.605","Text":"Over here we have cosine of k_xX, k_yY,"},{"Start":"05:58.605 ","End":"06:03.925","Text":"k_zZ minus Omega t. In other words,"},{"Start":"06:03.925 ","End":"06:14.440","Text":"if we take into account also the amplitude and this together we have just E_x,"},{"Start":"06:14.440 ","End":"06:20.220","Text":"the x component of our electric field."},{"Start":"06:20.900 ","End":"06:25.910","Text":"In other words, we can just write this as being equal to"},{"Start":"06:25.910 ","End":"06:34.320","Text":"negative k_x^2 multiplied by E_x."},{"Start":"06:37.340 ","End":"06:47.700","Text":"Now let\u0027s scroll down and it\u0027s of course the exact same thing for d_2E_x by dy^2."},{"Start":"06:48.440 ","End":"06:51.680","Text":"We\u0027re going to have the exact same thing except"},{"Start":"06:51.680 ","End":"06:54.080","Text":"instead of taking the second derivative with respect to"},{"Start":"06:54.080 ","End":"06:59.420","Text":"x we\u0027re taking the 2 derivatives with respect to y."},{"Start":"06:59.420 ","End":"07:05.300","Text":"Instead of having k_x^2 we\u0027ll have k_y^2."},{"Start":"07:05.300 ","End":"07:08.960","Text":"We do the exact same steps but we have"},{"Start":"07:08.960 ","End":"07:12.575","Text":"k_y^2 because we\u0027re taking the derivative with respect to y."},{"Start":"07:12.575 ","End":"07:14.630","Text":"What we\u0027ll have is that this is equal to"},{"Start":"07:14.630 ","End":"07:20.450","Text":"simply negative k_y^2E_x again"},{"Start":"07:20.450 ","End":"07:26.230","Text":"and of course d_2E_x by dz^2,"},{"Start":"07:26.230 ","End":"07:30.515","Text":"it\u0027s the same thing but taking the derivative with respect to z."},{"Start":"07:30.515 ","End":"07:40.405","Text":"We\u0027ll have negative kz^2E_x."},{"Start":"07:40.405 ","End":"07:43.305","Text":"Here we have all of these equations."},{"Start":"07:43.305 ","End":"07:46.610","Text":"Of course this is equal to this over here,"},{"Start":"07:46.610 ","End":"07:53.840","Text":"so 1 divided by c^2 of d_2E_x by dt^2."},{"Start":"07:53.840 ","End":"08:03.750","Text":"Now let\u0027s write out d_2E_x by dt^2."},{"Start":"08:03.750 ","End":"08:11.655","Text":"Now again we\u0027re taking the second derivative of this but with respect to t, to time."},{"Start":"08:11.655 ","End":"08:16.340","Text":"Just like before we\u0027re going to get that this is equal to the"},{"Start":"08:16.340 ","End":"08:22.238","Text":"negative which comes from taking the first derivative of cosine."},{"Start":"08:22.238 ","End":"08:28.620","Text":"Then it\u0027s just going to be the coefficient of the variable squared,"},{"Start":"08:28.620 ","End":"08:35.090","Text":"so the coefficient of t is negative Omega^2 but a negative squared is a positive,"},{"Start":"08:35.090 ","End":"08:41.576","Text":"so we\u0027re just going to have Omega^2 multiplied by E_x."},{"Start":"08:41.576 ","End":"08:43.700","Text":"Just like we\u0027ve seen before,"},{"Start":"08:43.700 ","End":"08:47.730","Text":"we\u0027ve done the exact same as what we\u0027ve done here."},{"Start":"08:48.680 ","End":"08:51.805","Text":"Now we take all of this,"},{"Start":"08:51.805 ","End":"08:56.295","Text":"so let\u0027s just show this in pink."},{"Start":"08:56.295 ","End":"08:59.794","Text":"We take this value, this value,"},{"Start":"08:59.794 ","End":"09:05.180","Text":"this value, and this value and we just plug it into this equation."},{"Start":"09:05.180 ","End":"09:10.860","Text":"Therefore, what we get is d_2E_x by dx^2,"},{"Start":"09:10.860 ","End":"09:18.390","Text":"so we have negative k_x^2E_x plus d_2E_x by dy^2,"},{"Start":"09:18.390 ","End":"09:24.162","Text":"so plus negative so we can just write negative,"},{"Start":"09:24.162 ","End":"09:28.065","Text":"k_y^2E_x plus negative,"},{"Start":"09:28.065 ","End":"09:33.435","Text":"so just negative kz^2E_x and"},{"Start":"09:33.435 ","End":"09:38.760","Text":"all of this is equal to 1 divided by c^2,"},{"Start":"09:38.760 ","End":"09:44.870","Text":"d_2E_x by dt^2 which is just this negative Omega^2 E_x,"},{"Start":"09:44.870 ","End":"09:51.830","Text":"so we can just write the negative over here, Omega squared E_x."},{"Start":"09:51.830 ","End":"09:55.900","Text":"Now what we can see is we can divide both sides by E_x."},{"Start":"09:55.900 ","End":"09:58.340","Text":"This cancels out, this cancels out, this cancels out,"},{"Start":"09:58.340 ","End":"10:02.645","Text":"this cancels out and by minus 1,"},{"Start":"10:02.645 ","End":"10:08.310","Text":"so all of these become positives and so now let\u0027s just rewrite this."},{"Start":"10:08.310 ","End":"10:13.175","Text":"We have k_x^2 plus k_y^2"},{"Start":"10:13.175 ","End":"10:21.565","Text":"plus k_z^2 which is equal to Omega^2 divided by c^2."},{"Start":"10:21.565 ","End":"10:24.395","Text":"Now what we can say is that,"},{"Start":"10:24.395 ","End":"10:31.070","Text":"what we have here is we just have our k vector squared."},{"Start":"10:31.070 ","End":"10:33.120","Text":"If our k vector was k_x,"},{"Start":"10:33.120 ","End":"10:34.536","Text":"k_y, k_z,"},{"Start":"10:34.536 ","End":"10:38.835","Text":"so if we square it we have k_x^2, k_y^2, k_z^2."},{"Start":"10:38.835 ","End":"10:47.620","Text":"What we have is our k vector squared which is equal to Omega^2 divided by c^2."},{"Start":"10:47.620 ","End":"10:52.490","Text":"Here, this is the same as just taking the magnitude of the k vector."},{"Start":"10:52.490 ","End":"10:56.780","Text":"I\u0027ve just written it like this and now if we take the square root of both sides,"},{"Start":"10:56.780 ","End":"11:01.470","Text":"so we\u0027re still left with the magnitude of our k vector,"},{"Start":"11:01.470 ","End":"11:05.750","Text":"so I\u0027ll leave it written like this as being equal to Omega divided"},{"Start":"11:05.750 ","End":"11:10.280","Text":"by c. Now if we multiply both sides by c we\u0027re left with"},{"Start":"11:10.280 ","End":"11:20.440","Text":"the relationship of Omega is equal to c multiplied by the magnitude of our k vector."},{"Start":"11:22.010 ","End":"11:26.255","Text":"Now we\u0027ve reached this which is"},{"Start":"11:26.255 ","End":"11:30.995","Text":"exactly the dispersion relationship which we wanted to show."},{"Start":"11:30.995 ","End":"11:33.635","Text":"This is how we get from"},{"Start":"11:33.635 ","End":"11:40.130","Text":"Maxwell\u0027s equations to the wave equation and from the wave equations,"},{"Start":"11:40.130 ","End":"11:44.020","Text":"we get to the dispersion relationship."},{"Start":"11:44.020 ","End":"11:47.830","Text":"That is the end of this lesson."}],"ID":22432},{"Watched":false,"Name":"Exercise","Duration":"17m 43s","ChapterTopicVideoID":21580,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21580.jpeg","UploadDate":"2020-05-07T18:53:35.0970000","DurationForVideoObject":"PT17M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"Hello. In this lesson,"},{"Start":"00:01.845 ","End":"00:04.590","Text":"we\u0027re going to be answering the following questions."},{"Start":"00:04.590 ","End":"00:09.180","Text":"So we\u0027re given a magnetic field that is equal to B naught multiplied by"},{"Start":"00:09.180 ","End":"00:16.365","Text":"cosine Ax minus 2Ay minus Omega t in the z direction."},{"Start":"00:16.365 ","End":"00:21.435","Text":"The first thing that we\u0027re being asked to do is to find this fields wave vector."},{"Start":"00:21.435 ","End":"00:22.935","Text":"What is wave vector?"},{"Start":"00:22.935 ","End":"00:26.500","Text":"The wave vector is just our k-vector."},{"Start":"00:26.600 ","End":"00:31.604","Text":"Our k-vector is any coefficient of x,"},{"Start":"00:31.604 ","End":"00:33.600","Text":"which over here is A,"},{"Start":"00:33.600 ","End":"00:35.700","Text":"any coefficient of y,"},{"Start":"00:35.700 ","End":"00:38.830","Text":"which over here is negative 2A,"},{"Start":"00:38.830 ","End":"00:46.050","Text":"and any coefficient of z at which we can see we have no variable here of z, so it\u0027s 0."},{"Start":"00:46.490 ","End":"00:51.785","Text":"Super easy. This is the answer to question Number 1."},{"Start":"00:51.785 ","End":"00:56.570","Text":"Question Number 2 is, what is this wave\u0027s frequency?"},{"Start":"00:56.570 ","End":"00:59.930","Text":"When we\u0027re asking in this type of question about frequency,"},{"Start":"00:59.930 ","End":"01:02.075","Text":"we\u0027re looking for Omega."},{"Start":"01:02.075 ","End":"01:09.980","Text":"We\u0027re going to use the equation linking Omega to our k vector or to our wave vector,"},{"Start":"01:09.980 ","End":"01:11.650","Text":"which we\u0027ve already found."},{"Start":"01:11.650 ","End":"01:15.830","Text":"We know that this is the dispersion relationship."},{"Start":"01:15.830 ","End":"01:18.670","Text":"We know that this is the equation,"},{"Start":"01:18.670 ","End":"01:25.345","Text":"Omega is equal to the speed of light multiplied by the magnitude of our k vector."},{"Start":"01:25.345 ","End":"01:27.720","Text":"We have C multiplied by,"},{"Start":"01:27.720 ","End":"01:31.210","Text":"so the magnitude of our k vector is just Pythagoras."},{"Start":"01:31.210 ","End":"01:37.440","Text":"We have our x component squared plus our y component squared,"},{"Start":"01:37.440 ","End":"01:40.500","Text":"so that\u0027s plus negative 2A^2,"},{"Start":"01:40.500 ","End":"01:46.630","Text":"which is just the same as 2A^2 and the square root of that,"},{"Start":"01:46.630 ","End":"01:50.830","Text":"which is just equal to c multiplied by,"},{"Start":"01:50.830 ","End":"01:55.225","Text":"so here we have just 5A^2."},{"Start":"01:55.225 ","End":"02:04.130","Text":"C multiplied by A multiplied by the square root of 5."},{"Start":"02:05.520 ","End":"02:08.005","Text":"That\u0027s the answer to Question 2."},{"Start":"02:08.005 ","End":"02:11.374","Text":"Again very easy. This is 5."},{"Start":"02:11.374 ","End":"02:14.650","Text":"Now let\u0027s take a look at Question 3,"},{"Start":"02:14.650 ","End":"02:17.620","Text":"calculate the corresponding electric field."},{"Start":"02:17.620 ","End":"02:24.715","Text":"We remember that if we have the magnetic field and we want to find the electric field,"},{"Start":"02:24.715 ","End":"02:28.141","Text":"we use Maxwell\u0027s 4th equation."},{"Start":"02:28.141 ","End":"02:34.045","Text":"I\u0027ll just write Maxwell\u0027s 4th equation."},{"Start":"02:34.045 ","End":"02:38.050","Text":"The equation is like so it says that"},{"Start":"02:38.050 ","End":"02:44.615","Text":"the rotor of our magnetic field is equal to Mu naught,"},{"Start":"02:44.615 ","End":"02:50.065","Text":"Epsilon naught multiplied by dE by dt,"},{"Start":"02:50.065 ","End":"02:53.720","Text":"the time derivative of our electric field."},{"Start":"02:53.720 ","End":"02:57.880","Text":"First of all, let\u0027s find the rotor of the magnetic field."},{"Start":"02:57.880 ","End":"03:00.110","Text":"The rotor, as we know,"},{"Start":"03:00.110 ","End":"03:02.715","Text":"is just partial derivative."},{"Start":"03:02.715 ","End":"03:09.068","Text":"D by dx, d by dy, d by dz."},{"Start":"03:09.068 ","End":"03:13.370","Text":"We know that we\u0027re going to get a vector back."},{"Start":"03:13.370 ","End":"03:19.145","Text":"Now our B fields we can see is only in the z direction,"},{"Start":"03:19.145 ","End":"03:22.455","Text":"so it only has a z component."},{"Start":"03:22.455 ","End":"03:24.735","Text":"X and y components are 0,"},{"Start":"03:24.735 ","End":"03:30.360","Text":"and then we just have B naught cosine of Ax minus"},{"Start":"03:30.360 ","End":"03:38.265","Text":"2Ay minus Omega t. Now,"},{"Start":"03:38.265 ","End":"03:43.930","Text":"we cross out the first and then we have d by dy of this."},{"Start":"03:43.930 ","End":"03:51.220","Text":"Again we get a vector. First we have d by dy of"},{"Start":"03:51.220 ","End":"04:00.580","Text":"B naught multiplied by cosine of Ax minus 2Ay minus Omega t,"},{"Start":"04:00.770 ","End":"04:04.340","Text":"minus d by dz of 0."},{"Start":"04:04.340 ","End":"04:09.035","Text":"Minus 0, or we can just leave it as this."},{"Start":"04:09.035 ","End":"04:14.375","Text":"Then we cross out the middle and then we flip it."},{"Start":"04:14.375 ","End":"04:21.420","Text":"We do d by dz of 0 is 0 minus d by dx,"},{"Start":"04:21.420 ","End":"04:28.605","Text":"so we have negative d by dx of B naught cosine"},{"Start":"04:28.605 ","End":"04:36.770","Text":"of Ax minus 2Ay minus Omega t. Then finally,"},{"Start":"04:36.770 ","End":"04:38.255","Text":"we cross out here,"},{"Start":"04:38.255 ","End":"04:43.520","Text":"d by dx of 0 is 0 minus d by dy of 0 is 0,"},{"Start":"04:43.520 ","End":"04:46.205","Text":"so we\u0027re just left with this."},{"Start":"04:46.205 ","End":"04:50.600","Text":"Now let us do these partial derivatives."},{"Start":"04:50.600 ","End":"04:57.365","Text":"Here in the x components we\u0027ll have"},{"Start":"04:57.365 ","End":"05:03.165","Text":"negative B naught sine"},{"Start":"05:03.165 ","End":"05:10.600","Text":"of Ax minus 2Ay minus Omega t,"},{"Start":"05:10.600 ","End":"05:15.215","Text":"multiplied by the inner derivative with respect to y,"},{"Start":"05:15.215 ","End":"05:17.480","Text":"which is just negative 2A,"},{"Start":"05:17.480 ","End":"05:19.745","Text":"so this becomes positive,"},{"Start":"05:19.745 ","End":"05:25.140","Text":"so we can just delete it, 2A."},{"Start":"05:25.390 ","End":"05:30.600","Text":"Then here we have the partial derivative with respect to x,"},{"Start":"05:30.600 ","End":"05:33.480","Text":"so we have negative already over here,"},{"Start":"05:33.480 ","End":"05:37.620","Text":"and then we have a negative from when cosine becomes sine,"},{"Start":"05:37.620 ","End":"05:41.435","Text":"so this just becomes a positive and negative and negative of,"},{"Start":"05:41.435 ","End":"05:43.940","Text":"so we\u0027ll just erase it."},{"Start":"05:43.940 ","End":"05:50.745","Text":"We have again B naught multiplied by sine"},{"Start":"05:50.745 ","End":"05:58.515","Text":"of Ax minus 2Ay minus Omega t multiplied by the inner derivative,"},{"Start":"05:58.515 ","End":"06:01.140","Text":"but again with respect to x over here."},{"Start":"06:01.140 ","End":"06:03.515","Text":"We\u0027re taking the derivative with respect to x."},{"Start":"06:03.515 ","End":"06:07.335","Text":"The coefficient of x over here is just A,"},{"Start":"06:07.335 ","End":"06:10.470","Text":"and here, of course, we have 0."},{"Start":"06:10.470 ","End":"06:15.470","Text":"What we can do is we can just rewrite this as"},{"Start":"06:15.470 ","End":"06:23.090","Text":"being equal to 2AB_0,"},{"Start":"06:23.090 ","End":"06:27.645","Text":"actually we\u0027ll write it with like terms."},{"Start":"06:27.645 ","End":"06:37.440","Text":"Everything has AB_0 sine of Ax minus 2Ay minus"},{"Start":"06:37.440 ","End":"06:42.870","Text":"Omega t. Then this is multiplied by 2 in"},{"Start":"06:42.870 ","End":"06:51.460","Text":"the x direction and 1 in the y direction."},{"Start":"06:54.230 ","End":"06:59.130","Text":"Now what we have to do,"},{"Start":"06:59.130 ","End":"07:00.920","Text":"this is our rotor,"},{"Start":"07:00.920 ","End":"07:08.220","Text":"and now we have to say that this is equal to Mu_0 Epsilon dE by dt."},{"Start":"07:09.530 ","End":"07:17.205","Text":"All of this is equal to Mu_0 Epsilon naught,"},{"Start":"07:17.205 ","End":"07:26.720","Text":"which is also the same as 1 divided by c^2 of dE by dt."},{"Start":"07:26.720 ","End":"07:31.085","Text":"Let\u0027s multiply both sides by c^2."},{"Start":"07:31.085 ","End":"07:41.670","Text":"What we have over here is c^2 AB_0 sine of"},{"Start":"07:41.670 ","End":"07:46.740","Text":"Ax minus 2Ay minus Omega t"},{"Start":"07:46.740 ","End":"07:53.795","Text":"multiplied by 2 in the x-direction plus 1 in the y-direction,"},{"Start":"07:53.795 ","End":"07:59.080","Text":"which is equal to dE by dt."},{"Start":"07:59.080 ","End":"08:02.670","Text":"Now, we\u0027re left with this differential equation."