1 00:00:00,060 --> 00:00:05,370 Now that we have good qualitative understanding where the Maxwell's equations come from, I want to 2 00:00:05,370 --> 00:00:06,690 show you something different. 3 00:00:08,190 --> 00:00:12,960 So far we have introduced to Maxwell's equations in the differential formulation. 4 00:00:13,800 --> 00:00:19,230 So as you can see on the right hand side of these equations, there are quantities like the charged 5 00:00:19,230 --> 00:00:26,760 density or the magnetic fields or also E and J and all of these depend not only on the time, but also 6 00:00:26,760 --> 00:00:27,960 on the position vector. 7 00:00:28,830 --> 00:00:35,100 So what this means is if we look at one particular position vector, we can calculate also the left 8 00:00:35,100 --> 00:00:42,180 hand side of these equations and we can determine E and B, and so we know what E and B look like at 9 00:00:42,180 --> 00:00:43,990 this particular position vector. 10 00:00:44,730 --> 00:00:47,660 So it's something like a local formulation. 11 00:00:49,020 --> 00:00:57,240 However, we can also integrate over these equations and so we are not restricted to one particular 12 00:00:57,240 --> 00:01:04,720 position vector, but it's more like an integral formulation, like it's the same equation, but from 13 00:01:04,720 --> 00:01:06,830 a different point of view, so to say. 14 00:01:07,800 --> 00:01:14,610 So in this lecture, I want to show you how we can derive these four integral formulations of the Maxwell 15 00:01:14,610 --> 00:01:15,210 equations. 16 00:01:16,740 --> 00:01:20,200 And for this, we need two laws from mathematics. 17 00:01:20,760 --> 00:01:22,590 The first is divorce law. 18 00:01:23,490 --> 00:01:27,290 And this is actually quite difficult to derive. 19 00:01:27,300 --> 00:01:32,940 So I hope that it's sufficient for you that you just believe me that this is true from the mathematical 20 00:01:32,940 --> 00:01:33,330 side. 21 00:01:33,330 --> 00:01:38,310 And if not, you can look up proofs for this law on the Internet or in a textbook. 22 00:01:38,610 --> 00:01:40,140 But it is quite difficult. 23 00:01:40,770 --> 00:01:46,260 But here, I want to just just show you the result and want to explain to you what this equation means. 24 00:01:47,520 --> 00:01:53,730 So Gaza's law tells us that when we integrate over the divergence of some vector. 25 00:01:54,480 --> 00:01:58,980 So this is here, the integral over in Cartesian coordinates. 26 00:01:59,010 --> 00:02:01,130 This would be the X divided. 27 00:02:01,620 --> 00:02:08,820 So I just tried D.V., which is integral over the volume, and we integrate over some volume that we 28 00:02:08,820 --> 00:02:09,760 have defined. 29 00:02:09,780 --> 00:02:14,790 For example, it could be such a shape, but it could also be any other smooth shape. 30 00:02:15,600 --> 00:02:17,740 So that's a three dimensional integral. 31 00:02:18,720 --> 00:02:26,940 So did that Virgin's of this vector A is equal to the Vector eight itself, but now it's not a volume 32 00:02:26,940 --> 00:02:31,770 integral anymore, but it's an integral over the surface of this volume. 33 00:02:32,130 --> 00:02:34,980 So this is what I have noted here with this. 34 00:02:35,460 --> 00:02:40,720 And here the we integrate over this area element. 35 00:02:40,770 --> 00:02:41,980 So this D. 36 00:02:41,990 --> 00:02:52,110 S means or is characterized by such an element of the surface, and it is characterized, of course, 37 00:02:52,110 --> 00:02:57,700 by the absolute value of this element and a normal vector, which is always perpendicular. 38 00:02:58,620 --> 00:03:05,790 So you could say you have your your your vector field, which depends, of course, on on our on the 39 00:03:05,790 --> 00:03:06,620 position vector. 40 00:03:06,900 --> 00:03:11,070 So it can be different inside of this volume and on the surface of this volume. 