1 00:00:00,180 --> 00:00:08,280 Let us now discuss our eight Maxwell equations in detail, so we started out with the differential formulations 2 00:00:08,280 --> 00:00:15,510 of the Maxwell equations, and I already told you that these four equations correspond to the statements 3 00:00:15,510 --> 00:00:21,790 that charges are the sources of the electric field while the magnetic field has no sources. 4 00:00:21,810 --> 00:00:23,640 So there are no magnetic monopoles. 5 00:00:24,660 --> 00:00:29,820 And then for the other two equations, we had that time dependent magnetic fields generate electric 6 00:00:29,820 --> 00:00:36,090 fields and vice versa, but also that currents can generate magnetic fields. 7 00:00:37,910 --> 00:00:44,490 The previous lecture, we have also derived integral formulations and so their interpretation can be 8 00:00:45,270 --> 00:00:46,280 a bit different. 9 00:00:46,290 --> 00:00:50,690 But of course, it's it's still the same equation just from a different point of view. 10 00:00:51,000 --> 00:00:56,520 So the physical consequences will be quite similar of these equations. 11 00:00:57,420 --> 00:01:00,440 So let us start with the first two equation equations. 12 00:01:00,450 --> 00:01:02,630 These are the Gauss laws. 13 00:01:03,600 --> 00:01:07,790 So here we have the differential formulation and the integrated formulation. 14 00:01:08,340 --> 00:01:13,170 And it means that charges are the sources of the electric field, which you can see, for example, 15 00:01:13,170 --> 00:01:19,350 here when we have two charges, both of them are sources or sinks of the electric field. 16 00:01:19,710 --> 00:01:24,750 And we have these lines of the electric field that connect to two sources. 17 00:01:26,370 --> 00:01:33,270 Then in the integrated formulation we have that the electric flux through a closed surface is determined 18 00:01:33,270 --> 00:01:35,160 by a deep and close charge. 19 00:01:35,700 --> 00:01:44,040 So when we have a volume here, the electric field or the electric flux is determined by the charge 20 00:01:44,040 --> 00:01:45,270 inside of the volume. 21 00:01:48,040 --> 00:01:54,730 Now, we can also formulate Koslov on magnetic fields, and it is kind of similar, but the right hand 22 00:01:54,730 --> 00:01:57,590 side of these equations is zero in both cases. 23 00:01:58,090 --> 00:02:04,900 So this means a magnetic field has no source or sinks, so monopoles do not exist. 24 00:02:05,740 --> 00:02:10,420 So there is not something like QM, like a magnetic charge just does not exist. 25 00:02:10,900 --> 00:02:17,080 We know, for example, when we have such a magnet, it always consists out of the north and the South 26 00:02:17,080 --> 00:02:17,430 Pole. 27 00:02:18,130 --> 00:02:25,590 So it is more of a vectorial quantity and not such a divergence in the sense of a monopole. 28 00:02:26,920 --> 00:02:32,770 And these magnetic fields, field lines, for example, they go through the magnets, as you can see 29 00:02:32,770 --> 00:02:33,190 here. 30 00:02:34,510 --> 00:02:40,390 Now, in the integrated version, it means that the magnetic flux vanishes when we integrate over every 31 00:02:40,390 --> 00:02:41,390 closed surface. 32 00:02:41,740 --> 00:02:48,100 So that's very interesting, I think, because you can take any magnetic system and you can take any 33 00:02:48,100 --> 00:02:53,360 volume, and if you just integrate the magnetic flux, it will always vanish. 34 00:02:55,810 --> 00:03:03,220 Now, in the very first section of this course, we have talked about early theories of electrodynamics 35 00:03:03,610 --> 00:03:09,160 and we have learned that people already knew about induction and the electromagnetic induction law. 36 00:03:10,000 --> 00:03:12,920 And this is the interpretation of the third Maxwell equation. 37 00:03:12,940 --> 00:03:20,260 So this is here the differential formulation and the integrated formulation and means that time dependent 38 00:03:20,260 --> 00:03:26,170 changes in the magnetic field give rise to electric fields and then the integrated formulation. 