1 00:00:00,480 --> 00:00:08,070 So far, we have talked about charges, Curran's fields and forces and the previous lecture, but one 2 00:00:08,070 --> 00:00:14,700 thing is missing, it's the energy because typically, for example, the classical mechanics, you can 3 00:00:14,700 --> 00:00:18,690 also solve physical problems using energy equations. 4 00:00:19,290 --> 00:00:26,550 For example, you can apply energy conservation laws where you transform potential energy into kinetic 5 00:00:26,550 --> 00:00:27,060 energy. 6 00:00:28,680 --> 00:00:32,550 So this is something we can also do for electrodynamics. 7 00:00:32,910 --> 00:00:35,730 And for this, we also need to Loren's force. 8 00:00:35,850 --> 00:00:38,670 So this is why I introduced it in the previous lecture. 9 00:00:39,270 --> 00:00:42,900 And once again, I really hope I did not confuse you too much. 10 00:00:44,730 --> 00:00:50,970 What I want to tell you is that the physicists in the 19th century could not derive this Lawrence force, 11 00:00:50,970 --> 00:00:52,040 but still used it. 12 00:00:52,260 --> 00:00:55,520 And this is how we want to handle the things in this course as well. 13 00:00:55,530 --> 00:00:56,860 We just want to use it here. 14 00:00:58,110 --> 00:01:07,380 So what we want to calculate here is energies and quantities that are related to energies, for example, 15 00:01:07,380 --> 00:01:13,290 to power, which is the energy generation or consumption over time. 16 00:01:13,500 --> 00:01:15,180 So the consumption rate. 17 00:01:16,150 --> 00:01:21,030 So this power W is equal to E of a T, as I just mentioned. 18 00:01:21,660 --> 00:01:30,840 And you can tell if you if you say the energy or the work that you need to apply is equal to the force 19 00:01:30,840 --> 00:01:36,580 times distance, then this becomes minus to four times the velocity. 20 00:01:36,870 --> 00:01:43,830 For example, consider such a geometry here when you want to pull some mass up a mountain. 21 00:01:44,160 --> 00:01:50,200 Then there were the power that you that you ordered, the energy that you consume. 22 00:01:50,250 --> 00:01:58,440 So that's the power which tells us the energy per second, for example, is equal to the force, so 23 00:01:58,720 --> 00:02:03,190 is proportional to the mass times, the velocity with which you are pulling. 24 00:02:03,210 --> 00:02:09,240 So if you are pulling faster than you require more power and if you're poor slower, then it's less 25 00:02:09,240 --> 00:02:09,620 power. 26 00:02:11,310 --> 00:02:15,320 Now, in our case, the force is the of course, the Lawrence force. 27 00:02:15,630 --> 00:02:21,900 So we have the power is equal to the negative charge of the particle times, the electric field, plus 28 00:02:21,900 --> 00:02:23,250 this vector product here. 29 00:02:23,760 --> 00:02:25,740 And then we have another velocity term. 30 00:02:26,760 --> 00:02:29,890 And now what's really interesting is we have two terms here. 31 00:02:30,210 --> 00:02:32,280 First one is minus cubed times. 32 00:02:32,280 --> 00:02:40,380 He taught V, but then we have another term which is proportional to the vector product of three times 33 00:02:40,380 --> 00:02:42,930 B and then scalar product with B. 34 00:02:44,670 --> 00:02:53,100 Now what happens here is that this or every every vector product of two vectors is always perpendicular 35 00:02:53,100 --> 00:02:54,540 to both of these vectors. 36 00:02:54,870 --> 00:03:02,340 So V times B is perpendicular to V, but when we then calculate the scalar product, we have to scale 37 00:03:02,340 --> 00:03:06,630 our product of two perpendicular vectors, which always is zero. 38 00:03:07,110 --> 00:03:14,160 So the second term does not contribute to the power and this is well known the power that there is no 39 00:03:14,160 --> 00:03:16,470 power consumption from magnetic fields. 40 00:03:17,340 --> 00:03:20,730 So we only need to consider the electric field contribution. 41 00:03:22,570 --> 00:03:27,710 Now, another quantity that we can introduce is the power density. 42 00:03:28,630 --> 00:03:35,160 This is our power minus Q e.V., but it's normalized to some volume. 