[{"Name":"Introduction to Maxwells Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"1 The Differential and Intergral Versions of the Equations","Duration":"16m 23s","ChapterTopicVideoID":21356,"CourseChapterTopicPlaylistID":99487,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/21356.jpeg","UploadDate":"2020-04-06T22:15:13.2870000","DurationForVideoObject":"PT16M23S","Description":null,"MetaTitle":"1 The Differential and Intergral Versions of the Equations: Video + Workbook | Proprep","MetaDescription":"Maxwells Equations - Introduction to Maxwells Equations. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/physics-2-electricity-and-magnetism/maxwells-equations/introduction-to-maxwells-equations/vid21436","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.070","Text":"Hello in this lesson we\u0027re going to be discussing Maxwell\u0027s equations."},{"Start":"00:06.070 ","End":"00:09.180","Text":"For us, this isn\u0027t anything new because all"},{"Start":"00:09.180 ","End":"00:11.870","Text":"of the laws that we\u0027ve been studying up until now,"},{"Start":"00:11.870 ","End":"00:15.140","Text":"have actually been Maxwell\u0027s equations."},{"Start":"00:15.140 ","End":"00:17.175","Text":"What we can see is we have"},{"Start":"00:17.175 ","End":"00:25.860","Text":"the differential version where here we have this Nabla or this Del,"},{"Start":"00:25.860 ","End":"00:28.920","Text":"so we can see that this is differential."},{"Start":"00:28.920 ","End":"00:31.830","Text":"Then we have the integral version where we can"},{"Start":"00:31.830 ","End":"00:34.710","Text":"see that we have all these integral symbols."},{"Start":"00:34.710 ","End":"00:40.770","Text":"Now of course, there\u0027s a way to show that these equations are the same,"},{"Start":"00:40.770 ","End":"00:44.180","Text":"where we convert between the differential version to"},{"Start":"00:44.180 ","End":"00:48.200","Text":"the integral version for all of these equations."},{"Start":"00:48.200 ","End":"00:53.910","Text":"Of course, that is what we\u0027re going to discuss in this lesson."},{"Start":"00:54.140 ","End":"00:57.560","Text":"Let\u0027s just speak about these equations first."},{"Start":"00:57.560 ","End":"01:00.920","Text":"We can see that here we have Div E,"},{"Start":"01:00.920 ","End":"01:04.685","Text":"which is equal to 1 divided by Epsilon naught Rho,"},{"Start":"01:04.685 ","End":"01:10.930","Text":"which is the equation that we saw when trying to find the charge density."},{"Start":"01:10.930 ","End":"01:16.355","Text":"Using this equation and then using Gauss\u0027s theorem,"},{"Start":"01:16.355 ","End":"01:18.875","Text":"we get this over here,"},{"Start":"01:18.875 ","End":"01:23.275","Text":"which of course, we\u0027ve already seen this is Gauss\u0027s law."},{"Start":"01:23.275 ","End":"01:26.930","Text":"Here we have the electric flux,"},{"Start":"01:26.930 ","End":"01:29.785","Text":"and over here we have Qin."},{"Start":"01:29.785 ","End":"01:33.740","Text":"The next equation that we have is this over here,"},{"Start":"01:33.740 ","End":"01:35.660","Text":"Div B, where B is of course,"},{"Start":"01:35.660 ","End":"01:38.360","Text":"the magnetic field which is equal to 0."},{"Start":"01:38.360 ","End":"01:41.975","Text":"Now, this equation we\u0027ve looked at slightly less,"},{"Start":"01:41.975 ","End":"01:44.930","Text":"but we have it anyway."},{"Start":"01:44.930 ","End":"01:50.060","Text":"It\u0027s less useful but just so that you know it and again, using Gauss\u0027s theorem,"},{"Start":"01:50.060 ","End":"01:53.300","Text":"we get the integral version,"},{"Start":"01:53.300 ","End":"01:57.700","Text":"and it\u0027s pretty much the same thing."