39 00:03:26,170 --> 00:03:32,450 It means that an electric field is generated by changes of the magnetic flux. 40 00:03:33,400 --> 00:03:41,350 So this means not only does the magnetic field have to change, but also the area of which we can integrate, 41 00:03:41,350 --> 00:03:45,310 all of which we have to integrate could also change. 42 00:03:45,910 --> 00:03:51,100 So this is something we have discussed earlier in terms of the electromagnetic induction, where we 43 00:03:51,100 --> 00:03:56,460 take a close to wire and bring it into a magnetic field or take it out of the magnetic field. 44 00:03:56,980 --> 00:04:03,490 And this leads to the generation of an electric field and therefore to an induction voltage that can 45 00:04:03,490 --> 00:04:10,710 be used, for example, to charge devices or to detect cars going over a red light, for example. 46 00:04:12,100 --> 00:04:15,290 And here you can see that this is the flux. 47 00:04:15,310 --> 00:04:19,630 So we have we have to integrate the magnetic field over an area. 48 00:04:20,260 --> 00:04:26,170 And if the magnetic field is constant in the area, then we can just ride the flux as equal to A times 49 00:04:26,170 --> 00:04:34,900 B and then this year is our induction voltage is equal to minus five minus the time derivative of fire. 50 00:04:34,900 --> 00:04:40,360 And this line integral here where we integrate over the surrounding of this area. 51 00:04:43,630 --> 00:04:47,950 Now, the last Maxwell equation is this one in the differential formulation. 52 00:04:47,950 --> 00:04:54,130 And here we have the integral formulation, and this is an extended or generalised version of unpause 53 00:04:54,130 --> 00:04:54,450 law. 54 00:04:55,390 --> 00:05:02,530 So we have encountered Amperes Law already in our first section about early theories of electrodynamics 55 00:05:03,130 --> 00:05:05,680 because in 18, 25, thirsted and Ampere. 56 00:05:06,040 --> 00:05:14,020 Notice that when you take a wire and apply the current, then a compass needle will always orient perpendicular 57 00:05:14,020 --> 00:05:14,740 to this wire. 58 00:05:15,580 --> 00:05:19,090 And this is because this is this current. 59 00:05:19,180 --> 00:05:23,210 So basically a moving charge will generate a magnetic field. 60 00:05:23,830 --> 00:05:26,740 So this was what I understood came up with. 61 00:05:27,290 --> 00:05:30,310 You can see it is kind of similar to this equation. 62 00:05:30,850 --> 00:05:37,270 But we have also learned that when we use this identity here, which is true for every vector. 63 00:05:37,270 --> 00:05:42,020 So it's also true for B that the divergence of the rotation is always zero. 64 00:05:42,640 --> 00:05:47,650 This means that the divergence of J is zero if we take this equation here. 65 00:05:48,760 --> 00:05:55,870 However, we also know that the charge must follow a continuity equation where we have on the right 66 00:05:55,870 --> 00:06:00,140 hand side some time derivative term of some charge density. 67 00:06:00,580 --> 00:06:02,860 So this term was missing in this equation. 68 00:06:03,490 --> 00:06:06,450 T it turns out in the Maxwell equation we have this term. 69 00:06:06,730 --> 00:06:13,820 So we have here a time derivative of music, zero, Epsilon Zero and the electric field. 70 00:06:14,680 --> 00:06:19,330 So in this amperes law, the continuity equation is finally fulfilled. 71 00:06:19,360 --> 00:06:27,340 So we know that this was a nice guess to describe this experiment, but it was not sufficient to really 72 00:06:27,340 --> 00:06:28,780 fulfill the whole theory. 73 00:06:29,740 --> 00:06:36,370 So this is the differential formulation that electric fields give rise or changes in the electric field 74 00:06:36,370 --> 00:06:38,730 or currents give rise to magnetic fields. 75 00:06:39,340 --> 00:06:47,250 And also we have here the the integrated version that this is also true if we integrate over the surrounding 76 00:06:47,260 --> 00:06:47,440 of. 77 00:06:47,580 --> 00:06:48,120 An area.