41 00:03:11,820 --> 00:03:21,750 Now you calculate here the the scale of product with this vector A and this surface, which is essentially 42 00:03:21,750 --> 00:03:28,890 just the normal of the surface, then you integrate over the whole surface and it turns out that the 43 00:03:28,890 --> 00:03:32,100 left hand side is equal to the right hand side of this equation. 44 00:03:33,850 --> 00:03:40,750 So now that has used his theorem and apply it to our Maxwell's equations, because maybe you remember 45 00:03:40,750 --> 00:03:49,690 this, the first two Maxwell equations were divergences of E and B, so let us first take the divergence 46 00:03:49,690 --> 00:03:57,940 of E, which was equal to a constant times, the charge density row of R so we can rewrite this. 47 00:03:57,940 --> 00:04:00,000 We just multiply by Epsilon zero. 48 00:04:00,010 --> 00:04:01,890 So nothing really happened here. 49 00:04:02,320 --> 00:04:09,220 And then of course, since this is true for every position vector, we can just we can just integrate 50 00:04:09,220 --> 00:04:10,790 over our. 51 00:04:10,840 --> 00:04:13,750 So basically we integrate over the whole volume. 52 00:04:14,620 --> 00:04:16,600 So again, this could be any volume. 53 00:04:16,600 --> 00:04:19,390 As long as it's smooth, it could be a sphere. 54 00:04:19,390 --> 00:04:20,340 It could be a cube. 55 00:04:20,830 --> 00:04:21,930 Doesn't really matter here. 56 00:04:21,940 --> 00:04:23,200 It's true for any volume. 57 00:04:23,590 --> 00:04:28,150 And everything that is done here is just adding the integral. 58 00:04:29,560 --> 00:04:33,310 And I think you you know, this an integral. 59 00:04:33,310 --> 00:04:41,400 You could also think of some of this integral here at several different positions inside of the volume. 60 00:04:41,770 --> 00:04:46,810 And if you make these divi volumes very, very small, then this becomes the integral. 61 00:04:48,100 --> 00:04:52,540 So this is, I think, quite straightforward to understand what happened here. 62 00:04:52,540 --> 00:04:57,610 I have just introduced the integral because it is true for every position. 63 00:04:57,620 --> 00:04:59,970 So it must also be true for the integral. 64 00:05:00,910 --> 00:05:08,560 And now we apply the Gauss Theorem where we say, OK, let the right hand side does not really change, 65 00:05:09,100 --> 00:05:16,420 but the left hand side here, we change the divergence of E to E itself and we change the volume integral 66 00:05:16,420 --> 00:05:17,740 to the surface integral. 67 00:05:19,030 --> 00:05:24,410 And the right hand side, um, is the integral over the charged density. 68 00:05:24,430 --> 00:05:25,660 This is the same as here. 69 00:05:26,120 --> 00:05:32,590 And if you think about it, if we integrate over the density of our whole volume, then this is just 70 00:05:32,590 --> 00:05:35,140 the total charge in our volume. 71 00:05:35,950 --> 00:05:44,470 So we know that a surface integral of the electric field is equal to the absolute charged in our volume. 72 00:05:44,470 --> 00:05:49,840 B is the integral version of this differential maximal equation. 73 00:05:51,950 --> 00:05:57,230 Now for the electric field, it is very similar, but it's a bit more easy, actually, because the 74 00:05:57,230 --> 00:06:00,230 right hand side is zero in this case. 75 00:06:00,960 --> 00:06:02,250 So we do the same thing. 76 00:06:02,270 --> 00:06:04,800 We know that this is true for every position vector. 77 00:06:05,150 --> 00:06:07,660 So it must also be true for the integral. 78 00:06:07,970 --> 00:06:10,840 And now we can apply Goussis theorem here. 79 00:06:11,000 --> 00:06:18,590 So the divergence of view becomes just beat and the volume integral becomes integral over the surface 80 00:06:18,590 --> 00:06:18,960 area. 81 00:06:20,720 --> 00:06:27,020 So we have already derived two of the integral versions of the Maxwell equations. 