43 00:03:35,350 --> 00:03:41,070 So it's we can also say it's the charge over the volume. 44 00:03:41,080 --> 00:03:43,050 So this is just a charge density. 45 00:03:44,200 --> 00:03:48,030 And now we have for you to charge density times the velocity. 46 00:03:48,370 --> 00:03:50,450 This is equal to a current density. 47 00:03:51,010 --> 00:03:57,700 So our power density, which is you could say the energy generation density is equal to minus the current 48 00:03:57,700 --> 00:04:03,640 density dot the electric field because it's a quantity that we will need in the following. 49 00:04:05,730 --> 00:04:12,810 Now, what I will do is I will once again consider a Maxwell equation, this time we take the equation 50 00:04:12,810 --> 00:04:19,170 that considers the rotation of the magnetic field is equal to this is actually a partial derivative 51 00:04:19,170 --> 00:04:21,210 of the electric field with respect to time. 52 00:04:21,870 --> 00:04:23,960 And then we have the currents as well. 53 00:04:24,990 --> 00:04:28,470 So the current itself is equal to these two terms. 54 00:04:28,500 --> 00:04:33,580 So I have just taken this turn, put it on the other side and you get something like this. 55 00:04:35,010 --> 00:04:40,620 Now, what we can do is we can multiply with a scale of product from the right hand side, so we multiply 56 00:04:40,620 --> 00:04:41,700 by the electric field. 57 00:04:42,930 --> 00:04:48,810 So we do it also on the right hand side, so here I have written this down and a bit of a different 58 00:04:48,810 --> 00:04:49,210 way. 59 00:04:50,010 --> 00:04:57,810 So the first term Epsilon zero idot times E can be written down like this to please follow along and 60 00:04:58,260 --> 00:04:59,880 see that this is really true. 61 00:05:00,570 --> 00:05:05,190 So what we have to do is we have to calculate a time derivative of each square. 62 00:05:06,150 --> 00:05:07,510 So what does that give us? 63 00:05:07,530 --> 00:05:16,380 It gives us two times E but don't forget the inner derivative gives us two times E times the time derivative 64 00:05:16,380 --> 00:05:25,210 of E, so it gives us in total Epsilon zero over two times, two times E times e dot. 65 00:05:25,470 --> 00:05:26,910 So absolute zero times. 66 00:05:26,910 --> 00:05:27,690 Eight times. 67 00:05:27,990 --> 00:05:29,660 So that's really the first term. 68 00:05:31,060 --> 00:05:37,300 Now, in the second term, we have this novela operator, which is basically the vector of partial derivatives, 69 00:05:38,230 --> 00:05:43,670 acting on B in terms of a rotation, but also it will act on it. 70 00:05:44,680 --> 00:05:48,770 So we have to apply we can apply a product rule. 71 00:05:50,200 --> 00:05:54,590 So what we can do is we can write this down in terms of two terms. 72 00:05:55,450 --> 00:06:00,610 So this is a bit more complicated, but you can really figure it out yourself if you write all of this 73 00:06:00,610 --> 00:06:06,880 down in terms of the components and you will see that it's equivalent to one of them zero times beat 74 00:06:07,210 --> 00:06:14,620 times to rotation of E, where at this time the novel operator only acts on E, and then we have heard 75 00:06:14,620 --> 00:06:21,910 this term, one of them use zero and then the divergence of this vector product of B. 76 00:06:21,910 --> 00:06:28,930 And so here you first have to calculate the the vector product and then calculate the divergence. 77 00:06:31,380 --> 00:06:34,900 OK, so what can we do here? 78 00:06:35,820 --> 00:06:39,270 What you can see here is the rotation of E! 79 00:06:39,930 --> 00:06:46,390 And if you think about a Maxwell equation, the rotation of E is equal to minus B dot. 80 00:06:47,040 --> 00:06:53,800 So we have here one over Muzio times B times B dot. 81 00:06:54,720 --> 00:06:57,120 So this is very similar to this one here. 82 00:06:57,120 --> 00:07:06,750 But we had E times e dot so we can use again such a trick, very write it down as to in terms of a derivative, 83 00:07:06,750 --> 00:07:09,560 with respect to time and some factors. 