1 00:00:00,060 --> 00:00:06,630 So in this lecture, we will introduce to Victor potential, which is somewhat similar to the electrostatic 2 00:00:06,630 --> 00:00:13,650 potential, and also we will derive an equation for the vector potential, which is a bit similar to 3 00:00:13,650 --> 00:00:15,960 the song equation from Electrostatic. 4 00:00:17,340 --> 00:00:23,460 So first, let us consider a quick recap of the electrostatic potential in electrostatic. 5 00:00:24,510 --> 00:00:30,420 So there we have considered this Maxwell equation where the rotation of the electric field is zero. 6 00:00:30,840 --> 00:00:38,940 And this led us to the possibility to introduce this electrostatic potential, because when we have 7 00:00:38,940 --> 00:00:42,510 the rotation of a gradient, this is always zero. 8 00:00:43,080 --> 00:00:49,050 So since we have this Maxwell equation here, this means that we will always have the possibility to 9 00:00:49,050 --> 00:00:53,720 express this electric field in terms of a gradient of some function. 10 00:00:54,930 --> 00:01:02,160 And also we could immediately understand that this electrostatic potential fire has certain properties. 11 00:01:02,280 --> 00:01:09,200 For example, we could add any possible constant without changing the physics. 12 00:01:10,050 --> 00:01:15,870 So this means that the electrostatic potential cannot really be a measurable quantity. 13 00:01:16,560 --> 00:01:22,590 But only differences in the electrostatic potential can be measured and the gradient can be measured 14 00:01:22,590 --> 00:01:24,420 because this is the electric field. 15 00:01:25,980 --> 00:01:32,880 Now, in Agneta Static's, we have the Maxwell equation that the divergence of the magnetic field is 16 00:01:32,880 --> 00:01:34,670 zero and not the rotation. 17 00:01:35,670 --> 00:01:40,680 So this again allows us to introduce a new potential, which we call the vector potential. 18 00:01:41,100 --> 00:01:46,860 And it has the probability that it's now are or the magnetic field has not a property, that it's the 19 00:01:46,860 --> 00:01:55,040 rotation of this vector potential, because this time we have the divergence of a rotation of a vector. 20 00:01:55,170 --> 00:01:59,480 And this identity is always zero for every vector. 21 00:02:00,300 --> 00:02:03,390 So it's the same logic, just like in a different order. 22 00:02:03,780 --> 00:02:07,260 So we know that the divergence of the magnetic field is zero. 23 00:02:07,680 --> 00:02:15,150 This means we must have the possibility to express the magnetic field in terms of a rotation of some 24 00:02:15,150 --> 00:02:16,160 other vector field. 25 00:02:18,470 --> 00:02:27,110 Now, since this magnetic field is expressed as the rotation of a this Eigg has no special properties, 26 00:02:28,340 --> 00:02:36,800 one of them is that the vector potentially can be shifted by any gradient of a function so we can introduce 27 00:02:36,800 --> 00:02:42,920 a new vector potential and it's the old vector potential plus the gradient of some function. 28 00:02:43,220 --> 00:02:46,900 And this function can even depend on the position vector. 29 00:02:48,110 --> 00:02:54,140 And while this means we have two different vector potentials, they will both give rise to the same 30 00:02:54,140 --> 00:03:01,290 magnetic field, which you can see in this quick calculation, because here we will get the idea, the 31 00:03:01,340 --> 00:03:05,290 rotation of A and then the other term, which is related to this one here. 32 00:03:05,300 --> 00:03:07,600 So we have to rotation of the gradient of chi. 33 00:03:08,030 --> 00:03:13,080 And so the rotation of a gradient of a circular function is always zero. 34 00:03:13,430 --> 00:03:16,860 So this means this B is equal to the old B. 35 00:03:18,170 --> 00:03:25,430 So again, this means that this vector potential, just like the electrostatic potential, is not really 36 00:03:25,850 --> 00:03:27,100 a measurable quantity. 37 00:03:27,860 --> 00:03:33,050 It's more that the rotation of this quantity can be measured because this is the magnetic field. 38 00:03:35,050 --> 00:03:44,440 Now we have again, like an electrostatic the possibility to tackle our problems in a different way, 39 00:03:44,770 --> 00:03:53,080 so instead of directly calculating it or B, we can first calculate fine or A and then from these properties 40 00:03:53,080 --> 00:03:57,850 can calculate B and by the gradient or by the rotation, respectively. 41 00:03:59,650 --> 00:04:04,660 Now, what what other advantage does it have to introduce such a vector potential? 42 00:04:05,410 --> 00:04:10,590 So we have introduced this vector potential and we can now use the other next equation. 43 00:04:10,990 --> 00:04:17,800 So here we have the rotation of the magnetic field is basically determined by the currents or by the 44 00:04:17,800 --> 00:04:19,580 current density of our. 45 00:04:21,070 --> 00:04:27,190 So this means if we take this relation here and put it in, we have the rotation of the rotation of 46 00:04:27,190 --> 00:04:31,330 A is equal to zero times J of our. 47 00:04:33,540 --> 00:04:41,430 Now we can use an identity, so that's really something you should practice yourself and please be warned, 48 00:04:41,430 --> 00:04:43,590 it's kind of difficult to show. 49 00:04:43,920 --> 00:04:48,600 So you should go ahead and consider all of the three components individually. 50 00:04:49,110 --> 00:04:53,830 Best would be you take the left hand side, write down what it is, you take the right hand side. 51 00:04:53,850 --> 00:04:56,880 So first this term, then this term and write down what it is. 52 00:04:56,880 --> 00:04:59,600 And then you will see that both of these sides are equal. 