},{"Start":"08:02.670 ","End":"08:08.550","Text":"What we have to do, is we have to integrate both sides with respect to t."},{"Start":"08:08.550 ","End":"08:12.724","Text":"Then what will be left with is the E field"},{"Start":"08:12.724 ","End":"08:17.765","Text":"over here on this side and now we just have to integrate here."},{"Start":"08:17.765 ","End":"08:23.390","Text":"What we\u0027ll have is the integral of sine is negative cosine,"},{"Start":"08:23.390 ","End":"08:30.830","Text":"so we have negative and then we have all of our c^2 AB_0, are constants,"},{"Start":"08:30.830 ","End":"08:37.835","Text":"and then we have cosine of Ax minus 2Ay minus"},{"Start":"08:37.835 ","End":"08:43.070","Text":"Omega t. Then these are"},{"Start":"08:43.070 ","End":"08:49.385","Text":"just the axes that we\u0027re going along,"},{"Start":"08:49.385 ","End":"08:51.440","Text":"and they\u0027re constants, 2 and 1,"},{"Start":"08:51.440 ","End":"08:58.260","Text":"so this is still 2 in the x plus 1 in the y direction."},{"Start":"08:58.260 ","End":"09:03.140","Text":"Then of course, we have to divide by the inner derivative"},{"Start":"09:03.140 ","End":"09:07.995","Text":"with respect to our variable which is t. The inner derivative,"},{"Start":"09:07.995 ","End":"09:10.265","Text":"we only have the variable t over here,"},{"Start":"09:10.265 ","End":"09:14.550","Text":"so the derivative of this is negative Omega,"},{"Start":"09:14.550 ","End":"09:18.575","Text":"the derivative of negative Omega t is negative Omega."},{"Start":"09:18.575 ","End":"09:24.050","Text":"All of this is divided by negative Omega,"},{"Start":"09:24.050 ","End":"09:27.575","Text":"so the minuses cancel out to make positives."},{"Start":"09:27.575 ","End":"09:31.655","Text":"What we\u0027re left with is an E field which is equal to"},{"Start":"09:31.655 ","End":"09:37.260","Text":"c^2 AB_0 divided by Omega multiplied by"},{"Start":"09:37.260 ","End":"09:46.940","Text":"cosine of Ax minus 2Ay minus Omega t. Then we"},{"Start":"09:46.940 ","End":"09:57.130","Text":"have 2 in the x direction and 1 in the y direction."},{"Start":"09:57.130 ","End":"09:59.605","Text":"This is our electric field,"},{"Start":"09:59.605 ","End":"10:03.910","Text":"and I\u0027m just going to copy it and put it next to our magnetic field so that we can"},{"Start":"10:03.910 ","End":"10:09.200","Text":"just go over some things that are similar."},{"Start":"10:10.200 ","End":"10:18.235","Text":"What we can see is that our electric field has the same argument,"},{"Start":"10:18.235 ","End":"10:21.070","Text":"also, just like the magnetic field,"},{"Start":"10:21.070 ","End":"10:24.250","Text":"cosine of Ax minus 2A minus Omega t,"},{"Start":"10:24.250 ","End":"10:27.460","Text":"the exact same thing over here."},{"Start":"10:27.460 ","End":"10:30.625","Text":"The major difference is just the amplitude."},{"Start":"10:30.625 ","End":"10:35.455","Text":"Here the amplitude of the magnetic field wave is B_0,"},{"Start":"10:35.455 ","End":"10:40.480","Text":"and here it\u0027s C^2 AB_0 divided by Omega,"},{"Start":"10:40.480 ","End":"10:42.955","Text":"and of course the direction."},{"Start":"10:42.955 ","End":"10:45.535","Text":"Here we can see that the magnetic field,"},{"Start":"10:45.535 ","End":"10:49.465","Text":"the amplitude is only in the z direction and"},{"Start":"10:49.465 ","End":"10:53.920","Text":"here we can see that it\u0027s in the x and y-direction."},{"Start":"10:53.920 ","End":"10:58.330","Text":"We can see that both waves are propagating in the same direction according to"},{"Start":"10:58.330 ","End":"11:03.940","Text":"the K vector A and negative 2A."},{"Start":"11:03.940 ","End":"11:08.590","Text":"We can also see that the 2 fields are perpendicular so"},{"Start":"11:08.590 ","End":"11:13.000","Text":"the magnetic field only has a component of amplitude,"},{"Start":"11:13.000 ","End":"11:14.545","Text":"in the z direction,"},{"Start":"11:14.545 ","End":"11:21.790","Text":"whereas the electric field has amplitude either in the x or the y or in both directions,"},{"Start":"11:21.790 ","End":"11:25.240","Text":"and both x and y are perpendicular to z."},{"Start":"11:25.240 ","End":"11:32.815","Text":"We can also check that this is correct by saying that we have no z component over here."},{"Start":"11:32.815 ","End":"11:35.980","Text":"This was our answer to question Number 3,"},{"Start":"11:35.980 ","End":"11:38.470","Text":"so now let\u0027s answer question Number 4."},{"Start":"11:38.470 ","End":"11:45.250","Text":"Calculate the force acting on a charge Q that is located at time t is equal to 0 at"},{"Start":"11:45.250 ","End":"11:52.285","Text":"the origin and the charge has a velocity of v_0 in the x-direction."},{"Start":"11:52.285 ","End":"11:57.280","Text":"In order to calculate the force acting on this particle or on this charge,"},{"Start":"11:57.280 ","End":"11:59.680","Text":"we\u0027re using Lorentz\u0027s law,"},{"Start":"11:59.680 ","End":"12:01.675","Text":"which is equal to Q,"},{"Start":"12:01.675 ","End":"12:04.990","Text":"the charge multiplied by the electric field,"},{"Start":"12:04.990 ","End":"12:14.240","Text":"plus Q multiplied by the velocity cross multiplied with the magnetic field."},{"Start":"12:15.800 ","End":"12:18.150","Text":"We know our charge,"},{"Start":"12:18.150 ","End":"12:19.800","Text":"we\u0027re given the charge, the velocity,"},{"Start":"12:19.800 ","End":"12:21.465","Text":"and t is equal to 0,"},{"Start":"12:21.465 ","End":"12:24.835","Text":"and we are told that it is at the origin."},{"Start":"12:24.835 ","End":"12:30.010","Text":"First of all, we want to find out what the electric field is at that point."},{"Start":"12:30.010 ","End":"12:32.080","Text":"Here we have the electric field,"},{"Start":"12:32.080 ","End":"12:33.640","Text":"now if we\u0027re at the origin,"},{"Start":"12:33.640 ","End":"12:35.460","Text":"that means x, y,"},{"Start":"12:35.460 ","End":"12:37.440","Text":"and z is equal to 0."},{"Start":"12:37.440 ","End":"12:40.365","Text":"Here we can see we have x and y,"},{"Start":"12:40.365 ","End":"12:43.530","Text":"so if they are equal to 0 and of course,"},{"Start":"12:43.530 ","End":"12:45.915","Text":"our t is equal to 0."},{"Start":"12:45.915 ","End":"12:49.260","Text":"That means that we have cosine of 0,"},{"Start":"12:49.260 ","End":"12:54.490","Text":"which is just 1, which means that the E field is just the amplitude."},{"Start":"12:54.490 ","End":"12:58.880","Text":"The E field is going to be equal to"},{"Start":"12:59.940 ","End":"13:08.950","Text":"2C^2 AB_0 divided by Omega in the x-direction plus 1 times this,"},{"Start":"13:08.950 ","End":"13:17.095","Text":"so just C^2 AB_0 divided by Omega in the y-direction."},{"Start":"13:17.095 ","End":"13:24.715","Text":"Then we know that our B field, our magnetic field,"},{"Start":"13:24.715 ","End":"13:28.030","Text":"we\u0027re again using the equation where x, y,"},{"Start":"13:28.030 ","End":"13:34.960","Text":"and z is equal to 0 and where t is also equal to 0 so our B field is this over here."},{"Start":"13:34.960 ","End":"13:38.110","Text":"Again, all of what\u0027s in the brackets here is equal to 0."},{"Start":"13:38.110 ","End":"13:40.210","Text":"We have cosine of 0,"},{"Start":"13:40.210 ","End":"13:43.600","Text":"which is 1 multiplied by B_0,"},{"Start":"13:43.600 ","End":"13:49.060","Text":"so we\u0027re just left with B_0 in the z-direction."},{"Start":"13:49.060 ","End":"13:58.855","Text":"Now what we want to do is we want to calculate what V cross B is equal to."},{"Start":"13:58.855 ","End":"14:07.540","Text":"V is only in the x-direction so we have V_0 00 cross multiplied with our B field,"},{"Start":"14:07.540 ","End":"14:10.825","Text":"which is 00, B_0 K,"},{"Start":"14:10.825 ","End":"14:14.350","Text":"because it\u0027s only in the z-direction."},{"Start":"14:14.350 ","End":"14:19.705","Text":"This, we can already see is just going to be equal to"},{"Start":"14:19.705 ","End":"14:28.820","Text":"V_0 B_0 in the negative y direction."},{"Start":"14:30.030 ","End":"14:34.660","Text":"Now, all we have to do is to just plug this into the equation."},{"Start":"14:34.660 ","End":"14:41.455","Text":"We have that F is equal to the charge Q multiplied by the E field."},{"Start":"14:41.455 ","End":"14:51.280","Text":"That is just going to be equal to 2C^2 AB_0 divided by"},{"Start":"14:51.280 ","End":"14:54.760","Text":"Omega in the x direction"},{"Start":"14:54.760 ","End":"15:02.175","Text":"plus C^2 AB_0 divided by Omega in the y-direction,"},{"Start":"15:02.175 ","End":"15:06.350","Text":"and this is plus our charge."},{"Start":"15:06.350 ","End":"15:09.850","Text":"Our charge Q multiplied by V cross B,"},{"Start":"15:09.850 ","End":"15:15.745","Text":"which is just V_0 B_0 in the negative y-direction."},{"Start":"15:15.745 ","End":"15:22.640","Text":"We can just have a y negative over here and y hat."},{"Start":"15:23.970 ","End":"15:27.895","Text":"Now that we\u0027ve seen the answer to Question 4,"},{"Start":"15:27.895 ","End":"15:29.980","Text":"let\u0027s do Question 5."},{"Start":"15:29.980 ","End":"15:32.830","Text":"Calculate the Poynting vector."},{"Start":"15:32.830 ","End":"15:37.960","Text":"The Poynting vector S is equal to 1 divided by"},{"Start":"15:37.960 ","End":"15:46.280","Text":"Mu_0 of the electric field cross multiplied with the magnetic field."},{"Start":"15:46.320 ","End":"15:54.160","Text":"This is just going to be equal to 1 divided by Mu naught of the electric field,"},{"Start":"15:54.160 ","End":"15:57.175","Text":"which we saw over here."},{"Start":"15:57.175 ","End":"16:05.770","Text":"We can take out all the constants C^2 AB_0 divided by Omega and then we"},{"Start":"16:05.770 ","End":"16:12.970","Text":"have cosine of Ax"},{"Start":"16:12.970 ","End":"16:17.530","Text":"minus 2Ay minus Omega t,"},{"Start":"16:17.530 ","End":"16:25.190","Text":"and this we have in the x direction and in the y-direction."},{"Start":"16:25.530 ","End":"16:30.519","Text":"This is cross multiplied with our B field,"},{"Start":"16:30.519 ","End":"16:36.355","Text":"which is equal to B_0."},{"Start":"16:36.355 ","End":"16:38.725","Text":"Let\u0027s just put these together."},{"Start":"16:38.725 ","End":"16:41.065","Text":"We have B_0 over here,"},{"Start":"16:41.065 ","End":"16:46.075","Text":"and again, cosine of the exact same thing."},{"Start":"16:46.075 ","End":"16:50.470","Text":"We can have cosine^2, however,"},{"Start":"16:50.470 ","End":"16:56.050","Text":"this time we\u0027re cross-multiplying this in the z-direction."},{"Start":"16:56.050 ","End":"17:05.845","Text":"What we\u0027re going to get is C^2 AB_0^2 divided by"},{"Start":"17:05.845 ","End":"17:13.840","Text":"Mu_0 Omega multiplied by cosine squared of Ax minus 2Ay"},{"Start":"17:13.840 ","End":"17:24.055","Text":"minus Omega t. Then here when we cross multiply 2x hat with z hat,"},{"Start":"17:24.055 ","End":"17:28.855","Text":"we\u0027re going to have negative 2y hat,"},{"Start":"17:28.855 ","End":"17:32.080","Text":"and when we cross multiply y with z,"},{"Start":"17:32.080 ","End":"17:37.255","Text":"we\u0027ll have plus x hat."},{"Start":"17:37.255 ","End":"17:41.140","Text":"This is the answer to Question number 5,"},{"Start":"17:41.140 ","End":"17:44.240","Text":"and that is the end of this lesson."}],"ID":22433},{"Watched":false,"Name":"Exercise - Calculate the Corresponding Magnetic Field, New Equation","Duration":"10m 50s","ChapterTopicVideoID":21581,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21581.jpeg","UploadDate":"2020-05-07T18:58:25.6670000","DurationForVideoObject":"PT10M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.785","Text":"Hello, in this lesson,"},{"Start":"00:01.785 ","End":"00:04.515","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.515 ","End":"00:09.480","Text":"We\u0027re being asked to calculate the magnetic field given the following electric fields."},{"Start":"00:09.480 ","End":"00:16.125","Text":"We have an electric field which is equal to E_0 with the vector 1, 1,"},{"Start":"00:16.125 ","End":"00:20.140","Text":"2 and then multiplied by e^i"},{"Start":"00:20.990 ","End":"00:28.725","Text":"multiplied by 2x minus z minus Omega t. This over here,"},{"Start":"00:28.725 ","End":"00:30.975","Text":"e^i with these brackets,"},{"Start":"00:30.975 ","End":"00:33.780","Text":"is the exact same function as"},{"Start":"00:33.780 ","End":"00:37.890","Text":"cosine of what\u0027s in the brackets or sine of what\u0027s in the brackets."},{"Start":"00:37.890 ","End":"00:42.055","Text":"It\u0027s the exact same thing just written in a different format."},{"Start":"00:42.055 ","End":"00:48.080","Text":"Up until now, we\u0027ve used Maxwell\u0027s equations in order to convert."},{"Start":"00:48.080 ","End":"00:53.255","Text":"Converting between the electric field to the magnetic field,"},{"Start":"00:53.255 ","End":"01:02.730","Text":"we would do Nabla cross multiplied by the E field is equal to negative dB by dt."},{"Start":"01:03.070 ","End":"01:06.290","Text":"This is how he solved it up until now."},{"Start":"01:06.290 ","End":"01:09.470","Text":"But in this lesson I\u0027m going to show you a new equation,"},{"Start":"01:09.470 ","End":"01:12.500","Text":"which might be a little bit easier."},{"Start":"01:12.500 ","End":"01:17.930","Text":"We calculate the B field by taking our k-vector."},{"Start":"01:17.930 ","End":"01:20.150","Text":"But the unit vector,"},{"Start":"01:20.150 ","End":"01:26.105","Text":"cross multiplied by the electric field and divide it by c,"},{"Start":"01:26.105 ","End":"01:28.410","Text":"the speed of light."},{"Start":"01:28.730 ","End":"01:34.995","Text":"This is the equation please put it in your equation sheets."},{"Start":"01:34.995 ","End":"01:37.650","Text":"Now let\u0027s begin calculating."},{"Start":"01:37.650 ","End":"01:41.775","Text":"First of all we want to know what our k-vector is,"},{"Start":"01:41.775 ","End":"01:44.205","Text":"so that we can do this."},{"Start":"01:44.205 ","End":"01:45.935","Text":"As we can see over here,"},{"Start":"01:45.935 ","End":"01:49.760","Text":"we have in the x direction,"},{"Start":"01:49.760 ","End":"01:55.710","Text":"2 and the y is 0 and then the z negative 1."},{"Start":"01:55.710 ","End":"01:59.210","Text":"Then k hat, the unit vector,"},{"Start":"01:59.210 ","End":"02:06.300","Text":"is just equal to our k-vector divided by its magnitude."},{"Start":"02:06.300 ","End":"02:10.030","Text":"This is going to be our k-vector."},{"Start":"02:10.420 ","End":"02:12.605","Text":"Write it over here."},{"Start":"02:12.605 ","End":"02:15.745","Text":"It\u0027s just 2, 0, negative 1."},{"Start":"02:15.745 ","End":"02:20.245","Text":"Then its magnitude, so 1 divided by its magnitude,"},{"Start":"02:20.245 ","End":"02:27.235","Text":"is just the square root of 2^2 plus 0^2 plus negative 1^2."},{"Start":"02:27.235 ","End":"02:28.920","Text":"2^2 is 4,"},{"Start":"02:28.920 ","End":"02:30.510","Text":"plus 0 plus 1,"},{"Start":"02:30.510 ","End":"02:33.810","Text":"so that\u0027s 5 and the square root of that."},{"Start":"02:33.810 ","End":"02:37.200","Text":"This is just Pythagoras."},{"Start":"02:37.200 ","End":"02:39.190","Text":"Now what we\u0027re going to do,"},{"Start":"02:39.190 ","End":"02:48.750","Text":"is we\u0027re going to cross multiply our k hat with our E. We have k hat cross"},{"Start":"02:48.750 ","End":"02:55.315","Text":"multiplied by E is"},{"Start":"02:55.315 ","End":"03:00.065","Text":"equal to 1 divided by the square root of 5."},{"Start":"03:00.065 ","End":"03:02.540","Text":"Then we have 2,"},{"Start":"03:02.540 ","End":"03:05.000","Text":"0, negative 1."},{"Start":"03:05.000 ","End":"03:08.700","Text":"All of this is cross multiplied by our E field,"},{"Start":"03:08.700 ","End":"03:11.895","Text":"which is E_0 1, 1,"},{"Start":"03:11.895 ","End":"03:20.705","Text":"2 multiplied by e^i(2x-z-Omega"},{"Start":"03:20.705 ","End":"03:26.045","Text":"t.) Now what we\u0027re going to do is we\u0027re going to put all the scalars to 1 side."