},{"Start":"01:58.970 ","End":"02:02.850","Text":"Here we have Del cross E,"},{"Start":"02:02.850 ","End":"02:07.550","Text":"the magnetic field, which is equal to negative dB by dt."},{"Start":"02:07.550 ","End":"02:11.060","Text":"This we also haven\u0027t used often,"},{"Start":"02:11.060 ","End":"02:18.270","Text":"but just so you know this equation and then we get to this equation via Stokes\u0027 law."},{"Start":"02:18.640 ","End":"02:23.945","Text":"Through Stokes\u0027 law, we get this equation where this equation we"},{"Start":"02:23.945 ","End":"02:30.060","Text":"already have seen when dealing with time dependent fields."},{"Start":"02:30.740 ","End":"02:38.465","Text":"This equation itself can be manipulated into Faraday\u0027s law,"},{"Start":"02:38.465 ","End":"02:45.270","Text":"where we have Epsilon is equal to negative Phi dot B,"},{"Start":"02:45.270 ","End":"02:50.795","Text":"the negative time derivative of the magnetic flux."},{"Start":"02:50.795 ","End":"02:53.390","Text":"We won\u0027t speak about this right now,"},{"Start":"02:53.390 ","End":"02:57.270","Text":"but just so you know this can also become Faraday\u0027s law."},{"Start":"02:57.560 ","End":"03:00.180","Text":"We\u0027ve already seen this equation."},{"Start":"03:00.180 ","End":"03:01.590","Text":"Again, over here,"},{"Start":"03:01.590 ","End":"03:05.195","Text":"so we have the rotor of the magnetic field,"},{"Start":"03:05.195 ","End":"03:07.775","Text":"which is equal to Mu naught J,"},{"Start":"03:07.775 ","End":"03:12.570","Text":"which is the differential Ampere\u0027s law,"},{"Start":"03:12.570 ","End":"03:16.095","Text":"so Ampere\u0027s differential law,"},{"Start":"03:16.095 ","End":"03:21.360","Text":"and we use this in order to find the current density."},{"Start":"03:22.400 ","End":"03:29.320","Text":"This over here is a little correction that later on we\u0027re going to see how to use this,"},{"Start":"03:29.320 ","End":"03:30.730","Text":"and this is of course,"},{"Start":"03:30.730 ","End":"03:35.780","Text":"Mu, Mu naught like over here."},{"Start":"03:35.820 ","End":"03:38.620","Text":"This is also Mu."},{"Start":"03:38.620 ","End":"03:41.065","Text":"Then by using Stokes\u0027 law,"},{"Start":"03:41.065 ","End":"03:42.640","Text":"we get this over here."},{"Start":"03:42.640 ","End":"03:47.410","Text":"This section over here is,"},{"Start":"03:47.410 ","End":"03:49.940","Text":"of course, Ampere\u0027s law."},{"Start":"03:49.940 ","End":"03:56.780","Text":"This addition over here is Maxwell\u0027s correction."},{"Start":"03:57.230 ","End":"04:02.005","Text":"These 2 equations are used in"},{"Start":"04:02.005 ","End":"04:07.450","Text":"electromagnetism and also in the various courses dealing with waves."},{"Start":"04:07.450 ","End":"04:11.000","Text":"Now we\u0027ve had a little introduction to these equations."},{"Start":"04:11.000 ","End":"04:15.335","Text":"I recommend that you stay and watch how we move"},{"Start":"04:15.335 ","End":"04:20.375","Text":"between these 2 equations or how we convert from here to here."},{"Start":"04:20.375 ","End":"04:25.130","Text":"But it\u0027s not necessary if you don\u0027t have time,"},{"Start":"04:25.130 ","End":"04:28.110","Text":"but I do recommend watching this."},{"Start":"04:29.150 ","End":"04:31.450","Text":"That\u0027s what I\u0027m going to do now."},{"Start":"04:31.450 ","End":"04:34.114","Text":"I\u0027m going to convert"},{"Start":"04:34.114 ","End":"04:42.300","Text":"the first and second differential equations into the integral version and then 3 and 4."},{"Start":"04:44.030 ","End":"04:49.820","Text":"Let\u0027s convert from the differential version to the integral version."},{"Start":"04:49.820 ","End":"04:52.655","Text":"First of all, I\u0027m going to rewrite this."},{"Start":"04:52.655 ","End":"04:57.750","Text":"We have Del dot E is"},{"Start":"04:57.750 ","End":"05:03.285","Text":"equal to 1 divided by Epsilon naught Rho."