84 00:07:09,570 --> 00:07:13,700 And then we write here B square and not E Square. 85 00:07:14,940 --> 00:07:22,060 So I done this here we have now another term, the time derivative with respect to time, as I, as 86 00:07:22,140 --> 00:07:22,590 I said. 87 00:07:22,980 --> 00:07:28,380 And then we have these three factors which are in this case, one of them use zero and then one half. 88 00:07:29,010 --> 00:07:34,800 And if we calculate the time derivative of this P squared term here, we had two times B times B dot. 89 00:07:35,310 --> 00:07:37,500 So this gives us the two councils. 90 00:07:37,500 --> 00:07:41,580 With this two, we get A B and we get B dot, which is this one. 91 00:07:42,270 --> 00:07:44,010 And this one is actually minus B dot. 92 00:07:44,280 --> 00:07:45,960 But here's minus, here's A plus. 93 00:07:46,350 --> 00:07:47,460 So it really works out. 94 00:07:48,390 --> 00:07:53,240 And here we get another term which where we can just swap around the order. 95 00:07:53,250 --> 00:07:57,350 So if you swap the two terms of a vector product, it gives us a negative sign. 96 00:07:57,720 --> 00:07:58,590 So this makes it here. 97 00:07:58,590 --> 00:08:07,680 Plus the divergence of this term here and now we can identify the shape of this equation with a continuity 98 00:08:07,680 --> 00:08:08,300 equation. 99 00:08:08,640 --> 00:08:15,030 So we have some term here, some scalar that is equal to the time derivative of some scalar because 100 00:08:15,030 --> 00:08:20,140 E squared, B square, both scalars and another term, which is a divergence. 101 00:08:20,760 --> 00:08:22,170 So this one is a vector. 102 00:08:23,640 --> 00:08:27,660 So we have here this is just from the previous slide to last lines. 103 00:08:28,050 --> 00:08:37,200 We have here a continuity equation for the term minus J times E, which is our power density as we have 104 00:08:37,200 --> 00:08:39,150 established in the beginning of this lecture. 105 00:08:40,170 --> 00:08:47,250 So we have established a continuity equation for the energy density where we have here a source or sink 106 00:08:47,250 --> 00:08:47,760 term. 107 00:08:47,990 --> 00:08:50,180 So this is how power can be generated. 108 00:08:50,730 --> 00:08:55,440 This is proportional to the current density times eat. 109 00:08:56,100 --> 00:09:03,510 Then we have the energy density itself, which is this term here and where we have to act. 110 00:09:03,810 --> 00:09:05,850 That's the time derivative act on. 111 00:09:06,240 --> 00:09:14,310 So this is Epsilon Zero Square and then we have another term proportional to be square and then we have 112 00:09:14,310 --> 00:09:20,790 another term, which is this divergence here, which is the where does this as vector? 113 00:09:20,790 --> 00:09:23,670 Here is the energy current density. 114 00:09:24,100 --> 00:09:25,710 This is the pointing vector. 115 00:09:25,710 --> 00:09:30,420 And you could say this tells us along which direction energy is transferred. 116 00:09:31,110 --> 00:09:37,020 And we will use this pointing vector in the next main section where we will discuss electromagnetic 117 00:09:37,020 --> 00:09:37,560 waves. 118 00:09:38,400 --> 00:09:40,400 So I will come back to these results here. 119 00:09:40,770 --> 00:09:47,190 But what we have done in this lecture is we have taken and one of the Maxwell equations and have shown 120 00:09:47,490 --> 00:09:53,070 that it can be transformed to a continuity equation for the energy density. 121 00:09:53,670 --> 00:09:59,820 And this allowed us to establish several terms, for example, the pointing vector or the energy density 122 00:10:00,180 --> 00:10:03,970 and show how they relate to the electric and magnetic fields. 123 00:10:04,330 --> 00:10:09,840 For example, you can see the energy density is proportional to the sum of each square and B square, 124 00:10:10,260 --> 00:10:17,850 and the pointing vector, which gives us the direction of energy transfer, is perpendicular to the 125 00:10:17,850 --> 00:10:21,240 electric field and perpendicular to the magnetic field. 126 00:10:21,330 --> 00:10:23,040 Due to this vector product, you.