},{"Start":"03:26.045 ","End":"03:31.355","Text":"This will be simply equal to E_0 as a scalar,"},{"Start":"03:31.355 ","End":"03:36.290","Text":"divided by the square root of 5."},{"Start":"03:36.290 ","End":"03:42.500","Text":"Then also this e^i of 2x minus z"},{"Start":"03:42.500 ","End":"03:50.040","Text":"minus Omega t. Then we have our vector."},{"Start":"03:50.040 ","End":"03:51.780","Text":"We have 2, 0,"},{"Start":"03:51.780 ","End":"03:58.600","Text":"negative 1 cross multiplied with 1, 1, 2."},{"Start":"04:00.230 ","End":"04:04.140","Text":"Let\u0027s just label all of this as in"},{"Start":"04:04.140 ","End":"04:08.900","Text":"the meantime A so that we don\u0027t have to rewrite everything."},{"Start":"04:08.900 ","End":"04:12.050","Text":"We can cross out the first line."},{"Start":"04:12.050 ","End":"04:14.485","Text":"We have 0 times 2,"},{"Start":"04:14.485 ","End":"04:22.605","Text":"which is 0 minus negative 1 times 1,"},{"Start":"04:22.605 ","End":"04:24.180","Text":"which is just negative 1."},{"Start":"04:24.180 ","End":"04:26.025","Text":"This becomes plus 1."},{"Start":"04:26.025 ","End":"04:29.055","Text":"Then we cross out the middle line."},{"Start":"04:29.055 ","End":"04:30.555","Text":"We switch the order,"},{"Start":"04:30.555 ","End":"04:35.280","Text":"so we have negative 1 times 1 is negative 1,"},{"Start":"04:35.280 ","End":"04:40.350","Text":"and then minus 2 times 2 which is 4."},{"Start":"04:40.350 ","End":"04:48.845","Text":"Then we cross out the final line."},{"Start":"04:48.845 ","End":"04:52.700","Text":"We have 2 times 1 is 2,"},{"Start":"04:52.700 ","End":"04:58.920","Text":"minus 0 times 1 is plus 0."},{"Start":"04:59.810 ","End":"05:06.300","Text":"Then what we get is A multiplied by the vector 1,"},{"Start":"05:06.300 ","End":"05:09.880","Text":"negative 5, 2."},{"Start":"05:12.230 ","End":"05:17.230","Text":"This is our k cross E. Therefore,"},{"Start":"05:17.230 ","End":"05:25.450","Text":"let\u0027s write out our magnetic field is equal to our k hat cross E,"},{"Start":"05:25.450 ","End":"05:27.875","Text":"which is what we calculated over here,"},{"Start":"05:27.875 ","End":"05:37.950","Text":"divided by c. We just take all of this and divide it by c. We have E_0 divided by root 5c"},{"Start":"05:39.160 ","End":"05:47.205","Text":"multiplied by e^i of 2x minus z minus Omega"},{"Start":"05:47.205 ","End":"05:57.310","Text":"t. Then multiplied by the vector 1 negative 5, 2."},{"Start":"05:58.520 ","End":"06:01.310","Text":"This is the answer to the question."},{"Start":"06:01.310 ","End":"06:09.720","Text":"Notice over here, root 5 is the magnitude of the k-vector."},{"Start":"06:10.070 ","End":"06:12.585","Text":"It\u0027s multiplied by c,"},{"Start":"06:12.585 ","End":"06:13.850","Text":"and as we\u0027ve seen before,"},{"Start":"06:13.850 ","End":"06:17.340","Text":"that is just equal to Omega."},{"Start":"06:17.870 ","End":"06:19.940","Text":"Instead of this over here,"},{"Start":"06:19.940 ","End":"06:21.440","Text":"we could have just written Omega."},{"Start":"06:21.440 ","End":"06:24.905","Text":"We can see now how we get here,"},{"Start":"06:24.905 ","End":"06:27.830","Text":"and that is the answer."},{"Start":"06:27.830 ","End":"06:30.635","Text":"Now, another thing that we can check is to make sure"},{"Start":"06:30.635 ","End":"06:35.600","Text":"that our magnetic field is perpendicular to the electric field."},{"Start":"06:35.600 ","End":"06:38.240","Text":"As we know, they always have to be perpendicular."},{"Start":"06:38.240 ","End":"06:45.080","Text":"In order to check that the electric field is perpendicular to the magnetic fields,"},{"Start":"06:45.080 ","End":"06:50.810","Text":"we do the dot product between the 2."},{"Start":"06:50.810 ","End":"06:56.505","Text":"What we\u0027ll have is this is in the 1, 1,"},{"Start":"06:56.505 ","End":"07:01.130","Text":"2 direction, dot product with the B field,"},{"Start":"07:01.130 ","End":"07:05.030","Text":"which is in the 1 negative 5, 2 direction."},{"Start":"07:05.030 ","End":"07:12.345","Text":"What we have is 1 times 1 is 1 plus 1 times minus 5."},{"Start":"07:12.345 ","End":"07:19.620","Text":"Negative 5 plus 2 times 2 is 4."},{"Start":"07:19.620 ","End":"07:23.655","Text":"1 plus 4 is 5 minus 5 is 0."},{"Start":"07:23.655 ","End":"07:31.390","Text":"Therefore, E and B are perpendicular."},{"Start":"07:31.430 ","End":"07:35.000","Text":"In addition to this equation here, of course,"},{"Start":"07:35.000 ","End":"07:38.345","Text":"we can see that if we have the electric field,"},{"Start":"07:38.345 ","End":"07:41.150","Text":"we can calculate the corresponding magnetic field,"},{"Start":"07:41.150 ","End":"07:42.650","Text":"as we did in this lesson."},{"Start":"07:42.650 ","End":"07:45.610","Text":"But what if we have the magnetic field,"},{"Start":"07:45.610 ","End":"07:51.035","Text":"and we want to calculate the electric fields using this equation."},{"Start":"07:51.035 ","End":"07:56.415","Text":"The electric field is equal to c,"},{"Start":"07:56.415 ","End":"07:57.900","Text":"which is the speed of light,"},{"Start":"07:57.900 ","End":"08:03.495","Text":"multiplied by B cross our k hat."},{"Start":"08:03.495 ","End":"08:07.510","Text":"The unit vector for our k-vector."},{"Start":"08:07.790 ","End":"08:11.290","Text":"Write this into your equation sheets as well."},{"Start":"08:11.290 ","End":"08:16.420","Text":"Remember this is if you\u0027re finding B and this is if you\u0027re finding E,"},{"Start":"08:16.420 ","End":"08:20.455","Text":"which is the opposite of what we saw in Maxwell\u0027s equations."},{"Start":"08:20.455 ","End":"08:24.685","Text":"Where here on the left side we put in the fields that we had,"},{"Start":"08:24.685 ","End":"08:27.700","Text":"not the field that we were trying to find."},{"Start":"08:27.700 ","End":"08:30.560","Text":"Just remember that."},{"Start":"08:32.270 ","End":"08:36.015","Text":"We\u0027ve seen that these are perpendicular,"},{"Start":"08:36.015 ","End":"08:38.725","Text":"and we can see that the argument over here,"},{"Start":"08:38.725 ","End":"08:41.395","Text":"E^i of what\u0027s in the brackets here,"},{"Start":"08:41.395 ","End":"08:47.100","Text":"is as usual the exact same as in the electric field."},{"Start":"08:47.100 ","End":"08:50.630","Text":"The only difference is that they\u0027re perpendicular to one another,"},{"Start":"08:50.630 ","End":"08:54.720","Text":"and that the amplitude is different."},{"Start":"08:54.980 ","End":"09:00.150","Text":"A short little notes on these new equations of why they work."},{"Start":"09:00.150 ","End":"09:05.885","Text":"We can imagine them as being in an axis."},{"Start":"09:05.885 ","End":"09:10.310","Text":"As we know, our k-vector,"},{"Start":"09:10.310 ","End":"09:16.205","Text":"k the direction of propagation is always perpendicular to both the electric field,"},{"Start":"09:16.205 ","End":"09:18.320","Text":"and the magnetic field."},{"Start":"09:18.320 ","End":"09:20.735","Text":"We know that the electric field and"},{"Start":"09:20.735 ","End":"09:23.540","Text":"the magnetic fields are always perpendicular to 1 another."},{"Start":"09:23.540 ","End":"09:26.540","Text":"We can think of this as being here,"},{"Start":"09:26.540 ","End":"09:27.965","Text":"the magnetic field,"},{"Start":"09:27.965 ","End":"09:31.310","Text":"here our k-vector,"},{"Start":"09:31.310 ","End":"09:32.855","Text":"the direction of propagation,"},{"Start":"09:32.855 ","End":"09:35.980","Text":"and here the electric field."},{"Start":"09:35.980 ","End":"09:39.895","Text":"As we can see that always perpendicular to one another."},{"Start":"09:39.895 ","End":"09:43.445","Text":"We can liken this to this being let\u0027s say the x,"},{"Start":"09:43.445 ","End":"09:46.505","Text":"this the y, and this the z axes."},{"Start":"09:46.505 ","End":"09:50.105","Text":"Just like with our Cartesian coordinates like so,"},{"Start":"09:50.105 ","End":"09:54.940","Text":"if we have x cross y, we get z."},{"Start":"09:54.940 ","End":"09:59.685","Text":"If we have y cross z, we get x."},{"Start":"09:59.685 ","End":"10:03.845","Text":"Similarly we can get all these different values."},{"Start":"10:03.845 ","End":"10:05.660","Text":"In the same way,"},{"Start":"10:05.660 ","End":"10:08.435","Text":"if we take B cross k,"},{"Start":"10:08.435 ","End":"10:13.455","Text":"we get E. B cross k we get E. If we do"},{"Start":"10:13.455 ","End":"10:19.170","Text":"k cross E we get B. k cross E, we get B."},{"Start":"10:19.170 ","End":"10:21.265","Text":"Exactly as we would,"},{"Start":"10:21.265 ","End":"10:26.095","Text":"if we were using these coordinates of x, y, z."},{"Start":"10:26.095 ","End":"10:28.135","Text":"The only thing to remember,"},{"Start":"10:28.135 ","End":"10:33.209","Text":"is that we have this value c for the speed of light, which is inside."},{"Start":"10:33.209 ","End":"10:35.645","Text":"When we\u0027re doing k cross E,"},{"Start":"10:35.645 ","End":"10:37.175","Text":"we divide by C,"},{"Start":"10:37.175 ","End":"10:39.430","Text":"and when we\u0027re doing B cross k,"},{"Start":"10:39.430 ","End":"10:47.465","Text":"we multiply it by c. I hope that\u0027s an explanation for why these equations work."},{"Start":"10:47.465 ","End":"10:50.910","Text":"That is the end of this lesson."}],"ID":22434},{"Watched":false,"Name":"Exercise - Wave Equations","Duration":"19m 13s","ChapterTopicVideoID":21582,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21582.jpeg","UploadDate":"2020-05-07T19:03:17.0430000","DurationForVideoObject":"PT19M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.815","Text":"Hello. In this lesson,"},{"Start":"00:01.815 ","End":"00:05.415","Text":"we\u0027re going to be answering the following question."},{"Start":"00:05.415 ","End":"00:12.585","Text":"We have here the general wave equation for all waves where we have nabla squared of Phi."},{"Start":"00:12.585 ","End":"00:17.190","Text":"The Laplacian of Phi is equal to 1 divided by v"},{"Start":"00:17.190 ","End":"00:23.460","Text":"squared multiplied by the second derivative with respect to time of Phi."},{"Start":"00:23.460 ","End":"00:25.520","Text":"Phi is, of course,"},{"Start":"00:25.520 ","End":"00:27.005","Text":"the wave function,"},{"Start":"00:27.005 ","End":"00:29.210","Text":"and v is the velocity of the wave."},{"Start":"00:29.210 ","End":"00:34.130","Text":"In general, the velocity of the wave is equal to Omega divided"},{"Start":"00:34.130 ","End":"00:39.440","Text":"by k. In this specific case of electromagnetic waves,"},{"Start":"00:39.440 ","End":"00:45.785","Text":"Phi is the function of the electric or magnetic field and v the velocity of the wave,"},{"Start":"00:45.785 ","End":"00:49.230","Text":"is equal to c, the speed of light."},{"Start":"00:49.240 ","End":"00:52.940","Text":"What we saw in previous lessons that instead of Phi,"},{"Start":"00:52.940 ","End":"00:57.945","Text":"we wrote here E the x component,"},{"Start":"00:57.945 ","End":"00:59.550","Text":"and then also here,"},{"Start":"00:59.550 ","End":"01:01.965","Text":"so this is what we\u0027ve seen up until now."},{"Start":"01:01.965 ","End":"01:07.800","Text":"But generally speaking, we\u0027ll write it out as Phi."},{"Start":"01:08.060 ","End":"01:12.120","Text":"Let\u0027s answer question number 1,"},{"Start":"01:12.120 ","End":"01:15.950","Text":"which is to show that Phi as a function of x and"},{"Start":"01:15.950 ","End":"01:20.525","Text":"t is equal to a cosine of kx multiplied by sine of"},{"Start":"01:20.525 ","End":"01:30.520","Text":"Omega t fulfills the wave equation and that it is a possible solution to the equation."},{"Start":"01:30.860 ","End":"01:34.670","Text":"The first thing that we\u0027re going to do is we\u0027re going to write"},{"Start":"01:34.670 ","End":"01:39.380","Text":"out the Laplacian of our Phi."},{"Start":"01:39.380 ","End":"01:48.995","Text":"The Laplacian is just the second derivative with respect to x of"},{"Start":"01:48.995 ","End":"01:57.065","Text":"our function Phi plus the second derivative with"},{"Start":"01:57.065 ","End":"02:01.760","Text":"respect to y of our function Phi plus"},{"Start":"02:01.760 ","End":"02:08.650","Text":"the second derivative with respect to z of our function Phi."},{"Start":"02:09.260 ","End":"02:17.970","Text":"Let\u0027s take the second derivative with respect to x so only here we have our variable x."},{"Start":"02:18.380 ","End":"02:23.485","Text":"When we take the first derivative we\u0027ll have A,"},{"Start":"02:23.485 ","End":"02:28.265","Text":"and the first derivative is negative sine."},{"Start":"02:28.265 ","End":"02:33.679","Text":"Let\u0027s actually just write this in blue over here that it\u0027s the first derivative."},{"Start":"02:33.679 ","End":"02:41.315","Text":"We\u0027ll have A of cosine the first derivative is negative sine of kx"},{"Start":"02:41.315 ","End":"02:45.720","Text":"multiplied by sine of Omega t"},{"Start":"02:45.720 ","End":"02:50.660","Text":"and this is multiplied by the inner derivative over here with respect to x,"},{"Start":"02:50.660 ","End":"02:57.210","Text":"which is k. We can just write it over here, negative kA."},{"Start":"02:58.670 ","End":"03:01.050","Text":"This was the first derivative."},{"Start":"03:01.050 ","End":"03:05.750","Text":"Now let\u0027s take the second derivative again with respect to x."},{"Start":"03:05.750 ","End":"03:12.830","Text":"We have negative kA and then with respect to x again,"},{"Start":"03:12.830 ","End":"03:19.610","Text":"so the derivative of sine is just going to be cosine."},{"Start":"03:19.610 ","End":"03:28.325","Text":"We have cosine of kx sine of Omega t. Then again,"},{"Start":"03:28.325 ","End":"03:31.580","Text":"we have to multiply by the inner derivative,"},{"Start":"03:31.580 ","End":"03:37.175","Text":"which over here again is k. We have k multiplied by k which is k squared."},{"Start":"03:37.175 ","End":"03:41.949","Text":"Now, I\u0027ll rub this out."},{"Start":"03:42.500 ","End":"03:46.700","Text":"Now let\u0027s add in the second derivative with respect to y."},{"Start":"03:46.700 ","End":"03:48.290","Text":"If we look at the equation,"},{"Start":"03:48.290 ","End":"03:52.490","Text":"we have no y-variable and similarly,"},{"Start":"03:52.490 ","End":"03:55.625","Text":"with respect to z, we have no z variable."},{"Start":"03:55.625 ","End":"03:59.550","Text":"This is just what we are left with."},{"Start":"03:59.780 ","End":"04:09.360","Text":"Now, let\u0027s work out the second derivative of our wave function with respect to time."},{"Start":"04:11.030 ","End":"04:15.245","Text":"That will be the second derivative of this."},{"Start":"04:15.245 ","End":"04:20.645","Text":"Again, only over here do we have this t variable."},{"Start":"04:20.645 ","End":"04:29.175","Text":"What we\u0027ll have is A cosine of kx,"},{"Start":"04:29.175 ","End":"04:34.820","Text":"and then if we take the first derivative of sine Omega t,"},{"Start":"04:34.820 ","End":"04:36.875","Text":"so we get cosine."},{"Start":"04:36.875 ","End":"04:40.970","Text":"We\u0027ll be left here with cosine of"},{"Start":"04:40.970 ","End":"04:46.760","Text":"Omega t multiplied by the inner derivative, which is Omega."},{"Start":"04:46.760 ","End":"04:51.140","Text":"Then when we take the second derivative of this,"},{"Start":"04:51.140 ","End":"04:56.510","Text":"so the cosine becomes negative sine so we just put here a"},{"Start":"04:56.510 ","End":"05:01.835","Text":"negative and we just change from cosine to sine."},{"Start":"05:01.835 ","End":"05:05.840","Text":"Then we again multiply by the inner derivative which is Omega."},{"Start":"05:05.840 ","End":"05:14.020","Text":"We\u0027re left with negative Omega squared A cosine of kx sine of Omega t. Now,"},{"Start":"05:14.020 ","End":"05:22.935","Text":"what we can see is A cosine sine is our original function Phi."