},{"Start":"05:03.285 ","End":"05:05.985","Text":"Now what I\u0027m going to do,"},{"Start":"05:05.985 ","End":"05:12.785","Text":"is I\u0027m going to integrate both sides with respect to some volume V,"},{"Start":"05:12.785 ","End":"05:18.750","Text":"so dV and also over here dV."},{"Start":"05:18.750 ","End":"05:24.935","Text":"What I\u0027ve done is I\u0027ve taken some kinds of axes,"},{"Start":"05:24.935 ","End":"05:30.900","Text":"and then over here I have some volume over here called V,"},{"Start":"05:30.900 ","End":"05:33.365","Text":"and I\u0027m just integrating along that volume."},{"Start":"05:33.365 ","End":"05:35.260","Text":"I\u0027ve done the same thing to both sides,"},{"Start":"05:35.260 ","End":"05:38.189","Text":"so it\u0027s the exact same equation."},{"Start":"05:39.110 ","End":"05:43.800","Text":"Using this, let\u0027s play around and see what we get."},{"Start":"05:43.800 ","End":"05:51.865","Text":"We have the integral of V of Del dot E dV."},{"Start":"05:51.865 ","End":"05:54.510","Text":"I\u0027ve just rewritten this over here."},{"Start":"05:54.510 ","End":"06:03.600","Text":"Using Gauss\u0027s theorem, it says that if we have Del dot E dV and we integrate,"},{"Start":"06:03.600 ","End":"06:11.400","Text":"so the integral of this differential dot with respect to volume,"},{"Start":"06:11.400 ","End":"06:20.925","Text":"then this is equal to the integral with respect to surface area of E dot ds."},{"Start":"06:20.925 ","End":"06:26.910","Text":"This move from here to here is using Gauss\u0027s theorem."},{"Start":"06:27.800 ","End":"06:32.374","Text":"If this side was integrating this entire volume,"},{"Start":"06:32.374 ","End":"06:35.780","Text":"so now I\u0027m doing a surface area integral,"},{"Start":"06:35.780 ","End":"06:41.940","Text":"so it\u0027s the surface area that encompasses this volume."},{"Start":"06:41.940 ","End":"06:45.460","Text":"That\u0027s this side over here."},{"Start":"06:48.110 ","End":"06:56.420","Text":"Obviously, if this surface area is the surface that is containing this volume so,"},{"Start":"06:56.420 ","End":"07:00.080","Text":"of course, the surface area must be a closed surface."},{"Start":"07:00.080 ","End":"07:03.395","Text":"That means that we just add in this circle over here."},{"Start":"07:03.395 ","End":"07:07.636","Text":"Now we can already see that this is equal to this side,"},{"Start":"07:07.636 ","End":"07:11.960","Text":"so this and this are the same."},{"Start":"07:11.960 ","End":"07:15.920","Text":"Then what we have over here 1 divided by Epsilon naught"},{"Start":"07:15.920 ","End":"07:19.970","Text":"Rho is simply the same thing as this."},{"Start":"07:19.970 ","End":"07:24.170","Text":"The Epsilon 1 divided by Epsilon naught is a"},{"Start":"07:24.170 ","End":"07:29.550","Text":"constant so we can take it outside of the integral."},{"Start":"07:29.930 ","End":"07:33.330","Text":"Imagine taking this outside of the integral,"},{"Start":"07:33.330 ","End":"07:34.915","Text":"so then it looks like so."},{"Start":"07:34.915 ","End":"07:36.730","Text":"Then we have Rho dV,"},{"Start":"07:36.730 ","End":"07:38.815","Text":"which is exactly this."},{"Start":"07:38.815 ","End":"07:42.045","Text":"We also call this,"},{"Start":"07:42.045 ","End":"07:45.010","Text":"and we\u0027re of course integrating along the volume."},{"Start":"07:45.010 ","End":"07:49.105","Text":"All of this is what up until now we\u0027ve called Qin."},{"Start":"07:49.105 ","End":"07:52.240","Text":"What we can see is that this is the magnetic,"},{"Start":"07:52.240 ","End":"07:55.340","Text":"the electric flux, sorry,"},{"Start":"07:55.340 ","End":"07:59.230","Text":"so this over here is the magnetic flux,"},{"Start":"07:59.230 ","End":"08:02.775","Text":"this entire equation over here."