82 00:06:27,440 --> 00:06:28,960 Now that is drive the other two. 83 00:06:29,960 --> 00:06:32,710 And for this we need a different mathematical law. 84 00:06:32,720 --> 00:06:33,980 And this is the Stokes law. 85 00:06:35,670 --> 00:06:41,640 So it is kind of similar to Kansas law, but also a bit different because here we do not have the diversions 86 00:06:41,640 --> 00:06:44,120 of a vector, but we have the rotation of a vector. 87 00:06:45,210 --> 00:06:51,420 And this whole integral is not a three dimensional integral of the volume, but it's a two dimensional 88 00:06:51,420 --> 00:06:55,660 integral of a some surface eight, for example, could be this one here. 89 00:06:57,060 --> 00:07:05,430 And so the two dimensional surface integral of this rotation of A becomes equal to the vector itself, 90 00:07:05,700 --> 00:07:10,090 integrated along the confining line. 91 00:07:10,110 --> 00:07:11,700 So does this align integral? 92 00:07:11,940 --> 00:07:19,980 You have this vector DL, which is always tangential to this area, eight so tangential to the confinement 93 00:07:20,410 --> 00:07:26,660 and you calculate here the the product of these two vectors and then you integrate over the closed loop. 94 00:07:27,810 --> 00:07:32,700 And if you do this, you will find that the left hand side is equal to the right hand side. 95 00:07:34,080 --> 00:07:41,850 So now we see here we have two dimensional integral over the rotation of a vector so that we can use 96 00:07:41,850 --> 00:07:49,650 this to formulate the integral version of the other two maximal equations, because there we have rotations 97 00:07:49,650 --> 00:07:51,590 of B and rotations of each. 98 00:07:53,220 --> 00:07:56,540 So first of all, and we'll just rewrite this a tiny bit. 99 00:07:56,550 --> 00:07:59,130 So I will just divide by Muzio. 100 00:08:00,000 --> 00:08:02,550 So nothing special happened here. 101 00:08:02,850 --> 00:08:05,790 And then we know that this is again true for every position. 102 00:08:05,800 --> 00:08:10,710 Makeda So it must also be true for an integral over these quantities. 103 00:08:12,210 --> 00:08:15,780 So we integrate over this one for this one and over this one. 104 00:08:16,210 --> 00:08:19,100 And in every case we integrate over the same area. 105 00:08:20,910 --> 00:08:23,280 And now we leave to the right hand side as it is. 106 00:08:23,790 --> 00:08:30,040 And we just use here stokes law because we have here a surface integral over the rotation of a vector. 107 00:08:30,540 --> 00:08:37,430 So this means it must be equal to the vector itself and the surface integral becomes an integral over 108 00:08:37,440 --> 00:08:39,320 the surrounding of the surface. 109 00:08:39,330 --> 00:08:40,980 So it's just a line integral. 110 00:08:42,360 --> 00:08:49,760 And the only other thing that I have done here is I have considered this integral over to current density. 111 00:08:50,190 --> 00:08:53,370 So it is just the total current in the area. 112 00:08:53,410 --> 00:09:01,980 A Those are our integral formulation for this maximal equation and for the electric field. 113 00:09:01,980 --> 00:09:03,040 It's very similar. 114 00:09:03,540 --> 00:09:09,650 We know that this is true for every position vector so we can integrate and this will remain true. 115 00:09:10,290 --> 00:09:13,730 And so now here we can apply Stokes law. 116 00:09:14,010 --> 00:09:18,480 So the rotation of E becomes E and this becomes a line integral. 117 00:09:20,280 --> 00:09:25,920 So this means we have derived the integral versions of these four different Maxwell's equations and 118 00:09:25,920 --> 00:09:31,460 we have come from a local definition of these relations to a global definition. 119 00:09:32,370 --> 00:09:38,790 And in the following lecture, I want to discuss both versions of these Maxwell's equations in more 120 00:09:38,790 --> 00:09:39,360 detail. 121 00:09:39,640 --> 00:09:45,810 And I want to show you how we can reestablish our well known results from the very first section of 122 00:09:45,810 --> 00:09:51,450 the scores, for example, the induction law, or also the umpire's law.