},{"Start":"05:22.935 ","End":"05:28.539","Text":"Also over here, A cosine sine is our original function."},{"Start":"05:29.030 ","End":"05:34.610","Text":"In that case, we can write this as negative k squared of"},{"Start":"05:34.610 ","End":"05:41.280","Text":"Phi and this as being equal to negative Omega squared of Phi."},{"Start":"05:41.280 ","End":"05:44.985","Text":"Now we want to make them equal,"},{"Start":"05:44.985 ","End":"05:53.465","Text":"so what we have is we say that negative k squared of Phi is equal to"},{"Start":"05:53.465 ","End":"06:02.700","Text":"1 divided by v squared of this over here which is negative Omega squared of Phi."},{"Start":"06:02.700 ","End":"06:06.495","Text":"We can cross out the Omega from each side and"},{"Start":"06:06.495 ","End":"06:09.240","Text":"the minus sign and then we\u0027re left"},{"Start":"06:09.240 ","End":"06:13.430","Text":"with k squared or let\u0027s take the square root of both sides,"},{"Start":"06:13.430 ","End":"06:19.610","Text":"we have k is equal to Omega divided by v. Now, if we look here,"},{"Start":"06:19.610 ","End":"06:22.670","Text":"we\u0027re given that v is equal to Omega k. So if"},{"Start":"06:22.670 ","End":"06:26.105","Text":"we multiply both sides by v and divide by k,"},{"Start":"06:26.105 ","End":"06:30.440","Text":"we get over here that v is equal to Omega divided by"},{"Start":"06:30.440 ","End":"06:36.760","Text":"k which we know is meant to be correct from what is given."},{"Start":"06:36.760 ","End":"06:42.305","Text":"That\u0027s how we show that this equation for Phi over here fulfills"},{"Start":"06:42.305 ","End":"06:48.715","Text":"the wave equation and therefore it is possible solution to the equation."},{"Start":"06:48.715 ","End":"06:52.305","Text":"Now let\u0027s answer question number 2."},{"Start":"06:52.305 ","End":"06:56.705","Text":"Question number 2 says that d\u0027Alembert solution to"},{"Start":"06:56.705 ","End":"07:00.890","Text":"the wave equation says that every solution must be of"},{"Start":"07:00.890 ","End":"07:09.720","Text":"the form f(x - vt) plus g(x + vt) where f and g are functions."},{"Start":"07:09.720 ","End":"07:13.555","Text":"We\u0027re being asked to show that the function from 1,"},{"Start":"07:13.555 ","End":"07:18.365","Text":"the function over here is also a solution according to"},{"Start":"07:18.365 ","End":"07:25.225","Text":"d\u0027Alembert and the hint for us to show this is to use trigonometric identities."},{"Start":"07:25.225 ","End":"07:28.880","Text":"For those of you that remember trigonometric identities"},{"Start":"07:28.880 ","End":"07:32.465","Text":"very well so that is very useful, but otherwise,"},{"Start":"07:32.465 ","End":"07:36.260","Text":"2 very useful identities to remember"},{"Start":"07:36.260 ","End":"07:42.900","Text":"are the following and I also suggest you write them in your equation sheets."},{"Start":"07:43.580 ","End":"07:54.480","Text":"The first is sine of Alpha plus sine of Beta is equal"},{"Start":"07:54.480 ","End":"08:01.390","Text":"to 2 sine of Alpha plus Beta over"},{"Start":"08:01.390 ","End":"08:10.270","Text":"2 multiplied by cosine of Alpha minus Beta over 2."},{"Start":"08:10.270 ","End":"08:14.935","Text":"This is the first one and the second one is sine of Alpha"},{"Start":"08:14.935 ","End":"08:22.390","Text":"minus sine of Beta is equal to 2 sine."},{"Start":"08:22.390 ","End":"08:24.610","Text":"Then the sines switch over here."},{"Start":"08:24.610 ","End":"08:30.145","Text":"Here it\u0027s Alpha minus Beta over 2 multiplied by"},{"Start":"08:30.145 ","End":"08:36.925","Text":"cosine of Alpha plus Beta over 2."},{"Start":"08:36.925 ","End":"08:39.595","Text":"Here if we have a positive,"},{"Start":"08:39.595 ","End":"08:42.370","Text":"it\u0027s like so plus then minus and if there\u0027s a negative,"},{"Start":"08:42.370 ","End":"08:45.310","Text":"it\u0027s negative then a plus."},{"Start":"08:45.310 ","End":"08:49.525","Text":"Now let\u0027s write out our function."},{"Start":"08:49.525 ","End":"09:00.330","Text":"We have Phi is equal to a cosine of kx"},{"Start":"09:00.330 ","End":"09:05.450","Text":"multiplied by sine of Omega t."},{"Start":"09:05.610 ","End":"09:14.180","Text":"Now what we want to do is we want to write this out in one of these formats over here."},{"Start":"09:14.580 ","End":"09:18.610","Text":"Taking into account this 2 over here."},{"Start":"09:18.610 ","End":"09:25.345","Text":"We can say that this is equal to a and then we\u0027ll divide and multiply by 2."},{"Start":"09:25.345 ","End":"09:28.270","Text":"The 2\u0027s will cancel out."},{"Start":"09:28.270 ","End":"09:33.610","Text":"But what this does, is this gives us this constant in relation to the 2 over here."},{"Start":"09:33.610 ","End":"09:37.345","Text":"We can equate it to this format over here."},{"Start":"09:37.345 ","End":"09:46.870","Text":"Then we have cosine of kx multiplied by"},{"Start":"09:46.870 ","End":"09:56.440","Text":"sine of Omega t. We can just show this."},{"Start":"09:56.440 ","End":"10:01.160","Text":"Now we can see that we\u0027re beginning to get this format."},{"Start":"10:05.490 ","End":"10:08.110","Text":"Which format are we going to use here?"},{"Start":"10:08.110 ","End":"10:16.240","Text":"We see that we want to have f of this plus g. We\u0027ll look at this format over here,"},{"Start":"10:16.240 ","End":"10:18.800","Text":"where we have a plus."},{"Start":"10:19.080 ","End":"10:23.215","Text":"We see that the cosine where we have"},{"Start":"10:23.215 ","End":"10:28.120","Text":"kx has to be equal"},{"Start":"10:28.120 ","End":"10:34.765","Text":"to Alpha minus beta divided by 2."},{"Start":"10:34.765 ","End":"10:39.835","Text":"Omega t has to be equal to the sine component,"},{"Start":"10:39.835 ","End":"10:44.935","Text":"which is Alpha plus Beta divided by 2."},{"Start":"10:44.935 ","End":"10:49.831","Text":"Let\u0027s call this a and let\u0027s call this b."},{"Start":"10:49.831 ","End":"10:52.540","Text":"These 2 equations."},{"Start":"10:52.540 ","End":"10:56.545","Text":"Now in order to find out what Alpha and Beta are."},{"Start":"10:56.545 ","End":"11:00.295","Text":"First, we can do a plus b."},{"Start":"11:00.295 ","End":"11:02.245","Text":"We add these 2 equations,"},{"Start":"11:02.245 ","End":"11:06.490","Text":"then we\u0027ll have Alpha over 2 plus Alpha over 2 is just Alpha."},{"Start":"11:06.490 ","End":"11:10.525","Text":"Then we have negative Beta over 2 plus Beta over 2."},{"Start":"11:10.525 ","End":"11:12.520","Text":"It cancels out."},{"Start":"11:12.520 ","End":"11:15.940","Text":"Alpha is equal to a plus b,"},{"Start":"11:15.940 ","End":"11:25.015","Text":"which is just kx plus Omega t. Then if we take a minus b,"},{"Start":"11:25.015 ","End":"11:30.550","Text":"we get half Alpha minus half Alpha is 0."},{"Start":"11:30.550 ","End":"11:34.460","Text":"Then we have negative half Beta,"},{"Start":"11:36.630 ","End":"11:40.450","Text":"which is a, negative b."},{"Start":"11:40.450 ","End":"11:44.470","Text":"Negative half Beta which is negative beta."},{"Start":"11:44.470 ","End":"11:49.135","Text":"We get that negative Beta is equal to a minus b,"},{"Start":"11:49.135 ","End":"11:56.980","Text":"so it\u0027s equal to kx minus b which is Omega t. Of course,"},{"Start":"11:56.980 ","End":"11:58.660","Text":"we can also flip these around,"},{"Start":"11:58.660 ","End":"12:00.970","Text":"it doesn\u0027t really make a difference."},{"Start":"12:00.970 ","End":"12:07.980","Text":"In other words, we can say that Beta is equal to Omega t minus kx."},{"Start":"12:07.980 ","End":"12:12.250","Text":"We multiply both sides by negative 1."},{"Start":"12:12.720 ","End":"12:17.230","Text":"In that case, we can say that our Phi,"},{"Start":"12:17.230 ","End":"12:19.405","Text":"back to this equation over here,"},{"Start":"12:19.405 ","End":"12:23.090","Text":"is equal to A divided by 2."},{"Start":"12:23.130 ","End":"12:28.085","Text":"Now we\u0027re just going to write it out instead of in this format over here,"},{"Start":"12:28.085 ","End":"12:30.045","Text":"which is what we have written over here,"},{"Start":"12:30.045 ","End":"12:33.100","Text":"we\u0027ll write it in this format here."},{"Start":"12:33.530 ","End":"12:38.800","Text":"Multiplied by sine of Alpha."},{"Start":"12:39.570 ","End":"12:42.055","Text":"Alpha, as we said,"},{"Start":"12:42.055 ","End":"12:47.410","Text":"was equal to kx plus Omega t,"},{"Start":"12:47.410 ","End":"12:50.390","Text":"plus sine of Beta,"},{"Start":"12:50.390 ","End":"13:00.500","Text":"where Beta is equal to Omega t minus kx."},{"Start":"13:03.480 ","End":"13:08.200","Text":"As we saw, we had here 2 cosine of something,"},{"Start":"13:08.200 ","End":"13:10.645","Text":"sine of something, which is like this,"},{"Start":"13:10.645 ","End":"13:13.645","Text":"2 cosine of something, sine of something."},{"Start":"13:13.645 ","End":"13:19.540","Text":"We wrote it in this format so that we can convert from this side to this side."},{"Start":"13:19.540 ","End":"13:23.665","Text":"That\u0027s why this 2 over here is important to us."},{"Start":"13:23.665 ","End":"13:27.355","Text":"Now, we\u0027re writing that it\u0027s equal to this over here."},{"Start":"13:27.355 ","End":"13:30.100","Text":"Now, if we look back into our question,"},{"Start":"13:30.100 ","End":"13:33.850","Text":"we have f(x) minus something as"},{"Start":"13:33.850 ","End":"13:37.690","Text":"a function of t. We have this x plus something as a function of t,"},{"Start":"13:37.690 ","End":"13:38.710","Text":"but that\u0027s okay,"},{"Start":"13:38.710 ","End":"13:42.655","Text":"it just means this something is just minus of this."},{"Start":"13:42.655 ","End":"13:50.540","Text":"That\u0027s fine, and then plus g(x) plus."},{"Start":"13:50.540 ","End":"13:54.640","Text":"Here, what we can do is we can use"},{"Start":"13:54.640 ","End":"14:01.120","Text":"the identity saying that sine of something,"},{"Start":"14:01.120 ","End":"14:02.845","Text":"so let\u0027s say we\u0027ll write it out here."},{"Start":"14:02.845 ","End":"14:09.145","Text":"Omega t minus kx is equal to negative sine"},{"Start":"14:09.145 ","End":"14:16.285","Text":"of kx minus Omega t. Therefore,"},{"Start":"14:16.285 ","End":"14:20.360","Text":"we can say that Phi"},{"Start":"14:20.820 ","End":"14:28.930","Text":"is equal to a over 2 of sine."},{"Start":"14:28.930 ","End":"14:31.975","Text":"Then we can write this over here."},{"Start":"14:31.975 ","End":"14:39.955","Text":"Sine of kx plus Omega t plus or rather minus over here,"},{"Start":"14:39.955 ","End":"14:45.249","Text":"minus sine of kx"},{"Start":"14:45.249 ","End":"14:52.250","Text":"minus Omega t from this identity over here."},{"Start":"14:56.490 ","End":"15:04.855","Text":"We\u0027re almost done. What we can see in d\u0027Alembert\u0027s is our x variable is alone."},{"Start":"15:04.855 ","End":"15:10.030","Text":"It doesn\u0027t have a coefficient."},{"Start":"15:10.030 ","End":"15:16.480","Text":"Here we have the coefficient k. We\u0027re going to take the k out."},{"Start":"15:16.480 ","End":"15:24.985","Text":"We\u0027re just going to rewrite this as A over 2 of sine,"},{"Start":"15:24.985 ","End":"15:26.350","Text":"and then we have k,"},{"Start":"15:26.350 ","End":"15:29.275","Text":"and then we\u0027ll have x plus Omega"},{"Start":"15:29.275 ","End":"15:36.745","Text":"over kt minus sine."},{"Start":"15:36.745 ","End":"15:40.510","Text":"I\u0027m just taking again k out as a factor,"},{"Start":"15:40.510 ","End":"15:44.650","Text":"so kx minus Omega over k,"},{"Start":"15:44.650 ","End":"15:51.135","Text":"t. As we saw before,"},{"Start":"15:51.135 ","End":"15:54.920","Text":"Omega over k is equal to v over here,"},{"Start":"15:54.920 ","End":"15:58.400","Text":"and we also saw it in question number 1 at the end."},{"Start":"15:58.400 ","End":"16:06.800","Text":"V is equal to Omega over k. We can just erase this and write this as v,"},{"Start":"16:06.800 ","End":"16:10.160","Text":"which is exactly also what we wanted here,"},{"Start":"16:10.160 ","End":"16:13.440","Text":"vt. Now we have vt."},{"Start":"16:14.160 ","End":"16:20.540","Text":"If we just multiply both by A divided by 2."},{"Start":"16:20.540 ","End":"16:30.856","Text":"We have a 2 multiplied by sine of kx plus vt plus,"},{"Start":"16:30.856 ","End":"16:33.240","Text":"and then let\u0027s write it minus,"},{"Start":"16:33.240 ","End":"16:34.755","Text":"I\u0027ll explain this in a second,"},{"Start":"16:34.755 ","End":"16:41.950","Text":"a divided by 2 of sine of kx minus vt."},{"Start":"16:42.390 ","End":"16:48.990","Text":"In that case, we can see that this term over here is,"},{"Start":"16:48.990 ","End":"16:51.320","Text":"because we have the plus over here,"},{"Start":"16:51.320 ","End":"16:57.580","Text":"so this is our g as a function of x plus vt."},{"Start":"16:57.580 ","End":"17:01.880","Text":"This term over here with the minus,"},{"Start":"17:01.880 ","End":"17:08.435","Text":"because here we have a minus is like our f. This is like a function of f,"},{"Start":"17:08.435 ","End":"17:11.380","Text":"which is x minus v_t."},{"Start":"17:11.380 ","End":"17:17.480","Text":"Now, we can see that we got from our original equation,"},{"Start":"17:17.480 ","End":"17:20.070","Text":"this one over here."},{"Start":"17:20.070 ","End":"17:22.565","Text":"Our original wave equation,"},{"Start":"17:22.565 ","End":"17:28.925","Text":"we use d\u0027Alembert\u0027s theory in order to manipulate it using"},{"Start":"17:28.925 ","End":"17:32.780","Text":"trigonometric identities in order to get it into this format"},{"Start":"17:32.780 ","End":"17:37.760","Text":"over here where it\u0027s the sum of 2 functions."},{"Start":"17:38.730 ","End":"17:41.545","Text":"We\u0027ve shown how to do this."},{"Start":"17:41.545 ","End":"17:43.530","Text":"This is the end of question number 2."},{"Start":"17:43.530 ","End":"17:46.160","Text":"I just want to speak about what type of wave this is,"},{"Start":"17:46.160 ","End":"17:48.635","Text":"because at the end of the day we started off with"},{"Start":"17:48.635 ","End":"17:51.860","Text":"a wave function that we just manipulated into a different format,"},{"Start":"17:51.860 ","End":"17:54.785","Text":"but it\u0027s still a wave function."},{"Start":"17:54.785 ","End":"17:58.760","Text":"What we can see, which is what we\u0027ve seen in previous videos,"},{"Start":"17:58.760 ","End":"18:01.205","Text":"is that because there\u0027s a plus over here,"},{"Start":"18:01.205 ","End":"18:05.690","Text":"that means that this wave is traveling in this negative direction."},{"Start":"18:05.690 ","End":"18:07.925","Text":"Because there\u0027s a minus over here,"},{"Start":"18:07.925 ","End":"18:14.280","Text":"that means that this wave is traveling in this positive direction."},{"Start":"18:14.940 ","End":"18:19.120","Text":"Along a certain axis,"},{"Start":"18:19.120 ","End":"18:21.245","Text":"this is what we can see,"},{"Start":"18:21.245 ","End":"18:23.800","Text":"that we have 2 waves,"},{"Start":"18:23.800 ","End":"18:30.200","Text":"where they\u0027re both equal in magnitude but they\u0027re traveling in different directions."},{"Start":"18:30.200 ","End":"18:35.305","Text":"What we get in a case like this is a standing wave."},{"Start":"18:35.305 ","End":"18:40.948","Text":"I\u0027m not going to go into too much detail with what a standing wave is right now."},{"Start":"18:40.948 ","End":"18:45.890","Text":"You\u0027ll learn it in later chapters and in your later studies."},{"Start":"18:45.890 ","End":"18:50.180","Text":"But this is just to give you a sense of what this means."},{"Start":"18:50.180 ","End":"18:54.410","Text":"We have 1 wave that has the exact same magnitude,"},{"Start":"18:54.410 ","End":"18:56.780","Text":"where one is traveling in one direction and the"},{"Start":"18:56.780 ","End":"19:00.449","Text":"other is traveling in the opposite direction."