},{"Start":"08:02.775 ","End":"08:07.035","Text":"Then here we have Qin divided by Epsilon naught."},{"Start":"08:07.035 ","End":"08:12.430","Text":"This is just Gauss\u0027s law as we\u0027ve seen."},{"Start":"08:12.860 ","End":"08:16.545","Text":"Now, of course, here for Qin,"},{"Start":"08:16.545 ","End":"08:21.680","Text":"what we have is the charge enclosed in a volume and, of course,"},{"Start":"08:21.680 ","End":"08:24.650","Text":"if we have just point charges so we just"},{"Start":"08:24.650 ","End":"08:28.310","Text":"add up the various point charges and if we have a surface area,"},{"Start":"08:28.310 ","End":"08:29.714","Text":"so we just change,"},{"Start":"08:29.714 ","End":"08:31.490","Text":"or a surface charge density,"},{"Start":"08:31.490 ","End":"08:34.970","Text":"so we just changed the integral to be over surface area."},{"Start":"08:34.970 ","End":"08:43.790","Text":"But what we\u0027re doing is we\u0027re summing or integrating along the charges enclosed in here."},{"Start":"08:43.790 ","End":"08:48.080","Text":"Whether it\u0027s a whole volume or just the surface area charge that is"},{"Start":"08:48.080 ","End":"08:53.805","Text":"enclosed in this red ball, if you want to call it."},{"Start":"08:53.805 ","End":"08:56.420","Text":"That\u0027s what we\u0027re doing here, and this is our Qin just like"},{"Start":"08:56.420 ","End":"09:00.060","Text":"we\u0027ve seen in previous chapters."},{"Start":"09:01.050 ","End":"09:08.000","Text":"Okay, so now let\u0027s take a look at equation number 2."},{"Start":"09:08.760 ","End":"09:12.880","Text":"We\u0027ll first show this side first."},{"Start":"09:12.880 ","End":"09:14.455","Text":"Okay, so we\u0027ll play around."},{"Start":"09:14.455 ","End":"09:16.285","Text":"Again, I\u0027ll rewrite it."},{"Start":"09:16.285 ","End":"09:20.550","Text":"We have Del.B,"},{"Start":"09:20.550 ","End":"09:23.595","Text":"which is equal to 0."},{"Start":"09:23.595 ","End":"09:25.440","Text":"Again, just like here,"},{"Start":"09:25.440 ","End":"09:29.460","Text":"I\u0027ll integrate with respect to volume and I\u0027m doing"},{"Start":"09:29.460 ","End":"09:34.570","Text":"the same thing to both sides so we have the exact same equation."},{"Start":"09:34.570 ","End":"09:38.515","Text":"Of course, if I\u0027m integrating along 0 dV,"},{"Start":"09:38.515 ","End":"09:40.675","Text":"this is equal to 0."},{"Start":"09:40.675 ","End":"09:44.035","Text":"This, we already get this over here,"},{"Start":"09:44.035 ","End":"09:47.140","Text":"and now we want to get from here to here."},{"Start":"09:47.140 ","End":"09:52.390","Text":"We\u0027re going to again use Gauss\u0027s theorem just like we did over here."},{"Start":"09:52.390 ","End":"09:57.895","Text":"Just like here, we saw the integral with respect to volume of"},{"Start":"09:57.895 ","End":"10:05.575","Text":"Del.E was equal to the closed circle integral with respect to surface area."},{"Start":"10:05.575 ","End":"10:13.510","Text":"Here we have this, an integral with respect to volume of Del dot a field DV."},{"Start":"10:13.510 ","End":"10:16.450","Text":"That means that this is going to be equal to"},{"Start":"10:16.450 ","End":"10:23.380","Text":"the closed circuit integral of surface area of the field."},{"Start":"10:23.380 ","End":"10:28.540","Text":"In this case, it\u0027s B.ds."},{"Start":"10:28.540 ","End":"10:33.380","Text":"Then here we get exactly this."},{"Start":"10:33.450 ","End":"10:38.695","Text":"These 2, we just use Gauss\u0027s theorem in order to"},{"Start":"10:38.695 ","End":"10:43.885","Text":"convert between the differential and the integral versions."