53 00:04:59,910 --> 00:05:05,790 So you will see that the rotation of the rotation of any vector must not be the vector. 54 00:05:05,790 --> 00:05:12,870 Potential can be really any vector that is are dependent is equal to the gradient of the divergence 55 00:05:12,870 --> 00:05:21,420 of the same vector minus the lab class operator acting on the vector and know something kind of difficult 56 00:05:21,720 --> 00:05:22,410 will happen. 57 00:05:22,410 --> 00:05:23,610 Or we will have to. 58 00:05:23,850 --> 00:05:24,190 Yeah. 59 00:05:24,540 --> 00:05:27,420 Do something difficult or something unintuitive. 60 00:05:28,380 --> 00:05:35,360 So I have already told you that this vector potentially can be shifted by any gradient of a function. 61 00:05:35,820 --> 00:05:43,050 So this means we can take our old vector potential and add gradient of some function which will give 62 00:05:43,050 --> 00:05:44,520 us a new vector potential. 63 00:05:44,850 --> 00:05:51,150 And we are allowed to do this because this does not change to physics, because this does not change 64 00:05:51,300 --> 00:05:52,260 the magnetic field. 65 00:05:52,260 --> 00:06:00,210 B And in particular, this means we always have the chance to choose a gauge. 66 00:06:00,210 --> 00:06:05,920 So it shows a gradient of chi in the way that the divergence of eight is zero. 67 00:06:06,480 --> 00:06:08,940 So maybe you can think of it like this. 68 00:06:09,300 --> 00:06:17,880 You first of all, calculate some vector potential A that fulfils this relation so that its rotation 69 00:06:17,880 --> 00:06:18,930 is the magnetic field. 70 00:06:19,290 --> 00:06:22,350 But then you figure out, OK, it's divergence is not zero. 71 00:06:22,960 --> 00:06:31,380 So what do you do then is you choose a function chi whose gradient is exactly the minus sign of this 72 00:06:31,380 --> 00:06:36,470 divergence of a and so your new AI has a divergence of zero. 73 00:06:37,680 --> 00:06:41,490 So that's something that's, as I mentioned, is a bit unintuitive. 74 00:06:41,490 --> 00:06:43,320 It's just a very clever trick. 75 00:06:44,130 --> 00:06:50,490 So it's not something to be worried about because probably the first physicist who has ever done it 76 00:06:51,240 --> 00:06:54,470 took many days or even weeks or month to figure this out. 77 00:06:54,480 --> 00:06:57,960 It's very it's really a very clever trick that we are using now. 78 00:06:58,650 --> 00:07:05,280 So we take our freedom to gauge our vector potential, to gauge it in such a way that the divergence 79 00:07:05,280 --> 00:07:06,690 of A is zero. 80 00:07:07,590 --> 00:07:09,180 So we are allowed to do this. 81 00:07:09,390 --> 00:07:10,650 But why should we do this? 82 00:07:11,160 --> 00:07:16,010 The answer is we do this because then this term here becomes zero. 83 00:07:16,710 --> 00:07:24,060 So this means this rotation of the rotation of A is then just minus the plus operator acting on a so 84 00:07:24,060 --> 00:07:29,880 we can write, mindful of how separate of A is equal to this one, or we write it like this. 85 00:07:31,440 --> 00:07:38,050 And now if you remember the same equation from electrostatic, this equation looks kind of similar. 86 00:07:38,670 --> 00:07:45,970 So here we had the lipless operator acting on five is equal to minus a constant times row of R. 87 00:07:47,040 --> 00:07:53,520 So it's basically the same thing, just a different constant and it's a vector equation. 88 00:07:53,610 --> 00:07:59,570 So here we have three components of this equation and here we have a scalar differential equation. 89 00:08:00,510 --> 00:08:08,730 So what we can do now is we can solve the equation for every individual component and the solution will 90 00:08:08,730 --> 00:08:11,880 look just like the solution for the equation. 91 00:08:12,300 --> 00:08:18,390 So we can take all of our results from electrostatic and carry it over to Manado Static's. 92 00:08:19,900 --> 00:08:26,320 So that's really the reason why we have considered this difficult looking trick here, where we have 93 00:08:26,320 --> 00:08:30,040 chosen a gauge such that the divergence of eight is zero. 94 00:08:30,820 --> 00:08:36,310 So here all the magic happens because this allows us to simplify our. 95 00:08:36,610 --> 00:08:42,370 Yeah, our differential equation for a in such a way that it looks like the differential equation for 96 00:08:42,370 --> 00:08:48,050 the electrostatic potential, and this allows us to carry over all of the old results. 97 00:08:49,600 --> 00:08:56,530 So one more comment before I continue with the next lecture is that it's not necessary to do this trick. 98 00:08:56,830 --> 00:09:01,390 We do not have to use a gauge so that the divergence of A is zero. 99 00:09:02,260 --> 00:09:07,540 But if we don't do this, we will get this additional term here that also would appear here in this 100 00:09:07,540 --> 00:09:08,120 equation. 101 00:09:08,800 --> 00:09:17,290 So if we do not choose such a very clever gauge, then our differential equation for a looks more difficult 102 00:09:17,500 --> 00:09:22,150 and then it would be more difficult to solve it because then we would have to start again from scratch 103 00:09:22,570 --> 00:09:26,680 and we could not carry over our old results from electrostatic. 104 00:09:27,130 --> 00:09:33,430 And this is why I've considered it here so that we now know the solutions from our previous calculations.