},{"Start":"19:00.660 ","End":"19:02.930","Text":"This is a standing wave."},{"Start":"19:02.930 ","End":"19:09.380","Text":"This might give you a little bit more intuition later in more advanced wave courses."},{"Start":"19:09.380 ","End":"19:13.559","Text":"That is the end of this lesson."}],"ID":22435},{"Watched":false,"Name":"Refraction Through a Pane of Glass","Duration":"12m 20s","ChapterTopicVideoID":21374,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21374.jpeg","UploadDate":"2020-04-06T22:37:50.3530000","DurationForVideoObject":"PT12M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.085","Text":"Hello. In this lesson,"},{"Start":"00:02.085 ","End":"00:08.280","Text":"we\u0027re going to be speaking about refraction through different media."},{"Start":"00:08.280 ","End":"00:11.565","Text":"In 1 of the previous lessons,"},{"Start":"00:11.565 ","End":"00:18.735","Text":"we spoke about light passing through a rectangular cube made out of prospects,"},{"Start":"00:18.735 ","End":"00:24.470","Text":"so we\u0027re just going to go over this example in a slightly easier way,"},{"Start":"00:24.470 ","End":"00:27.055","Text":"just in case it wasn\u0027t clear."},{"Start":"00:27.055 ","End":"00:33.080","Text":"Let\u0027s imagine that we have some sheet over here,"},{"Start":"00:33.080 ","End":"00:36.665","Text":"and this is made out of glass,"},{"Start":"00:36.665 ","End":"00:43.425","Text":"where glass has a refractive index of 1.5."},{"Start":"00:43.425 ","End":"00:48.390","Text":"On either side we have air with a refractive index of 1."},{"Start":"00:49.040 ","End":"00:55.505","Text":"We can already see that we have a rarer or a less dense medium,"},{"Start":"00:55.505 ","End":"01:00.810","Text":"a denser medium, and back to the rarer medium over here."},{"Start":"01:01.610 ","End":"01:09.330","Text":"Now let\u0027s draw the normal to the surface."},{"Start":"01:09.330 ","End":"01:14.965","Text":"We have an incident ray coming in like so with"},{"Start":"01:14.965 ","End":"01:22.715","Text":"an angle of 30 degrees and I\u0027m reminding you it\u0027s an angle of 30 degrees to the normal."},{"Start":"01:22.715 ","End":"01:26.535","Text":"It hits the interface and through Snell\u0027s law,"},{"Start":"01:26.535 ","End":"01:35.670","Text":"we know that n1 sine of Theta_1 is equal to n_2 sine of Theta_2."},{"Start":"01:35.670 ","End":"01:39.430","Text":"In this case, our first medium is through air,"},{"Start":"01:39.430 ","End":"01:48.140","Text":"so our refractive index is 1 multiplied by sine of the angle of incidence,"},{"Start":"01:48.140 ","End":"01:51.495","Text":"which is 30 degrees."},{"Start":"01:51.495 ","End":"01:53.810","Text":"This is equal to n_2,"},{"Start":"01:53.810 ","End":"01:57.320","Text":"the refractive index of the second medium, which is glass."},{"Start":"01:57.320 ","End":"02:05.450","Text":"That\u0027s 1.5 multiplied by sine of the angle of refraction,"},{"Start":"02:05.450 ","End":"02:09.010","Text":"which we\u0027re trying to calculate."},{"Start":"02:09.010 ","End":"02:11.465","Text":"Once we do this,"},{"Start":"02:11.465 ","End":"02:15.050","Text":"what we will get is that Theta_2 and we plug this into"},{"Start":"02:15.050 ","End":"02:22.230","Text":"our calculator is equal to 19.5 degrees."},{"Start":"02:22.820 ","End":"02:26.565","Text":"It\u0027s 19.5 degrees again to the normal."},{"Start":"02:26.565 ","End":"02:31.030","Text":"As we know, we could have guessed that it would be less than 30"},{"Start":"02:31.030 ","End":"02:35.660","Text":"because we can see that we\u0027re going from a rarer medium to a denser medium,"},{"Start":"02:35.660 ","End":"02:42.915","Text":"which means that we knew that our ray would bend towards the normal."},{"Start":"02:42.915 ","End":"02:50.155","Text":"This angle over here is 19.5 degrees to the normal."},{"Start":"02:50.155 ","End":"02:55.450","Text":"The ray continues and it propagates until this point over here,"},{"Start":"02:55.450 ","End":"03:01.840","Text":"the boundary between glass and air."},{"Start":"03:03.020 ","End":"03:07.170","Text":"Let\u0027s call this section over here,"},{"Start":"03:07.170 ","End":"03:09.660","Text":"instead of using n_1 and n_2,"},{"Start":"03:09.660 ","End":"03:11.970","Text":"we\u0027ll use n_3 and n_4."},{"Start":"03:11.970 ","End":"03:21.540","Text":"n_3 sine of Theta_3 is equal to n_4 sine of Theta_4."},{"Start":"03:21.540 ","End":"03:23.695","Text":"It\u0027s the exact same calculation,"},{"Start":"03:23.695 ","End":"03:30.370","Text":"I\u0027ve just changed the indices to reflect this second case now over here."},{"Start":"03:30.370 ","End":"03:34.911","Text":"We want what is n_3?"},{"Start":"03:34.911 ","End":"03:40.955","Text":"The first medium that the ray is propagating through over here is glass."},{"Start":"03:40.955 ","End":"03:44.170","Text":"The refractive index of glass is 1.5."},{"Start":"03:44.170 ","End":"03:49.475","Text":"N_3 is equal to 1.5 multiplied by sine of Theta_3."},{"Start":"03:49.475 ","End":"03:50.810","Text":"Theta_3 is, of course,"},{"Start":"03:50.810 ","End":"03:52.805","Text":"the angle of incidence in this case."},{"Start":"03:52.805 ","End":"03:55.110","Text":"What is the angle?"},{"Start":"03:55.550 ","End":"04:00.830","Text":"We can see that we have z angles, or alternating angles,"},{"Start":"04:00.830 ","End":"04:06.835","Text":"which means that the angle over here is equal to the angle over here."},{"Start":"04:06.835 ","End":"04:11.830","Text":"It\u0027s 19.5 degrees."},{"Start":"04:14.360 ","End":"04:17.250","Text":"This is equal to n_4,"},{"Start":"04:17.250 ","End":"04:20.075","Text":"the refractive index of the fourth medium,"},{"Start":"04:20.075 ","End":"04:21.680","Text":"which is this over here,"},{"Start":"04:21.680 ","End":"04:29.315","Text":"it\u0027s air or rather the refractive index of the second medium in the second case."},{"Start":"04:29.315 ","End":"04:35.180","Text":"We\u0027re doing the equation for the second time when we\u0027re looking at the second medium,"},{"Start":"04:35.180 ","End":"04:38.930","Text":"the refractive index is that of the second medium over here which is air,"},{"Start":"04:38.930 ","End":"04:45.825","Text":"so that\u0027s 1 multiplied by sine of Theta_4 which we are trying to calculate."},{"Start":"04:45.825 ","End":"04:48.724","Text":"Once we plug all of this into our calculator,"},{"Start":"04:48.724 ","End":"04:53.550","Text":"we\u0027ll get the Theta_4 is equal to 30 degrees."},{"Start":"04:53.550 ","End":"04:55.760","Text":"As we could\u0027ve also guessed,"},{"Start":"04:55.760 ","End":"05:00.595","Text":"we\u0027re going from a denser medium to a rarer medium so"},{"Start":"05:00.595 ","End":"05:10.020","Text":"the ray will bend away from the normal, so 30 degrees."},{"Start":"05:10.020 ","End":"05:15.660","Text":"Now, notice that we have just the same case in mirror image."},{"Start":"05:16.020 ","End":"05:21.455","Text":"The ray traveled through air at 30 degrees and again,"},{"Start":"05:21.455 ","End":"05:24.865","Text":"on the other side is traveling through air at 30 degrees."},{"Start":"05:24.865 ","End":"05:29.095","Text":"The ray was diffracted at 19.5 degrees."},{"Start":"05:29.095 ","End":"05:34.865","Text":"Over here, it\u0027s also 19.5 degrees to the normal,"},{"Start":"05:34.865 ","End":"05:42.350","Text":"resulting in again, a refraction of 30 degrees."},{"Start":"05:42.350 ","End":"05:45.215","Text":"This is exactly what we were speaking about."},{"Start":"05:45.215 ","End":"05:48.020","Text":"Whether the ray comes in at 30 degrees,"},{"Start":"05:48.020 ","End":"05:51.200","Text":"given these refractive indexes,"},{"Start":"05:51.200 ","End":"05:56.855","Text":"we\u0027re going to get this angle of 19.5 and whether the ray comes in at 19.5"},{"Start":"05:56.855 ","End":"06:02.675","Text":"in either direction that you look when it comes into contact with the same medium."},{"Start":"06:02.675 ","End":"06:05.780","Text":"Again, it\u0027s going to come out at the exact same angle."},{"Start":"06:05.780 ","End":"06:10.945","Text":"This is what we\u0027ve been speaking about throughout these past few lessons"},{"Start":"06:10.945 ","End":"06:18.200","Text":"and it makes sense because the whole point is the interaction between the beam of light,"},{"Start":"06:18.200 ","End":"06:22.010","Text":"between these 2 different media, and of course,"},{"Start":"06:22.010 ","End":"06:24.410","Text":"these 2 different media are the same,"},{"Start":"06:24.410 ","End":"06:29.710","Text":"so there\u0027s no reason why we should have these different angles over here."},{"Start":"06:29.710 ","End":"06:35.195","Text":"Another way that we could have looked at this is to see that 1.5 sine"},{"Start":"06:35.195 ","End":"06:40.614","Text":"of 19.5 is pretty much,"},{"Start":"06:40.614 ","End":"06:43.320","Text":"or rather, let\u0027s write this,"},{"Start":"06:43.320 ","End":"06:50.040","Text":"we could have said that n_3 sine of Theta_3"},{"Start":"06:50.040 ","End":"06:58.350","Text":"is going to be exactly the same as n_2 sine of Theta_2 in this example."},{"Start":"06:58.820 ","End":"07:04.145","Text":"I\u0027m reminding you n_3 we said was the refractive index of the glass and"},{"Start":"07:04.145 ","End":"07:09.335","Text":"also n_2 was the refractive index of the glass remember."},{"Start":"07:09.335 ","End":"07:12.065","Text":"Sine of Theta_3,"},{"Start":"07:12.065 ","End":"07:17.780","Text":"we already saw is 19.5 degrees because of alternate angles."},{"Start":"07:17.780 ","End":"07:26.455","Text":"We already saw due to our first calculation of Snell\u0027s law that Theta_2 was 19.5."},{"Start":"07:26.455 ","End":"07:28.365","Text":"These are exactly the same."},{"Start":"07:28.365 ","End":"07:35.175","Text":"As we know, n_2 sine Theta_2 is equal to n_1 sine of Theta_1."},{"Start":"07:35.175 ","End":"07:43.125","Text":"This is just regular Snell\u0027s law as we\u0027ve seen throughout the whole lesson."},{"Start":"07:43.125 ","End":"07:47.450","Text":"This, of course, we saw that we just changed the indices."},{"Start":"07:47.450 ","End":"07:53.305","Text":"It\u0027s equal to n_4 sine of Theta_4."},{"Start":"07:53.305 ","End":"07:56.630","Text":"We saw this as exactly the same as Snell\u0027s law."},{"Start":"07:56.630 ","End":"07:59.105","Text":"Over here we just changed 1 to 3,"},{"Start":"07:59.105 ","End":"08:00.635","Text":"and 2 to 4."},{"Start":"08:00.635 ","End":"08:03.965","Text":"That\u0027s it. It\u0027s the exact same calculation."},{"Start":"08:03.965 ","End":"08:13.290","Text":"Therefore, we can say that n_4 sine 4 is the same as n_1 sine Theta_1."},{"Start":"08:13.290 ","End":"08:17.485","Text":"We especially know this because we\u0027ve already seen that n_4,"},{"Start":"08:17.485 ","End":"08:22.490","Text":"the refractive index of the second medium in the second equation,"},{"Start":"08:22.490 ","End":"08:27.170","Text":"the second time we\u0027re using Snell\u0027s law is 1 for air."},{"Start":"08:27.170 ","End":"08:35.240","Text":"Similarly, we saw already in this that n_1 is also here."},{"Start":"08:35.240 ","End":"08:37.370","Text":"The first medium which has also air,"},{"Start":"08:37.370 ","End":"08:38.690","Text":"which is equal to 1."},{"Start":"08:38.690 ","End":"08:47.055","Text":"Therefore, we can see that we can cancel both of them out because n_1 is equal to n_4."},{"Start":"08:47.055 ","End":"08:55.440","Text":"Therefore, sine of Theta_4 is equal to sine of Theta_1."},{"Start":"08:55.440 ","End":"09:01.605","Text":"Therefore Theta_4 is equal to Theta_1."},{"Start":"09:01.605 ","End":"09:06.020","Text":"That is why the angle incidence over here is 30,"},{"Start":"09:06.020 ","End":"09:11.930","Text":"and the angle of diffraction on the other side of the glass is also equal to 30."},{"Start":"09:11.930 ","End":"09:19.730","Text":"Of course, the same can be applied between these 2 because n_3 and n_2 are equivalent."},{"Start":"09:19.730 ","End":"09:23.105","Text":"Therefore, Theta_3 must be equal to Theta_2,"},{"Start":"09:23.105 ","End":"09:29.225","Text":"which is exactly what we saw and it also corresponds to the law of alternate angles."},{"Start":"09:29.225 ","End":"09:35.045","Text":"Now, something that\u0027s interesting is that if we have someone who\u0027s looking"},{"Start":"09:35.045 ","End":"09:40.900","Text":"over here and this ray reaches them like so, so they can see."},{"Start":"09:40.900 ","End":"09:49.270","Text":"If we follow this line back like so,"},{"Start":"09:49.270 ","End":"09:53.140","Text":"we can see that these 2 lines over here,"},{"Start":"09:53.140 ","End":"09:58.335","Text":"the initial incident ray and this, they are parallel."},{"Start":"09:58.335 ","End":"10:01.395","Text":"They are parallel, these 2 lines."},{"Start":"10:01.395 ","End":"10:07.210","Text":"That\u0027s why we see this bend over here or this shift in"},{"Start":"10:07.210 ","End":"10:13.365","Text":"perspective when a ray of light undergoes refraction."},{"Start":"10:13.365 ","End":"10:16.735","Text":"That\u0027s why if you put a pencil in the glass of water,"},{"Start":"10:16.735 ","End":"10:19.910","Text":"it appears as if the pencil has bent."},{"Start":"10:19.910 ","End":"10:21.355","Text":"This is exactly why,"},{"Start":"10:21.355 ","End":"10:26.515","Text":"because we can see that the ray that is reflected to us looks as if it should"},{"Start":"10:26.515 ","End":"10:33.405","Text":"originate from a point that is higher than the initial incident ray."},{"Start":"10:33.405 ","End":"10:37.190","Text":"That\u0027s why we can see that if this was"},{"Start":"10:37.190 ","End":"10:41.610","Text":"the glass with a pencil in it or we can just draw it."},{"Start":"10:41.650 ","End":"10:46.840","Text":"If the pencil comes in like so,"},{"Start":"10:46.840 ","End":"10:49.565","Text":"so inside the water,"},{"Start":"10:49.565 ","End":"10:52.790","Text":"it will be bent, something like this."},{"Start":"10:52.790 ","End":"10:58.820","Text":"That\u0027s why we see this looping going on."},{"Start":"10:58.820 ","End":"11:00.680","Text":"Exactly because of this."},{"Start":"11:00.680 ","End":"11:05.345","Text":"People who have glasses will especially notice this."},{"Start":"11:05.345 ","End":"11:09.500","Text":"The thicker that the glass is or the higher"},{"Start":"11:09.500 ","End":"11:14.180","Text":"the prescription when using the cheaper materials for the glasses,"},{"Start":"11:14.180 ","End":"11:17.180","Text":"the thicker the glasses lens,"},{"Start":"11:17.180 ","End":"11:20.940","Text":"which causes more of this bending."},{"Start":"11:22.010 ","End":"11:26.865","Text":"This bending over here, this distance,"},{"Start":"11:26.865 ","End":"11:29.900","Text":"this difference that we see over here d,"},{"Start":"11:29.900 ","End":"11:33.470","Text":"is as a function of this width of the glass."},{"Start":"11:33.470 ","End":"11:36.860","Text":"People with a high prescription using"},{"Start":"11:36.860 ","End":"11:41.630","Text":"cheaper materials and their glasses will have thicker lenses,"},{"Start":"11:41.630 ","End":"11:46.190","Text":"which will cause this increased refraction,"},{"Start":"11:46.190 ","End":"11:49.430","Text":"which causes a shift in what they can see,"},{"Start":"11:49.430 ","End":"11:50.990","Text":"which is very difficult."},{"Start":"11:50.990 ","End":"11:53.405","Text":"That\u0027s why the higher the prescription,"},{"Start":"11:53.405 ","End":"11:58.310","Text":"you need to use more expensive materials in order"},{"Start":"11:58.310 ","End":"12:03.665","Text":"to maintain a thinner glasses lens to get less of this shift,"},{"Start":"12:03.665 ","End":"12:08.240","Text":"to decrease this difference d in order to try and see"},{"Start":"12:08.240 ","End":"12:11.060","Text":"the pencil in this example as straight as"},{"Start":"12:11.060 ","End":"12:15.840","Text":"possible if you use this as a little case study."