},{"Start":"10:43.885 ","End":"10:46.450","Text":"Now we\u0027re going to look at equations 3 and 4,"},{"Start":"10:46.450 ","End":"10:49.660","Text":"where we do pretty much the same trick except instead of"},{"Start":"10:49.660 ","End":"10:54.530","Text":"using Gauss\u0027s theorem, we use Stokes\u0027."},{"Start":"10:55.260 ","End":"10:59.095","Text":"This is equation number 3,"},{"Start":"10:59.095 ","End":"11:01.120","Text":"and as we\u0027ll see,"},{"Start":"11:01.120 ","End":"11:06.340","Text":"it\u0027s exactly like what we saw previously."},{"Start":"11:06.340 ","End":"11:10.240","Text":"However, here we just use Stokes\u0027 law instead."},{"Start":"11:10.240 ","End":"11:17.905","Text":"Again, I\u0027m going to write out this side of the equation first."},{"Start":"11:17.905 ","End":"11:24.070","Text":"This is equal to the negative time"},{"Start":"11:24.070 ","End":"11:30.450","Text":"derivative of the magnetic field and then I\u0027m just going to integrate,"},{"Start":"11:30.450 ","End":"11:34.620","Text":"but this time with respect to surface area."},{"Start":"11:34.620 ","End":"11:39.770","Text":"We\u0027re integrating with respect to surface area."},{"Start":"11:39.770 ","End":"11:48.670","Text":"Also over here, ds done the same to both sides of the equations haven\u0027t changed."},{"Start":"11:48.670 ","End":"11:55.900","Text":"Okay, so it\u0027s as if we look over here at these axes and we\u0027ve"},{"Start":"11:55.900 ","End":"12:02.740","Text":"taken some kind of surface area and we\u0027re integrating with respect to it."},{"Start":"12:02.740 ","End":"12:05.890","Text":"Now according to Stokes\u0027 law,"},{"Start":"12:05.890 ","End":"12:12.790","Text":"a surface integral of the rotor of some kind of field."},{"Start":"12:12.790 ","End":"12:16.870","Text":"This is the surface integral of the rotor,"},{"Start":"12:16.870 ","End":"12:18.790","Text":"of the electric field,"},{"Start":"12:18.790 ","End":"12:24.310","Text":"is equal to the line"},{"Start":"12:24.310 ","End":"12:32.560","Text":"integral of that same vector field dot dL."},{"Start":"12:32.560 ","End":"12:36.370","Text":"What we\u0027ve done is we\u0027re just"},{"Start":"12:36.370 ","End":"12:42.310","Text":"taking the outskirts of the surface area and we\u0027re looking at"},{"Start":"12:42.310 ","End":"12:47.710","Text":"the tangent to this over"},{"Start":"12:47.710 ","End":"12:53.830","Text":"here that is encompassing the surface area so it\u0027s the line integral."},{"Start":"12:53.830 ","End":"12:56.605","Text":"That\u0027s all we have over here, and of course,"},{"Start":"12:56.605 ","End":"12:59.590","Text":"it\u0027s a closed integral because it encloses"},{"Start":"12:59.590 ","End":"13:05.390","Text":"this original surface so we put a circle over here."},{"Start":"13:06.450 ","End":"13:11.800","Text":"All right, so this was just done via Stokes\u0027 law and then we get"},{"Start":"13:11.800 ","End":"13:17.515","Text":"to this side of the equation over here."},{"Start":"13:17.515 ","End":"13:20.665","Text":"Of course, what we did to both sides over here."},{"Start":"13:20.665 ","End":"13:24.970","Text":"Here we have the integral with respect to surface area,"},{"Start":"13:24.970 ","End":"13:26.995","Text":"which is the same as over here,"},{"Start":"13:26.995 ","End":"13:33.640","Text":"of the negative time derivative of the magnetic field ds."},{"Start":"13:33.640 ","End":"13:38.815","Text":"This is the exact same thing as this."},{"Start":"13:38.815 ","End":"13:43.435","Text":"Okay, so now we\u0027ve showed through Stokes\u0027 law how we get from this side,"},{"Start":"13:43.435 ","End":"13:44.995","Text":"the differential side,"},{"Start":"13:44.995 ","End":"13:48.160","Text":"to that integral side and now we\u0027re"},{"Start":"13:48.160 ","End":"13:51.775","Text":"going to do the exact same thing with equation number 4."