},{"Start":"12:16.060 ","End":"12:19.919","Text":"That is the end of this lesson."}],"ID":21464},{"Watched":false,"Name":"Refraction in a Semicircle","Duration":"8m 13s","ChapterTopicVideoID":21375,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21375.jpeg","UploadDate":"2020-04-06T22:39:46.5770000","DurationForVideoObject":"PT8M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.905","Text":"Hello, in this lesson,"},{"Start":"00:01.905 ","End":"00:04.725","Text":"we\u0027re going to speak about refraction."},{"Start":"00:04.725 ","End":"00:10.360","Text":"But this time, we\u0027re going to speak about refraction through a semicircle."},{"Start":"00:11.090 ","End":"00:15.570","Text":"Imagine that this is a perfect semicircle,"},{"Start":"00:15.570 ","End":"00:19.035","Text":"where this is the origin or the center."},{"Start":"00:19.035 ","End":"00:26.835","Text":"Now, let\u0027s imagine that I take a laser beam and they shine it right at the origin."},{"Start":"00:26.835 ","End":"00:30.045","Text":"Here I\u0027m just drawing the normal,"},{"Start":"00:30.045 ","End":"00:39.975","Text":"and let\u0027s say I shine it at an angle of 40 degrees to the normal."},{"Start":"00:39.975 ","End":"00:43.635","Text":"Now, I want to see which direction it\u0027s going to refract."},{"Start":"00:43.635 ","End":"00:48.365","Text":"Just quickly, let\u0027s say that this material over here is perspex,"},{"Start":"00:48.365 ","End":"00:51.766","Text":"so its index of refraction is 1.5,"},{"Start":"00:51.766 ","End":"00:57.390","Text":"and here we have air with index of refraction is 1."},{"Start":"00:58.040 ","End":"01:03.720","Text":"Now we wanted to see what will happen to the beam."},{"Start":"01:03.720 ","End":"01:09.080","Text":"Using Snell\u0027s Law, we have n_1 sine of Theta_1,"},{"Start":"01:09.080 ","End":"01:14.285","Text":"which is equal to n_2 sine of Theta_2."},{"Start":"01:14.285 ","End":"01:16.100","Text":"N_1, the refractive index of the first medium."},{"Start":"01:16.100 ","End":"01:21.170","Text":"We\u0027ve air, so that\u0027s 1 multiplied by sine of the angle of incidence,"},{"Start":"01:21.170 ","End":"01:24.150","Text":"which is 40 degrees."},{"Start":"01:24.150 ","End":"01:28.160","Text":"This is equal to the refractive index of the second medium,"},{"Start":"01:28.160 ","End":"01:32.990","Text":"which is 1.5 multiplied by the sine of the angle,"},{"Start":"01:32.990 ","End":"01:34.580","Text":"which is what we\u0027re trying to find."},{"Start":"01:34.580 ","End":"01:37.925","Text":"Once we plug everything into our calculator,"},{"Start":"01:37.925 ","End":"01:43.760","Text":"we\u0027ll get that Theta_2 is equal to"},{"Start":"01:43.760 ","End":"01:51.935","Text":"the arc sine of 40 degrees divided by 1.5,"},{"Start":"01:51.935 ","End":"01:59.430","Text":"which comes out to be 25.4 degrees."},{"Start":"02:00.200 ","End":"02:02.990","Text":"We can see, as well as expected,"},{"Start":"02:02.990 ","End":"02:06.395","Text":"we\u0027re going from a rarer medium,"},{"Start":"02:06.395 ","End":"02:11.880","Text":"air into a denser medium, perspex."},{"Start":"02:12.790 ","End":"02:14.900","Text":"We knew already,"},{"Start":"02:14.900 ","End":"02:17.690","Text":"when going from a less dense to a more dense medium,"},{"Start":"02:17.690 ","End":"02:21.685","Text":"that the beam of light is going to bend towards the normal."},{"Start":"02:21.685 ","End":"02:27.170","Text":"We can really see that this is what is happening like so and the beam, of course,"},{"Start":"02:27.170 ","End":"02:28.760","Text":"continues on like this,"},{"Start":"02:28.760 ","End":"02:35.395","Text":"where this angle over here is 25.4 degrees."},{"Start":"02:35.395 ","End":"02:40.010","Text":"As we said, now the ray continues through the perspex until it"},{"Start":"02:40.010 ","End":"02:45.770","Text":"reaches the next interface between perspex and air."},{"Start":"02:45.770 ","End":"02:48.480","Text":"Because we\u0027re dealing with a semicircle,"},{"Start":"02:48.480 ","End":"02:50.900","Text":"what we have here is a rounded edge."},{"Start":"02:50.900 ","End":"02:57.755","Text":"In order to find the surface over here,"},{"Start":"02:57.755 ","End":"03:02.503","Text":"we consider this point and we draw a line to it."},{"Start":"03:02.503 ","End":"03:04.130","Text":"Here draw a tangent."},{"Start":"03:04.130 ","End":"03:05.960","Text":"I\u0027m sure you\u0027ve heard this before."},{"Start":"03:05.960 ","End":"03:10.220","Text":"Here\u0027s our tangent to this point."},{"Start":"03:10.220 ","End":"03:13.925","Text":"There\u0027s an important rule that says,"},{"Start":"03:13.925 ","End":"03:15.800","Text":"and this is why it was important,"},{"Start":"03:15.800 ","End":"03:19.805","Text":"that the beam goes through the center of the semicircle."},{"Start":"03:19.805 ","End":"03:26.555","Text":"The angle between the radius of a circle and its tangent is 90 degrees."},{"Start":"03:26.555 ","End":"03:32.140","Text":"Because over here our beam went through at the center of the circle,"},{"Start":"03:32.140 ","End":"03:34.070","Text":"and it reaches all the way to the edge."},{"Start":"03:34.070 ","End":"03:39.155","Text":"That means that this is exactly the radius of the circle."},{"Start":"03:39.155 ","End":"03:42.365","Text":"Here we\u0027ve drawn our tangent line,"},{"Start":"03:42.365 ","End":"03:46.175","Text":"which means that the angle between the 2 is 90 degrees,"},{"Start":"03:46.175 ","End":"03:52.075","Text":"or the radius is at 90 degrees or at right angles to its tangent."},{"Start":"03:52.075 ","End":"03:54.845","Text":"This is something important to remember."},{"Start":"03:54.845 ","End":"04:03.665","Text":"Now we can see that our array is reaching the interface at 90 degrees to it."},{"Start":"04:03.665 ","End":"04:12.049","Text":"When we\u0027ve already seen that when a beam of light is at 90 degrees to its surface,"},{"Start":"04:12.049 ","End":"04:13.700","Text":"or in other words,"},{"Start":"04:13.700 ","End":"04:17.275","Text":"at an angle of 0 degrees to the normal."},{"Start":"04:17.275 ","End":"04:22.600","Text":"Because this will just be the normal line, like so."},{"Start":"04:22.600 ","End":"04:27.005","Text":"It\u0027s at 90 degrees to the surface,"},{"Start":"04:27.005 ","End":"04:32.670","Text":"or in other words, 0 degrees to the normal."},{"Start":"04:33.160 ","End":"04:39.500","Text":"Therefore, what we get over here is that there is no refraction."},{"Start":"04:39.500 ","End":"04:42.710","Text":"So in other words, like we\u0027ve seen in previous videos,"},{"Start":"04:42.710 ","End":"04:47.655","Text":"the ray just continues out straight, like so."},{"Start":"04:47.655 ","End":"04:57.275","Text":"Its angle is still at this angle of 25.4 degrees to the normal over here."},{"Start":"04:57.275 ","End":"05:02.870","Text":"This is the rule when we\u0027re"},{"Start":"05:02.870 ","End":"05:08.075","Text":"dealing with a semicircle and specifically when the ray goes through the center,"},{"Start":"05:08.075 ","End":"05:13.830","Text":"and please remember this over here."},{"Start":"05:13.830 ","End":"05:17.265","Text":"Now, what can we use this for?"},{"Start":"05:17.265 ","End":"05:26.110","Text":"If we take a protractor and place the protractor over here."},{"Start":"05:26.180 ","End":"05:32.594","Text":"Then we set this over here to be 90 degrees,"},{"Start":"05:32.594 ","End":"05:34.183","Text":"this is 0,"},{"Start":"05:34.183 ","End":"05:38.180","Text":"and this we can also set a 0 and this at 90."},{"Start":"05:38.180 ","End":"05:42.155","Text":"It depends what protractor you\u0027re using."},{"Start":"05:42.155 ","End":"05:46.559","Text":"Equally, this could have been 0,"},{"Start":"05:46.559 ","End":"05:48.050","Text":"90,180, 270,"},{"Start":"05:48.050 ","End":"05:50.290","Text":"and then all the way around."},{"Start":"05:50.290 ","End":"05:52.920","Text":"It\u0027s the same thing."},{"Start":"05:52.920 ","End":"05:55.260","Text":"What we can do is we can measure."},{"Start":"05:55.260 ","End":"05:59.675","Text":"Here we know that this is 40,"},{"Start":"05:59.675 ","End":"06:01.310","Text":"so 10, 20,"},{"Start":"06:01.310 ","End":"06:02.360","Text":"30, 40,"},{"Start":"06:02.360 ","End":"06:03.665","Text":"50, 60,"},{"Start":"06:03.665 ","End":"06:05.460","Text":"70, 80,"},{"Start":"06:05.460 ","End":"06:07.740","Text":"90, and so on."},{"Start":"06:07.740 ","End":"06:12.660","Text":"Then here this is going to be 10, 20,"},{"Start":"06:12.660 ","End":"06:14.700","Text":"30, 40, 50,"},{"Start":"06:14.700 ","End":"06:17.715","Text":"60, 70, 80, approximately."},{"Start":"06:17.715 ","End":"06:22.085","Text":"Then this really does look like 25.4, something like that."},{"Start":"06:22.085 ","End":"06:29.375","Text":"What we can do without doing this calculation using this semicircle made out of"},{"Start":"06:29.375 ","End":"06:32.540","Text":"this material or any other material that is"},{"Start":"06:32.540 ","End":"06:38.045","Text":"transparent and has a higher density than air."},{"Start":"06:38.045 ","End":"06:42.710","Text":"We can really measure this angle of refraction inside"},{"Start":"06:42.710 ","End":"06:49.370","Text":"the perspex because this corresponds to the same thing."},{"Start":"06:49.370 ","End":"06:52.520","Text":"Because as the ray leaves the semicircle,"},{"Start":"06:52.520 ","End":"06:56.630","Text":"it isn\u0027t refracted, it just continues in a straight line."},{"Start":"06:56.630 ","End":"07:03.330","Text":"This is a way to measure the refraction within this second medium."},{"Start":"07:03.330 ","End":"07:08.690","Text":"Now, just regarding this shining at the origin."},{"Start":"07:08.690 ","End":"07:14.659","Text":"If a ray would have come in over here lower than the origin,"},{"Start":"07:14.659 ","End":"07:20.555","Text":"even if it was still at 40 degrees to the normal,"},{"Start":"07:20.555 ","End":"07:26.380","Text":"it would have diffracted like"},{"Start":"07:26.380 ","End":"07:32.885","Text":"so at the exact same angle of 25.4 degrees."},{"Start":"07:32.885 ","End":"07:40.220","Text":"But the angle over here wouldn\u0027t be 90 degrees to the tangent."},{"Start":"07:40.220 ","End":"07:44.585","Text":"In which case, the ray wouldn\u0027t continue straight out,"},{"Start":"07:44.585 ","End":"07:48.500","Text":"but it would be potentially internally reflected"},{"Start":"07:48.500 ","End":"07:54.950","Text":"or further refracted depending on the critical angle."},{"Start":"07:54.950 ","End":"07:56.825","Text":"But the point is that here,"},{"Start":"07:56.825 ","End":"07:58.610","Text":"there isn\u0027t a 90-degree angle,"},{"Start":"07:58.610 ","End":"08:02.285","Text":"which means that we can use this rule over here."},{"Start":"08:02.285 ","End":"08:10.685","Text":"This is specifically when we\u0027re going through the origin of a circle or of a semicircle."},{"Start":"08:10.685 ","End":"08:13.320","Text":"That\u0027s the end of this lesson."}],"ID":21465},{"Watched":false,"Name":"Dispersion Prism","Duration":"17m 58s","ChapterTopicVideoID":21376,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21376.jpeg","UploadDate":"2020-04-06T22:45:26.9100000","DurationForVideoObject":"PT17M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello. In this lesson,"},{"Start":"00:01.755 ","End":"00:07.830","Text":"we\u0027re going to be dealing with refraction through a triangular prism, and dispersion."},{"Start":"00:07.830 ","End":"00:11.130","Text":"This is an image taken from Wikipedia."},{"Start":"00:11.130 ","End":"00:15.285","Text":"Here we can see that there is a dispersion prism,"},{"Start":"00:15.285 ","End":"00:19.770","Text":"some triangular prism made out of"},{"Start":"00:19.770 ","End":"00:27.065","Text":"a clear material that has a refractive index that is higher than air."},{"Start":"00:27.065 ","End":"00:32.585","Text":"What we can see is that when white light is shone through it,"},{"Start":"00:32.585 ","End":"00:34.895","Text":"a laser beam with white light,"},{"Start":"00:34.895 ","End":"00:42.320","Text":"the result is that we see the colors of the rainbow come out."},{"Start":"00:42.320 ","End":"00:45.785","Text":"We can see suddenly the white light split."},{"Start":"00:45.785 ","End":"00:50.165","Text":"In this lesson, we\u0027re going to be speaking a little bit about this."},{"Start":"00:50.165 ","End":"00:53.045","Text":"First of all, let\u0027s imagine that we have"},{"Start":"00:53.045 ","End":"01:00.485","Text":"a triangular prism and we shine a red laser beam through it,"},{"Start":"01:00.485 ","End":"01:04.710","Text":"let\u0027s say over here"},{"Start":"01:05.470 ","End":"01:12.265","Text":"at some angle to the normal."},{"Start":"01:12.265 ","End":"01:16.135","Text":"I\u0027m not going to do any calculations because that\u0027s not the point of this video."},{"Start":"01:16.135 ","End":"01:22.960","Text":"Because we\u0027re going from a less dense medium air into a denser medium,"},{"Start":"01:22.960 ","End":"01:29.290","Text":"so we would expect our beam to bend towards the normal."},{"Start":"01:29.290 ","End":"01:31.885","Text":"Then as it reaches here,"},{"Start":"01:31.885 ","End":"01:41.590","Text":"we would expect it to continue out like so at the same angle to the"},{"Start":"01:41.590 ","End":"01:43.915","Text":"normal as when it"},{"Start":"01:43.915 ","End":"01:53.400","Text":"approached the prism, something like, so."},{"Start":"01:53.920 ","End":"01:57.080","Text":"Something like this. This is what we would expect."},{"Start":"01:57.080 ","End":"02:01.585","Text":"However, when we shine white light,"},{"Start":"02:01.585 ","End":"02:04.155","Text":"instead of just one color,"},{"Start":"02:04.155 ","End":"02:07.695","Text":"so this is white light,"},{"Start":"02:07.695 ","End":"02:12.485","Text":"then as it reaches the prism, the colors,"},{"Start":"02:12.485 ","End":"02:15.755","Text":"each wavelength that makes up white light,"},{"Start":"02:15.755 ","End":"02:20.180","Text":"white light is made up of all the colors of the rainbow and of course,"},{"Start":"02:20.180 ","End":"02:22.700","Text":"each color has a different wavelength,"},{"Start":"02:22.700 ","End":"02:25.460","Text":"which is why we see the colors differently."},{"Start":"02:25.460 ","End":"02:27.380","Text":"Each color, in this case,"},{"Start":"02:27.380 ","End":"02:29.360","Text":"going through the triangular prism,"},{"Start":"02:29.360 ","End":"02:34.940","Text":"refracts slightly differently at a different angle,"},{"Start":"02:34.940 ","End":"02:38.735","Text":"both inside the prism and when exiting."},{"Start":"02:38.735 ","End":"02:44.240","Text":"Therefore what we get is this rainbow effect over here."},{"Start":"02:45.290 ","End":"02:48.950","Text":"If we just shine a red laser as we saw,"},{"Start":"02:48.950 ","End":"02:51.450","Text":"it\u0027s going to refract,"},{"Start":"02:51.490 ","End":"02:58.075","Text":"just as we have seen in all the previous lessons of this chapter."},{"Start":"02:58.075 ","End":"03:02.100","Text":"However, if we shine a white light it\u0027s going to be"},{"Start":"03:02.100 ","End":"03:05.415","Text":"completely different. Let\u0027s look at this."},{"Start":"03:05.415 ","End":"03:06.770","Text":"White light, as we know,"},{"Start":"03:06.770 ","End":"03:09.195","Text":"is made out of all colors."},{"Start":"03:09.195 ","End":"03:11.850","Text":"We can start from the red."},{"Start":"03:11.850 ","End":"03:14.900","Text":"Here we have a red beam."},{"Start":"03:14.900 ","End":"03:21.035","Text":"Let\u0027s try it over here that approaches the triangular prism."},{"Start":"03:21.035 ","End":"03:26.390","Text":"And we look at the normal to the surface,"},{"Start":"03:26.390 ","End":"03:29.360","Text":"which is something like so."},{"Start":"03:29.360 ","End":"03:31.430","Text":"It\u0027s at this angle."