},{"Start":"13:51.775 ","End":"13:58.345","Text":"Again, converting between the differential to the integral using Stokes\u0027 law."},{"Start":"13:58.345 ","End":"14:02.350","Text":"Again, we\u0027re going to write out the left side over here."},{"Start":"14:02.350 ","End":"14:07.915","Text":"Here we have the rotor of the magnetic field,"},{"Start":"14:07.915 ","End":"14:11.365","Text":"which is equal to Mu."},{"Start":"14:11.365 ","End":"14:17.208","Text":"These are Mu\u0027s, Mu naught J plus Mu"},{"Start":"14:17.208 ","End":"14:23.810","Text":"naught Epsilon naught dE by dt."},{"Start":"14:23.820 ","End":"14:30.385","Text":"What I\u0027m going to do again is I\u0027m going to integrate with respect to surface area."},{"Start":"14:30.385 ","End":"14:35.838","Text":"Again, then all of this ds."},{"Start":"14:35.838 ","End":"14:40.765","Text":"First of all, if we open up these brackets,"},{"Start":"14:40.765 ","End":"14:45.715","Text":"so we take out the Mu naught outside because this is a common factor,"},{"Start":"14:45.715 ","End":"14:52.705","Text":"and it\u0027s a constant so we can take it out of the integration sign,"},{"Start":"14:52.705 ","End":"14:58.555","Text":"and then we\u0027ll have Jds plus again the Mu naught,"},{"Start":"14:58.555 ","End":"15:02.380","Text":"the integral of Epsilon naught dE by dt,"},{"Start":"15:02.380 ","End":"15:04.329","Text":"dE by dt ds."},{"Start":"15:04.329 ","End":"15:08.830","Text":"We can see already that if we just open up the brackets,"},{"Start":"15:08.830 ","End":"15:10.795","Text":"this is equal to this side."},{"Start":"15:10.795 ","End":"15:16.030","Text":"Now we just want to get this side over here on the left to be equal this."},{"Start":"15:16.030 ","End":"15:25.990","Text":"Again, we look"},{"Start":"15:25.990 ","End":"15:30.595","Text":"at an axes and we have some kind of surface area,"},{"Start":"15:30.595 ","End":"15:35.755","Text":"so again, we\u0027re going to look at the line integral around it."},{"Start":"15:35.755 ","End":"15:38.020","Text":"According to Stokes\u0027,"},{"Start":"15:38.020 ","End":"15:42.100","Text":"this integral on the surface area is going to be equal"},{"Start":"15:42.100 ","End":"15:47.152","Text":"to a closed circuit integral along the line integral"},{"Start":"15:47.152 ","End":"15:52.660","Text":"enclosing this surface area and it is just going to"},{"Start":"15:52.660 ","End":"15:58.510","Text":"be equal to the vector field dot dL."},{"Start":"15:58.510 ","End":"16:07.850","Text":"This is from Stokes\u0027 and then we get that this over here is equal to this."},{"Start":"16:08.070 ","End":"16:12.940","Text":"Now we\u0027ve shown how to convert from each of"},{"Start":"16:12.940 ","End":"16:16.975","Text":"Maxwell\u0027s differential equations into"},{"Start":"16:16.975 ","End":"16:23.390","Text":"the integral form and so that means that this is the end of this lesson."}],"ID":21436},{"Watched":false,"Name":"2 Maxwell_s Third Equation and Faraday_s Law","Duration":"6m 44s","ChapterTopicVideoID":21357,"CourseChapterTopicPlaylistID":99487,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"Hello. In the previous lesson,"},{"Start":"00:02.010 ","End":"00:04.950","Text":"we were looking at Maxwell\u0027s equations and how to"},{"Start":"00:04.950 ","End":"00:09.240","Text":"convert from the differential form into the integral form."},{"Start":"00:09.240 ","End":"00:13.950","Text":"Maxwell\u0027s equation Number 3 had another version where we"},{"Start":"00:13.950 ","End":"00:18.405","Text":"could convert from the integral form into Faraday\u0027s law."},{"Start":"00:18.405 ","End":"00:22.420","Text":"This video is going to discuss this."},{"Start":"00:22.520 ","End":"00:27.000","Text":"First of all, I\u0027m going to cross this out because we\u0027re not looking at this,"},{"Start":"00:27.000 ","End":"00:31.605","Text":"we\u0027re just looking at this conversion from here to here."},{"Start":"00:31.605 ","End":"00:35.000","Text":"First of all, here we have the time"},{"Start":"00:35.000 ","End":"00:38.735","Text":"derivative or the negative time derivative of the magnetic flux."},{"Start":"00:38.735 ","End":"00:42.215","Text":"First of all, let\u0027s see what the magnetic flux is."},{"Start":"00:42.215 ","End":"00:47.550","Text":"It\u0027s equal to the integral of B.ds,"},{"Start":"00:47.550 ","End":"00:49.530","Text":"so the surface integral."},{"Start":"00:49.530 ","End":"00:53.030","Text":"Now if we want to take the time derivative of this,"},{"Start":"00:53.030 ","End":"00:54.185","Text":"we\u0027re going to have,"},{"Start":"00:54.185 ","End":"01:01.145","Text":"this is equal to d by dt of the magnetic flux,"},{"Start":"01:01.145 ","End":"01:07.370","Text":"which is going to give us d by dt of this over here,"},{"Start":"01:07.370 ","End":"01:11.310","Text":"the integral of B.ds."},{"Start":"01:13.370 ","End":"01:16.190","Text":"As we can see over here,"},{"Start":"01:16.190 ","End":"01:20.795","Text":"we have the d by dt outside of the integral sign."},{"Start":"01:20.795 ","End":"01:24.440","Text":"However, over here, we can see that the d by dt,"},{"Start":"01:24.440 ","End":"01:27.770","Text":"the time derivative, is within the integral sign."},{"Start":"01:27.770 ","End":"01:29.350","Text":"Now this isn\u0027t trivial,"},{"Start":"01:29.350 ","End":"01:33.065","Text":"we can just write this out inside the integral sign,"},{"Start":"01:33.065 ","End":"01:35.240","Text":"so we have to change it."},{"Start":"01:35.240 ","End":"01:40.115","Text":"By moving this inside to make this equation,"},{"Start":"01:40.115 ","End":"01:43.680","Text":"that means we\u0027re changing this side."},{"Start":"01:43.720 ","End":"01:47.415","Text":"To get from here to here,"},{"Start":"01:47.415 ","End":"01:51.360","Text":"so that means moving the d by dt outside,"},{"Start":"01:51.360 ","End":"01:56.670","Text":"we have to add some correction over here to this side."},{"Start":"01:56.670 ","End":"02:04.095","Text":"What we\u0027re going to get is we\u0027re going from here to here."},{"Start":"02:04.095 ","End":"02:09.950","Text":"What we\u0027ll have is the closed loop integral of E"},{"Start":"02:09.950 ","End":"02:16.530","Text":"plus v cross B dl,"},{"Start":"02:16.530 ","End":"02:21.020","Text":"and this is going to be equal to this over here,"},{"Start":"02:21.020 ","End":"02:22.115","Text":"but with a minus."},{"Start":"02:22.115 ","End":"02:30.900","Text":"Negative d by dt integral of B.ds."},{"Start":"02:33.950 ","End":"02:41.730","Text":"We went from this step and we took the d by dt out to make it look like this,"},{"Start":"02:41.730 ","End":"02:47.665","Text":"so we continued the minus and because we take the d by dt out of the integration sign,"},{"Start":"02:47.665 ","End":"02:51.190","Text":"we add on this correction over here,"},{"Start":"02:51.190 ","End":"02:55.200","Text":"and then this is our EMF,"},{"Start":"02:55.200 ","End":"02:58.150","Text":"and then we have the minus that we carry on over here,"},{"Start":"02:58.150 ","End":"03:02.275","Text":"and then we have d by dt of B.ds,"},{"Start":"03:02.275 ","End":"03:05.050","Text":"which is d by dt of the magnetic flux,"},{"Start":"03:05.050 ","End":"03:10.410","Text":"or in other words, the time derivative of the magnetic flux."},{"Start":"03:10.410 ","End":"03:14.650","Text":"Then what we can see is we got to this equation over here."},{"Start":"03:14.650 ","End":"03:22.520","Text":"Now, I\u0027m going to show the mathematics of how we get this correction over here,"},{"Start":"03:22.520 ","End":"03:24.095","Text":"this v cross B."