},{"Start":"03:31.430 ","End":"03:38.100","Text":"Then, because we\u0027re going from a rarer medium to a denser medium,"},{"Start":"03:38.810 ","End":"03:42.705","Text":"again, the normal extends in here."},{"Start":"03:42.705 ","End":"03:47.690","Text":"We will expect the red to bend slightly towards the normal,"},{"Start":"03:47.690 ","End":"03:51.650","Text":"and then as it reaches the next interface,"},{"Start":"03:51.650 ","End":"03:56.280","Text":"where this is the normal,"},{"Start":"03:56.280 ","End":"04:00.710","Text":"as it comes from a denser medium to a rarer medium,"},{"Start":"04:00.710 ","End":"04:04.835","Text":"we will expect it to bend away from the normal,"},{"Start":"04:04.835 ","End":"04:11.810","Text":"like so, going like this."},{"Start":"04:11.810 ","End":"04:18.380","Text":"Then another component of white light is the green wavelength."},{"Start":"04:18.380 ","End":"04:23.270","Text":"It comes in right on top of the red because it\u0027s white light,"},{"Start":"04:23.270 ","End":"04:26.115","Text":"so they come in together like so."},{"Start":"04:26.115 ","End":"04:28.265","Text":"Then as it enters the prism,"},{"Start":"04:28.265 ","End":"04:32.065","Text":"it refracts at a slightly different angle."},{"Start":"04:32.065 ","End":"04:37.235","Text":"It will be, let\u0027s say like this."},{"Start":"04:37.235 ","End":"04:40.025","Text":"Then over here again,"},{"Start":"04:40.025 ","End":"04:45.600","Text":"we have our normal, at this point."},{"Start":"04:45.600 ","End":"04:54.030","Text":"It too will refract again in a slightly different way,"},{"Start":"04:54.030 ","End":"04:55.760","Text":"or rather a slightly different angle."},{"Start":"04:55.760 ","End":"04:58.140","Text":"Then on top of this,"},{"Start":"04:58.140 ","End":"05:00.710","Text":"we have the blue wavelength,"},{"Start":"05:00.710 ","End":"05:03.590","Text":"which again comes in like so."},{"Start":"05:03.590 ","End":"05:08.015","Text":"Again, it will refract slightly differently"},{"Start":"05:08.015 ","End":"05:15.040","Text":"inside and refract at a slightly different angle as it exits the prism."},{"Start":"05:15.040 ","End":"05:16.890","Text":"Finally the purple."},{"Start":"05:16.890 ","End":"05:18.230","Text":"I\u0027m just going to draw it in pink,"},{"Start":"05:18.230 ","End":"05:20.210","Text":"but this is meant to be purple."},{"Start":"05:20.210 ","End":"05:25.430","Text":"A similar situation as with the other wavelengths,"},{"Start":"05:25.430 ","End":"05:28.130","Text":"and it will come out like this."},{"Start":"05:28.130 ","End":"05:36.275","Text":"Then what will happen is that if I place a screen over here,"},{"Start":"05:36.275 ","End":"05:40.880","Text":"what I will see is that the red wavelength or"},{"Start":"05:40.880 ","End":"05:48.320","Text":"the red waves will be shone over here like so."},{"Start":"05:48.320 ","End":"05:54.200","Text":"The green color will be shone over here like so."},{"Start":"05:54.200 ","End":"06:00.770","Text":"Similarly, with the blue and similarly with the purple,"},{"Start":"06:00.770 ","End":"06:02.180","Text":"we\u0027re imagining that this is purple."},{"Start":"06:02.180 ","End":"06:05.735","Text":"Of course, over here in between these gaps,"},{"Start":"06:05.735 ","End":"06:08.300","Text":"there\u0027s lots of other different colors."},{"Start":"06:08.300 ","End":"06:10.730","Text":"As we can see in this picture,"},{"Start":"06:10.730 ","End":"06:15.905","Text":"there\u0027s also yellows and oranges in-between here,"},{"Start":"06:15.905 ","End":"06:20.190","Text":"and different magentas and everything."},{"Start":"06:20.190 ","End":"06:29.565","Text":"Here we have all the colors of the rainbow. This is great."},{"Start":"06:29.565 ","End":"06:33.280","Text":"What this is called is dispersion."},{"Start":"06:34.010 ","End":"06:38.110","Text":"I\u0027ll write down the definition."},{"Start":"06:38.360 ","End":"06:45.260","Text":"Dispersion is the splitting of white light into its components via refraction,"},{"Start":"06:45.260 ","End":"06:49.000","Text":"exactly what we\u0027ve seen up until now. Why does this happen?"},{"Start":"06:49.000 ","End":"06:52.270","Text":"This can be explained by Snell\u0027s law,"},{"Start":"06:52.270 ","End":"07:01.605","Text":"so n_1 sine of Theta 1 = n_2 sine of Theta 2."},{"Start":"07:01.605 ","End":"07:04.845","Text":"Now, when we\u0027re coming from air,"},{"Start":"07:04.845 ","End":"07:06.870","Text":"when air is our first medium,"},{"Start":"07:06.870 ","End":"07:09.775","Text":"so n_1, the refractive index of air,"},{"Start":"07:09.775 ","End":"07:13.870","Text":"is approximately equal to 1."},{"Start":"07:14.150 ","End":"07:19.070","Text":"This is the exact same refractive index for red,"},{"Start":"07:19.070 ","End":"07:26.105","Text":"blue, green, purple, and all the colors."},{"Start":"07:26.105 ","End":"07:29.630","Text":"It\u0027s equal to approximately 1."},{"Start":"07:29.630 ","End":"07:40.275","Text":"Therefore, Theta 1 is going to be the same for red, blue,"},{"Start":"07:40.275 ","End":"07:44.850","Text":"green, this is purple, and so on,"},{"Start":"07:44.850 ","End":"07:47.055","Text":"all the other colors,"},{"Start":"07:47.055 ","End":"07:51.540","Text":"because our refractive index is the same for all of them."},{"Start":"07:52.400 ","End":"07:59.105","Text":"This is going to be equal to the refractive index in the second medium."},{"Start":"07:59.105 ","End":"08:04.775","Text":"Now, what we can see is that when the medium is different to air,"},{"Start":"08:04.775 ","End":"08:09.700","Text":"so any other medium aside from air and a vacuum,"},{"Start":"08:09.700 ","End":"08:13.420","Text":"the refractive index is different,"},{"Start":"08:13.420 ","End":"08:18.110","Text":"very slightly, but it is different for each and every color."},{"Start":"08:18.110 ","End":"08:21.750","Text":"There\u0027s going to be n_2 red,"},{"Start":"08:21.750 ","End":"08:25.740","Text":"n_2 blue, and so on and so forth."},{"Start":"08:25.740 ","End":"08:27.695","Text":"They\u0027re going to be slightly different,"},{"Start":"08:27.695 ","End":"08:32.790","Text":"even if they\u0027re in the same medium of let\u0027s say perspex."},{"Start":"08:33.650 ","End":"08:38.615","Text":"Therefore, if the refractive index for each color,"},{"Start":"08:38.615 ","End":"08:47.270","Text":"which is caused by a slightly different wavelength,"},{"Start":"08:47.270 ","End":"08:50.390","Text":"that\u0027s why we see red as red because it has"},{"Start":"08:50.390 ","End":"08:52.790","Text":"a certain wavelength and blue has"},{"Start":"08:52.790 ","End":"08:55.715","Text":"a different wavelength and that\u0027s why we see it as blue."},{"Start":"08:55.715 ","End":"08:59.795","Text":"In order to maintain this equation,"},{"Start":"08:59.795 ","End":"09:05.240","Text":"in order to keep this equality as being true,"},{"Start":"09:05.240 ","End":"09:13.160","Text":"that means that Theta 2 is going to be slightly different for red and Theta 2 for blue,"},{"Start":"09:13.160 ","End":"09:16.800","Text":"and so on and so forth for all the colors."},{"Start":"09:17.090 ","End":"09:20.660","Text":"Basically, in dispersion,"},{"Start":"09:20.660 ","End":"09:22.834","Text":"when we have a white light,"},{"Start":"09:22.834 ","End":"09:25.445","Text":"it has to be white light,"},{"Start":"09:25.445 ","End":"09:28.445","Text":"as it travels through air,"},{"Start":"09:28.445 ","End":"09:33.755","Text":"the refractive index is uniform, it\u0027s the same."},{"Start":"09:33.755 ","End":"09:37.175","Text":"However, when it passes through another medium,"},{"Start":"09:37.175 ","End":"09:38.950","Text":"that is not air,"},{"Start":"09:38.950 ","End":"09:42.995","Text":"the refractive index is slightly different for each color."},{"Start":"09:42.995 ","End":"09:46.370","Text":"Then, in order to maintain Snell\u0027s law,"},{"Start":"09:46.370 ","End":"09:50.075","Text":"Theta 2, the angle of refraction,"},{"Start":"09:50.075 ","End":"09:54.170","Text":"has to change in order to maintain this equation,"},{"Start":"09:54.170 ","End":"09:59.900","Text":"in order to maintain the equality between both sides."},{"Start":"09:59.900 ","End":"10:03.815","Text":"That is why we get these slightly different angles over here."},{"Start":"10:03.815 ","End":"10:09.750","Text":"That is why the colors of the rainbow split out of white light."},{"Start":"10:10.220 ","End":"10:12.890","Text":"Let\u0027s give a little example."},{"Start":"10:12.890 ","End":"10:14.540","Text":"Let\u0027s say we\u0027re dealing with perspex."},{"Start":"10:14.540 ","End":"10:18.890","Text":"Up until now when we\u0027ve been dealing with a laser with one color of light,"},{"Start":"10:18.890 ","End":"10:24.275","Text":"we\u0027ve said that the refractive index for perspex is approximately 1.5,"},{"Start":"10:24.275 ","End":"10:26.660","Text":"but if we\u0027re going to be a little bit more accurate,"},{"Start":"10:26.660 ","End":"10:33.570","Text":"it\u0027s approximately closer to 1.51, for perspex."},{"Start":"10:33.570 ","End":"10:38.975","Text":"However, what we\u0027ll see is that the refractive index for perspex"},{"Start":"10:38.975 ","End":"10:47.655","Text":"for the red color is equal to approximately 1.50."},{"Start":"10:47.655 ","End":"10:53.755","Text":"Then the refractive index for prospects for the other side,"},{"Start":"10:53.755 ","End":"10:57.295","Text":"the other end, if here is red,"},{"Start":"10:57.295 ","End":"10:59.695","Text":"so red is over here."},{"Start":"10:59.695 ","End":"11:04.310","Text":"At the opposite end of the rainbow, we have purple."},{"Start":"11:04.700 ","End":"11:07.694","Text":"The n for purple,"},{"Start":"11:07.694 ","End":"11:09.360","Text":"the refractive index for purple,"},{"Start":"11:09.360 ","End":"11:13.840","Text":"and prospects is approximately 1.52."},{"Start":"11:14.670 ","End":"11:18.490","Text":"What we\u0027ll see is that for every wavelength between,"},{"Start":"11:18.490 ","End":"11:22.960","Text":"so this is red, it\u0027s going to be this."},{"Start":"11:22.960 ","End":"11:30.490","Text":"Then for green, it\u0027s going to be somewhere in-between 1.50 and 1.52."},{"Start":"11:30.490 ","End":"11:37.480","Text":"Similarly, for blue, its n in prospects is going to be slightly different,"},{"Start":"11:37.480 ","End":"11:44.992","Text":"and up until we finally reach purple at the other end of the spectrum to this value."},{"Start":"11:44.992 ","End":"11:48.070","Text":"This very, very slight difference in"},{"Start":"11:48.070 ","End":"11:52.405","Text":"the refractive index for the same material but for different wavelengths,"},{"Start":"11:52.405 ","End":"11:56.980","Text":"is what causes this different angle of diffraction,"},{"Start":"11:56.980 ","End":"12:00.740","Text":"which is what splits up the white light."},{"Start":"12:01.860 ","End":"12:07.450","Text":"If we plug in these numbers to Snell\u0027s equation."},{"Start":"12:07.450 ","End":"12:11.920","Text":"If we have n_1 being of air,"},{"Start":"12:11.920 ","End":"12:15.235","Text":"1 multiplied by sine of,"},{"Start":"12:15.235 ","End":"12:17.440","Text":"let\u0027s say this is 30 degrees."},{"Start":"12:17.440 ","End":"12:20.665","Text":"The white light is approaching at 30 degrees,"},{"Start":"12:20.665 ","End":"12:24.760","Text":"so sine of 30 degrees is equal to n_2."},{"Start":"12:24.760 ","End":"12:27.250","Text":"Now we\u0027re working it out for red,"},{"Start":"12:27.250 ","End":"12:31.750","Text":"so let\u0027s write this for red light, so n_2,"},{"Start":"12:31.750 ","End":"12:39.865","Text":"we can see is equal to 1.5 approximately multiplied by sine of Theta_ 2."},{"Start":"12:39.865 ","End":"12:45.850","Text":"Once we plug this into the calculator we\u0027ll see that Theta_ 2 for the red,"},{"Start":"12:45.850 ","End":"12:50.950","Text":"so the angle of diffraction for the red light is equal"},{"Start":"12:50.950 ","End":"12:58.370","Text":"to approximately 19.471 degrees."},{"Start":"12:58.890 ","End":"13:05.350","Text":"However, if we do the same equation for a purple, so 1,"},{"Start":"13:05.350 ","End":"13:10.360","Text":"the refractive index for air multiplied by sine,"},{"Start":"13:10.360 ","End":"13:14.110","Text":"it\u0027s still 30 degrees because we\u0027re still at the white light."},{"Start":"13:14.110 ","End":"13:16.195","Text":"Then we look over here."},{"Start":"13:16.195 ","End":"13:21.820","Text":"This is going to be equal to the refractive index and prospects for it purple,"},{"Start":"13:21.820 ","End":"13:27.670","Text":"so 1.52 multiplied by sine of Theta_ 2,"},{"Start":"13:27.670 ","End":"13:30.205","Text":"so this is for purple."},{"Start":"13:30.205 ","End":"13:40.540","Text":"Therefore, I will get that Theta_ 2 purple is equal to approximately 19.2 degrees."},{"Start":"13:40.540 ","End":"13:49.105","Text":"The difference between these 2 values is about 0.27 degrees,"},{"Start":"13:49.105 ","End":"13:54.620","Text":"which is a lot if we take a further distance away."},{"Start":"13:56.850 ","End":"14:01.525","Text":"Because if I take the smallest angle between something here,"},{"Start":"14:01.525 ","End":"14:03.445","Text":"there\u0027s no difference barely."},{"Start":"14:03.445 ","End":"14:07.750","Text":"However, if I continue these lines like so,"},{"Start":"14:07.750 ","End":"14:10.540","Text":"suddenly at a distance,"},{"Start":"14:10.540 ","End":"14:13.615","Text":"we get even bigger because this isn\u0027t straight."},{"Start":"14:13.615 ","End":"14:15.565","Text":"This huge distance over here,"},{"Start":"14:15.565 ","End":"14:20.905","Text":"which explains why we can see the different colors so differently."},{"Start":"14:20.905 ","End":"14:23.800","Text":"When we\u0027re dealing with a pane of glass,"},{"Start":"14:23.800 ","End":"14:27.100","Text":"the edges of the glass."},{"Start":"14:27.100 ","End":"14:30.190","Text":"This is the site of the pane."},{"Start":"14:30.190 ","End":"14:32.905","Text":"Are parallel to one another."},{"Start":"14:32.905 ","End":"14:37.300","Text":"Just like we\u0027ve seen before if my ray hits like this,"},{"Start":"14:37.300 ","End":"14:39.670","Text":"it will bend slightly like so,"},{"Start":"14:39.670 ","End":"14:44.200","Text":"and it will come out in the same direction."},{"Start":"14:44.200 ","End":"14:48.850","Text":"Because here everything is parallel and the pain is so thin,"},{"Start":"14:48.850 ","End":"14:52.300","Text":"so the difference over here is so"},{"Start":"14:52.300 ","End":"14:58.315","Text":"small that when it comes out the other side of the pane of glass,"},{"Start":"14:58.315 ","End":"15:02.680","Text":"the different wave colors will be so"},{"Start":"15:02.680 ","End":"15:06.250","Text":"close to one another that my eye won\u0027t be able to tell the difference,"},{"Start":"15:06.250 ","End":"15:10.105","Text":"and it won\u0027t be able to see the splitting of the colors."},{"Start":"15:10.105 ","End":"15:16.164","Text":"However, over here, when we have this situation where we have different angles,"},{"Start":"15:16.164 ","End":"15:25.570","Text":"the normal over here is not parallel to the normal on this side of the prism."},{"Start":"15:25.570 ","End":"15:30.520","Text":"It enhances this very small change in"},{"Start":"15:30.520 ","End":"15:36.385","Text":"angle caused by the different refractive indexes for the different colors,"},{"Start":"15:36.385 ","End":"15:38.020","Text":"so it enhances it."},{"Start":"15:38.020 ","End":"15:41.470","Text":"Also over here hitting slightly,"},{"Start":"15:41.470 ","End":"15:46.780","Text":"very slight different points along the the other edge of the prism."},{"Start":"15:46.780 ","End":"15:53.320","Text":"Then, in turn, these points are refracted slightly differently over here,"},{"Start":"15:53.320 ","End":"15:55.630","Text":"because again it\u0027s at a different angle,"},{"Start":"15:55.630 ","End":"16:01.000","Text":"so we get this much greater difference."},{"Start":"16:01.000 ","End":"16:03.505","Text":"We always have the same difference,"},{"Start":"16:03.505 ","End":"16:08.880","Text":"but because over here the sides are not parallel to one another,"},{"Start":"16:08.880 ","End":"16:18.105","Text":"it enhances the distance between each of the colors diverted propagation,"},{"Start":"16:18.105 ","End":"16:20.480","Text":"which is why we can then see it,"},{"Start":"16:20.480 ","End":"16:23.200","Text":"see this change much further away."},{"Start":"16:23.200 ","End":"16:25.525","Text":"Whereas if we\u0027re looking through a window,"},{"Start":"16:25.525 ","End":"16:28.480","Text":"this effect does not happen,"},{"Start":"16:28.480 ","End":"16:32.960","Text":"or at least it\u0027s not visible to the naked eye."},{"Start":"16:33.810 ","End":"16:40.040","Text":"This is also the reason why we see rainbows in clouds."},{"Start":"16:40.080 ","End":"16:43.940","Text":"The Sun emits white light."},{"Start":"16:45.180 ","End":"16:48.730","Text":"This is the sun,"},{"Start":"16:48.730 ","End":"16:52.825","Text":"and it emits white light."},{"Start":"16:52.825 ","End":"16:56.065","Text":"Now, if it\u0027s being a dry day,"},{"Start":"16:56.065 ","End":"17:00.355","Text":"so the white light just comes to our eye and that\u0027s it,"},{"Start":"17:00.355 ","End":"17:02.470","Text":"we don\u0027t see a rainbow."},{"Start":"17:02.470 ","End":"17:04.525","Text":"If it\u0027s a cloudy day,"},{"Start":"17:04.525 ","End":"17:07.750","Text":"then we won\u0027t really see the sun\u0027s light properly,"},{"Start":"17:07.750 ","End":"17:09.640","Text":"and that\u0027s also why we don\u0027t see a rainbow."},{"Start":"17:09.640 ","End":"17:11.575","Text":"However, if it\u0027s rained,"},{"Start":"17:11.575 ","End":"17:15.055","Text":"then there\u0027s going to be microscopic,"},{"Start":"17:15.055 ","End":"17:18.850","Text":"very tiny water droplets in the air."},{"Start":"17:18.850 ","End":"17:20.710","Text":"If the clouds clear away,"},{"Start":"17:20.710 ","End":"17:23.860","Text":"whilst the air still has this moisture in,"},{"Start":"17:23.860 ","End":"17:28.780","Text":"the white light is refracted in each of these droplets,"},{"Start":"17:28.780 ","End":"17:33.010","Text":"so we get red light,"},{"Start":"17:33.010 ","End":"17:37.225","Text":"green light refracted, blue,"},{"Start":"17:37.225 ","End":"17:39.370","Text":"and all the way to purple."},{"Start":"17:39.370 ","End":"17:42.550","Text":"Of course, we have all the different colors in between,"},{"Start":"17:42.550 ","End":"17:47.245","Text":"and these reach our eye and that is how we see a rainbow."},{"Start":"17:47.245 ","End":"17:52.015","Text":"Whereas if there\u0027s no water droplets does nothing to properly refract the light,"},{"Start":"17:52.015 ","End":"17:53.680","Text":"the white light coming from the sun,"},{"Start":"17:53.680 ","End":"17:56.485","Text":"and therefore, we do not see a rainbow."},{"Start":"17:56.485 ","End":"17:59.360","Text":"That\u0027s the end of this lesson."}],"ID":21466},{"Watched":false,"Name":"Refraction in a Cylinder","Duration":"9m 32s","ChapterTopicVideoID":21377,"CourseChapterTopicPlaylistID":99492,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21377.jpeg","UploadDate":"2020-04-06T22:48:02.3930000","DurationForVideoObject":"PT9M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"Hello. In this lesson we\u0027re going to be learning about refraction."},{"Start":"00:04.620 ","End":"00:10.050","Text":"When we are looking at refraction in a cylinder."},{"Start":"00:10.050 ","End":"00:13.170","Text":"For example, in a cup."},{"Start":"00:13.170 ","End":"00:17.895","Text":"Let\u0027s imagine that right now we\u0027re looking at a cylinder."},{"Start":"00:17.895 ","End":"00:22.920","Text":"This is the picture of the cylinder from the side."},{"Start":"00:22.920 ","End":"00:26.910","Text":"Now we\u0027re going to look from a bird\u0027s-eye view."},{"Start":"00:26.910 ","End":"00:31.980","Text":"Our eye is over here looking down on the cylinder."},{"Start":"00:32.530 ","End":"00:36.769","Text":"Now let\u0027s imagine that on the bottom,"},{"Start":"00:36.769 ","End":"00:39.350","Text":"we have something over here."},{"Start":"00:39.350 ","End":"00:43.480","Text":"We can imagine that it\u0027s a coin, whatever."},{"Start":"00:43.480 ","End":"00:45.575","Text":"Coming from the coin,"},{"Start":"00:45.575 ","End":"00:50.690","Text":"we\u0027re going to have these light rays that are reflected back from it."},{"Start":"00:50.690 ","End":"00:52.805","Text":"From the center of the coin,"},{"Start":"00:52.805 ","End":"00:54.860","Text":"a light ray will propagate like this in"},{"Start":"00:54.860 ","End":"01:00.305","Text":"a straight line and it will hit the interface over here."},{"Start":"01:00.305 ","End":"01:02.630","Text":"This is a cylinder with water."},{"Start":"01:02.630 ","End":"01:06.150","Text":"Let\u0027s just write that there\u0027s water in here."},{"Start":"01:06.150 ","End":"01:13.235","Text":"It will hit the interface between water and air and a 90 degrees to the interface,"},{"Start":"01:13.235 ","End":"01:14.300","Text":"or in other words,"},{"Start":"01:14.300 ","End":"01:17.270","Text":"at a 0 degree angle to the normal, 0 degrees to the normal."},{"Start":"01:17.270 ","End":"01:27.225","Text":"Which means that there\u0027s no refraction and it just continues to the eye like so."},{"Start":"01:27.225 ","End":"01:28.815","Text":"On the other hand,"},{"Start":"01:28.815 ","End":"01:34.055","Text":"from a slightly different angle of reflection,"},{"Start":"01:34.055 ","End":"01:40.895","Text":"another ray will propagate, something like so."},{"Start":"01:40.895 ","End":"01:49.340","Text":"It will reach at some angle to the normal and because we\u0027re going from water to air,"},{"Start":"01:49.340 ","End":"01:54.290","Text":"we can assume that out here we have air unless we\u0027re told otherwise."},{"Start":"01:54.290 ","End":"01:57.890","Text":"We\u0027re going from a denser medium to a rarer medium,"},{"Start":"01:57.890 ","End":"02:01.820","Text":"which means that the light ray"},{"Start":"02:01.820 ","End":"02:08.640","Text":"will bend away from the normal,"},{"Start":"02:08.640 ","End":"02:11.510","Text":"slightly, something like so,"},{"Start":"02:11.510 ","End":"02:13.355","Text":"and it will reach the eye here."},{"Start":"02:13.355 ","End":"02:16.850","Text":"Similarly, over here on this side,"},{"Start":"02:16.850 ","End":"02:22.865","Text":"it will bend away and reach the eye, like so."},{"Start":"02:22.865 ","End":"02:32.230","Text":"Of course, we can just draw another line over here that looks something like this."},{"Start":"02:32.660 ","End":"02:36.330","Text":"This is what is going on."},{"Start":"02:36.330 ","End":"02:40.230","Text":"All these rays have reached the eye,"},{"Start":"02:40.230 ","End":"02:42.290","Text":"and once that happens,"},{"Start":"02:42.290 ","End":"02:47.135","Text":"the eye traces them back as straight lines."},{"Start":"02:47.135 ","End":"02:51.930","Text":"These straight lines are called lines of sight."},{"Start":"02:51.980 ","End":"02:55.470","Text":"Lines of sight, I\u0027m going to draw them in pink."},{"Start":"02:55.470 ","End":"03:01.130","Text":"What we have is the eye traces them back like so."},{"Start":"03:01.130 ","End":"03:05.645","Text":"Through this in pink,"},{"Start":"03:05.645 ","End":"03:10.970","Text":"it goes something like so."},{"Start":"03:10.970 ","End":"03:14.790","Text":"Similarly, with everything else,"},{"Start":"03:14.790 ","End":"03:20.110","Text":"imagine the pink is straight down, like so."},{"Start":"03:21.740 ","End":"03:26.850","Text":"What we have in pink are these lines of sight,"},{"Start":"03:26.850 ","End":"03:29.115","Text":"I\u0027m going to continue."},{"Start":"03:29.115 ","End":"03:32.675","Text":"This line of sight goes like this."},{"Start":"03:32.675 ","End":"03:37.520","Text":"This line of sight goes like so and continuous at a straight line."},{"Start":"03:37.520 ","End":"03:44.115","Text":"All of them are going like this and this line of sight or so continuous, like so."},{"Start":"03:44.115 ","End":"03:49.580","Text":"What we can see is that they all meet in this area over here."},{"Start":"03:49.580 ","End":"03:57.620","Text":"What happens is that we see the coin not as if it is"},{"Start":"03:57.620 ","End":"04:05.360","Text":"at its actual depth inside the cup or inside this cylinder filled with water."},{"Start":"04:05.360 ","End":"04:09.540","Text":"Instead, we see the coin over here."},{"Start":"04:09.540 ","End":"04:14.590","Text":"This is called apparent depth."},{"Start":"04:16.190 ","End":"04:24.325","Text":"The apparent depth is when we have some object submerged at a certain depth underwater."},{"Start":"04:24.325 ","End":"04:26.120","Text":"Here we have a coin and the glass,"},{"Start":"04:26.120 ","End":"04:30.410","Text":"but it could also be corals in the ocean."},{"Start":"04:30.410 ","End":"04:33.305","Text":"If anyone\u0027s been to"},{"Start":"04:33.305 ","End":"04:38.450","Text":"a beautiful paradise beach and been able to look when you\u0027re outside of the water,"},{"Start":"04:38.450 ","End":"04:39.620","Text":"down into the water,"},{"Start":"04:39.620 ","End":"04:43.840","Text":"and see the corals it looks like the corals a much, much shallower."},{"Start":"04:43.840 ","End":"04:48.320","Text":"It looks like the corals are closer to the water surface."},{"Start":"04:48.320 ","End":"04:57.605","Text":"That is exactly because of the refraction,"},{"Start":"04:57.605 ","End":"05:01.490","Text":"and the eye uses these lines of sight,"},{"Start":"05:01.490 ","End":"05:06.690","Text":"which means that the coin submerge in the cup or the corals"},{"Start":"05:06.690 ","End":"05:13.260","Text":"in the ocean have an apparent depth, they look shallower."},{"Start":"05:14.180 ","End":"05:17.115","Text":"This was the bird\u0027s-eye view."},{"Start":"05:17.115 ","End":"05:19.775","Text":"Now let\u0027s give the example."},{"Start":"05:19.775 ","End":"05:23.850","Text":"If we\u0027re looking from a side view."},{"Start":"05:25.130 ","End":"05:28.575","Text":"I\u0027ll draw the eye soon."},{"Start":"05:28.575 ","End":"05:33.960","Text":"Let\u0027s just say that here we have our cup."},{"Start":"05:34.520 ","End":"05:37.265","Text":"At this point over here,"},{"Start":"05:37.265 ","End":"05:39.620","Text":"I have my coin that I\u0027ve put,"},{"Start":"05:39.620 ","End":"05:44.740","Text":"and of course the cup is filled with, let\u0027s say water."},{"Start":"05:44.740 ","End":"05:46.590","Text":"What will happen?"},{"Start":"05:46.590 ","End":"05:48.105","Text":"I\u0027ll draw the rays."},{"Start":"05:48.105 ","End":"05:52.760","Text":"So 1 ray will come out from here and reach this point."},{"Start":"05:52.760 ","End":"05:56.890","Text":"Of course at this point we draw the tangent,"},{"Start":"05:56.890 ","End":"06:00.180","Text":"can draw the tangent in gray."},{"Start":"06:00.180 ","End":"06:05.990","Text":"Here we have the tangent and then we can draw like this,"},{"Start":"06:05.990 ","End":"06:09.185","Text":"the normal to the tangent."},{"Start":"06:09.185 ","End":"06:11.900","Text":"Because we\u0027re coming from water to air,"},{"Start":"06:11.900 ","End":"06:14.870","Text":"so we\u0027re going from a denser to rarer medium."},{"Start":"06:14.870 ","End":"06:25.475","Text":"We would expect the ray to be refracted away from the normal, like so."},{"Start":"06:25.475 ","End":"06:27.290","Text":"Then from a different,"},{"Start":"06:27.290 ","End":"06:33.895","Text":"so here we have another light ray that reaches this point over here."},{"Start":"06:33.895 ","End":"06:38.180","Text":"Now we have to draw the tangent to this point, which is like so."},{"Start":"06:38.180 ","End":"06:43.350","Text":"Then the normal to this tangent is as such."},{"Start":"06:43.350 ","End":"06:47.190","Text":"Again, the light ray is refracted,"},{"Start":"06:47.190 ","End":"06:50.900","Text":"and it\u0027s refracted away from the normal because we\u0027re going from a denser medium,"},{"Start":"06:50.900 ","End":"06:55.450","Text":"to water into a rarer medium, the air."},{"Start":"06:55.450 ","End":"06:59.260","Text":"It will propagate like this."},{"Start":"06:59.870 ","End":"07:03.630","Text":"Let\u0027s just leave this as these 2 points."},{"Start":"07:03.630 ","End":"07:10.160","Text":"Let\u0027s say that here is our eye looking from the side of the glass at the coin."},{"Start":"07:10.160 ","End":"07:13.280","Text":"Again, just like here we have these lines of sight,"},{"Start":"07:13.280 ","End":"07:14.660","Text":"which I\u0027ll draw it in pink."},{"Start":"07:14.660 ","End":"07:19.650","Text":"The eye follows this ray that reached it,"},{"Start":"07:19.650 ","End":"07:26.615","Text":"and then continues in a straight line as a line of sight, like so."},{"Start":"07:26.615 ","End":"07:28.550","Text":"Similarly with this ray,"},{"Start":"07:28.550 ","End":"07:33.285","Text":"straight line as the line of sight,"},{"Start":"07:33.285 ","End":"07:37.205","Text":"and we can see that in the intersection of these 2 lines,"},{"Start":"07:37.205 ","End":"07:42.420","Text":"this is where we\u0027ll think that we see the coin."},{"Start":"07:42.420 ","End":"07:46.880","Text":"To someone looking from the side of the glass,"},{"Start":"07:46.880 ","End":"07:48.350","Text":"the side of the cylinder."},{"Start":"07:48.350 ","End":"07:53.220","Text":"The coin will appear at a different location."},{"Start":"07:54.110 ","End":"07:59.585","Text":"This of course, also happens when I have, let\u0027s say,"},{"Start":"07:59.585 ","End":"08:02.000","Text":"my glass of water,"},{"Start":"08:02.000 ","End":"08:05.630","Text":"and I put inside a long straight object,"},{"Start":"08:05.630 ","End":"08:08.615","Text":"for instance, a straw or a pencil."},{"Start":"08:08.615 ","End":"08:14.870","Text":"So if I place my straw or pencil like so"},{"Start":"08:14.870 ","End":"08:22.485","Text":"into the right now we\u0027ll just draw in blue the water."},{"Start":"08:22.485 ","End":"08:24.420","Text":"That it\u0027s hopefully clear."},{"Start":"08:24.420 ","End":"08:29.389","Text":"This will be where the pencil actually continues."},{"Start":"08:29.389 ","End":"08:32.630","Text":"Of course, it just continues straight down in a straight line."},{"Start":"08:32.630 ","End":"08:38.675","Text":"However, due to this refraction and what we already saw that we\u0027ve seen here,"},{"Start":"08:38.675 ","End":"08:42.155","Text":"where when looking from the side"},{"Start":"08:42.155 ","End":"08:46.570","Text":"of the glass that the coin appears in a different location."},{"Start":"08:46.570 ","End":"08:51.045","Text":"Similarly here, what we\u0027ll see is"},{"Start":"08:51.045 ","End":"08:56.240","Text":"that the pencil is at a different location submerged in the water."},{"Start":"08:56.240 ","End":"08:59.025","Text":"Let\u0027s say that it\u0027s over here."},{"Start":"08:59.025 ","End":"09:01.970","Text":"Then I can erase all of this because we won\u0027t"},{"Start":"09:01.970 ","End":"09:05.195","Text":"actually see this section because of the refraction."},{"Start":"09:05.195 ","End":"09:06.995","Text":"Then if I\u0027m looking from the side,"},{"Start":"09:06.995 ","End":"09:09.365","Text":"it will look like the pencil goes in here and it\u0027s somehow"},{"Start":"09:09.365 ","End":"09:12.795","Text":"bent and continue straight down like this."},{"Start":"09:12.795 ","End":"09:18.860","Text":"In this lesson we\u0027ve seen how we will see an object submerged in water,"},{"Start":"09:18.860 ","End":"09:21.625","Text":"both from a bird\u0027s-eye view and from the side view,"},{"Start":"09:21.625 ","End":"09:29.045","Text":"and why we will get this apparent depth and why we get this bending effect."},{"Start":"09:29.045 ","End":"09:31.920","Text":"That\u0027s the end of this lesson."}],"ID":21467}],"Thumbnail":null,"ID":99492}]

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