},{"Start":"03:24.095 ","End":"03:25.370","Text":"If you don\u0027t want to see it,"},{"Start":"03:25.370 ","End":"03:28.380","Text":"you can move on to the next video."},{"Start":"03:28.760 ","End":"03:32.565","Text":"Let\u0027s show this addition."},{"Start":"03:32.565 ","End":"03:35.705","Text":"If we have this equation over here,"},{"Start":"03:35.705 ","End":"03:37.610","Text":"let\u0027s write it out over here,"},{"Start":"03:37.610 ","End":"03:47.610","Text":"we have d by dt of the integral of B.ds."},{"Start":"03:49.310 ","End":"03:54.695","Text":"Let\u0027s assume that our magnetic field is constant."},{"Start":"03:54.695 ","End":"04:00.065","Text":"If we do that, we can just write this as d by dt,"},{"Start":"04:00.065 ","End":"04:02.825","Text":"and if we have a constant magnetic field,"},{"Start":"04:02.825 ","End":"04:11.290","Text":"this will be B dot the total surface area."},{"Start":"04:14.000 ","End":"04:25.220","Text":"Then we can rewrite this using the chain rule as dB by dt"},{"Start":"04:25.220 ","End":"04:32.820","Text":"multiplied by the surface area plus B multiplied"},{"Start":"04:32.820 ","End":"04:41.460","Text":"by the ds by dt."},{"Start":"04:41.460 ","End":"04:49.620","Text":"This dB by dt multiplied by the surface area is just this over here."},{"Start":"04:50.720 ","End":"04:54.780","Text":"We can see how we got this equation over here,"},{"Start":"04:54.780 ","End":"05:04.140","Text":"but now we can see that we have this additional element in our equation."},{"Start":"05:04.140 ","End":"05:07.115","Text":"What is this? Here we have the magnetic field"},{"Start":"05:07.115 ","End":"05:11.060","Text":"multiplied by the time derivative of the surface area."},{"Start":"05:11.060 ","End":"05:15.990","Text":"In other words, the change in the surface area."},{"Start":"05:16.610 ","End":"05:24.980","Text":"This change in surface area is expressed by this over here, this velocity vector."},{"Start":"05:24.980 ","End":"05:27.750","Text":"This v is for velocity."},{"Start":"05:28.550 ","End":"05:36.380","Text":"The velocity of the edges of whatever surface we\u0027re integrating according to it,"},{"Start":"05:36.380 ","End":"05:42.380","Text":"so that is equal to the change in surface area over a period of time."},{"Start":"05:42.380 ","End":"05:45.649","Text":"Now of course, if the surface area isn\u0027t changing,"},{"Start":"05:45.649 ","End":"05:52.805","Text":"then this term will cancel out and we\u0027ll be left with the exact same equation."},{"Start":"05:52.805 ","End":"05:56.329","Text":"However, if the surface area is changing,"},{"Start":"05:56.329 ","End":"05:58.400","Text":"then we have this addition over here,"},{"Start":"05:58.400 ","End":"06:02.395","Text":"which is expressed as v cross B."},{"Start":"06:02.395 ","End":"06:10.745","Text":"This over here is the same as v cross B,"},{"Start":"06:10.745 ","End":"06:14.310","Text":"where of course v is the velocity."},{"Start":"06:14.780 ","End":"06:18.290","Text":"The moment that I take out this derivative,"},{"Start":"06:18.290 ","End":"06:22.190","Text":"this d by dt outside of the integral sign,"},{"Start":"06:22.190 ","End":"06:26.464","Text":"then I have to do it this equation."},{"Start":"06:26.464 ","End":"06:28.250","Text":"It will look like this,"},{"Start":"06:28.250 ","End":"06:30.230","Text":"and then via the chain rule,"},{"Start":"06:30.230 ","End":"06:35.935","Text":"I can open it up to get this and that is why we have this correction over here."},{"Start":"06:35.935 ","End":"06:42.185","Text":"That\u0027s how we get from this stage over to here, Faraday\u0027s law."},{"Start":"06:42.185 ","End":"06:45.240","Text":"That\u0027s the end of this lesson."}],"ID":21437}],"Thumbnail":null,"ID":99487}]

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[21436,21437];

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