1 00:00:00,000 --> 00:00:25,000 2 00:00:25,000 --> 00:00:30,878 There is energy in an electric field and there is energy in a 3 00:00:30,878 --> 00:00:35,092 magnetic field. You remember that from 8.02. 4 00:00:35,092 --> 00:00:39,893 And the energy density, mind the units in terms of 5 00:00:39,893 --> 00:00:45,184 joules per cubic meter, for the electric field we write 6 00:00:45,184 --> 00:00:49,397 with that u equals one-half epsilon zero E2. 7 00:00:49,397 --> 00:00:53,121 There is no such thing as a free lunch. 8 00:00:53,121 --> 00:00:59,000 You have to do work to create an electric field. 9 00:00:59,000 --> 00:01:03,044 You have to assemble charges and bring them together. 10 00:01:03,044 --> 00:01:06,622 That means work that creates an electric field. 11 00:01:06,622 --> 00:01:09,733 And the same is true for magnetic fields. 12 00:01:09,733 --> 00:01:14,166 When you have a solenoid and you create the magnetic field 13 00:01:14,166 --> 00:01:17,122 inside the solenoid that caused energy. 14 00:01:17,122 --> 00:01:20,622 And the energy density, mind the word density, 15 00:01:20,622 --> 00:01:24,355 it is per cubic meter, for a magnetic field is B2 16 00:01:24,355 --> 00:01:29,678 divided by 2mu zero. Now traveling EM wave, 17 00:01:29,678 --> 00:01:36,619 traveling electromagnetic wave, so this is now a traveling 18 00:01:36,619 --> 00:01:44,047 wave, we know that the magnitude of B is the magnitude of E at 19 00:01:44,047 --> 00:01:51,597 any moment in time divided by c. I can write this as E2 divided 20 00:01:51,597 --> 00:01:57,929 by 2mu zero times c2. But c2 is one over epsilon zero 21 00:01:57,929 --> 00:02:02,509 mu zero. This is also one-half epsilon 22 00:02:02,509 --> 00:02:05,451 zero E2. And what you see now is so 23 00:02:05,451 --> 00:02:10,038 wonderful, so beautifully symmetric in electromagnetic 24 00:02:10,038 --> 00:02:13,067 waves. One is the same as the other. 25 00:02:13,067 --> 00:02:16,009 One cannot exist without the other. 26 00:02:16,009 --> 00:02:20,855 And look at the energy density in the electric field of a 27 00:02:20,855 --> 00:02:24,663 traveling wave. It is exactly the same as the 28 00:02:24,663 --> 00:02:27,778 energy density in the magnetic field. 29 00:02:27,778 --> 00:02:32,365 And so the total energy density is the sum of the two, 30 00:02:32,365 --> 00:02:40,000 is epsilon zero times E2. But you can also write for that 31 00:02:40,000 --> 00:02:46,322 epsilon zero times E times Bc, if you prefer that. 32 00:02:46,322 --> 00:02:52,387 This, of course, is only true in vacuum when the 33 00:02:52,387 --> 00:03:00,000 speed of propagation is c. Now it is a traveling wave. 34 00:03:00,000 --> 00:03:04,767 And this traveling wave moves, and so it carries energy with 35 00:03:04,767 --> 00:03:07,272 it. Now the question is how much 36 00:03:07,272 --> 00:03:10,666 energy flows through an area which is, say, 37 00:03:10,666 --> 00:03:14,787 one square meter area perpendicular to the direction 38 00:03:14,787 --> 00:03:18,343 of propagation? Suppose I have a box here and 39 00:03:18,343 --> 00:03:22,949 this side is one square meter, and I want to know how much 40 00:03:22,949 --> 00:03:26,343 radiation comes out of there in one second. 41 00:03:26,343 --> 00:03:32,000 The radiation is flowing in this direction with speed c. 42 00:03:32,000 --> 00:03:36,181 In one second this box, it's quite a large box, 43 00:03:36,181 --> 00:03:41,272 3 times 10 to the 8 meters. And all the energy that is in 44 00:03:41,272 --> 00:03:47,000 there will flow through this one square meter in the time of one 45 00:03:47,000 --> 00:03:50,272 second. And so the dimensions that we 46 00:03:50,272 --> 00:03:55,727 are talking about now are joules per second per square meter, 47 00:03:55,727 --> 00:04:00,000 which is also watts per square meter. 48 00:04:00,000 --> 00:04:05,139 And that now is, of course, the total energy 49 00:04:05,139 --> 00:04:10,398 density times c, the speed of light for which 50 00:04:10,398 --> 00:04:16,494 you can write epsilon zero times E times B times c2, 51 00:04:16,494 --> 00:04:22,111 if you like that. But you can also write that EB 52 00:04:22,111 --> 00:04:30,000 divided by mu zero because c2 is one over epsilon zero mu. 53 00:04:30,000 --> 00:04:36,397 And this should remind you of something that is in your far 54 00:04:36,397 --> 00:04:43,014 distant past which is what we earlier have called in 8.02 the 55 00:04:43,014 --> 00:04:48,198 poynting vector. And the poynting vector S was E 56 00:04:48,198 --> 00:04:54,264 cross B divided by mu zero, and the units were watts per 57 00:04:54,264 --> 00:05:00,000 square meter, which is exactly what this is. 58 00:05:00,000 --> 00:05:05,226 And the reason why the cross disappears here is that with 59 00:05:05,226 --> 00:05:11,106 electromagnetic traveling waves, E is always perpendicular to B. 60 00:05:11,106 --> 00:05:16,240 That takes care of the cross. Now, both E and B are time 61 00:05:16,240 --> 00:05:19,973 variable. And so the poynting vector will 62 00:05:19,973 --> 00:05:25,759 obviously also be time variable. E is going to be proportional, 63 00:05:25,759 --> 00:05:31,173 or you can write E is some E0 times cosine omega t and B is 64 00:05:31,173 --> 00:05:36,701 some B0 times cosine omega t. In the poynting vector, 65 00:05:36,701 --> 00:05:39,363 you get the cosine square of omega t. 66 00:05:39,363 --> 00:05:42,988 But, since we are never interested in the poynting 67 00:05:42,988 --> 00:05:47,500 vector on a timescale smaller than the period of oscillations, 68 00:05:47,500 --> 00:05:51,198 we want to know the average over many oscillations. 69 00:05:51,198 --> 00:05:55,266 What matters there is the average value of cosine square 70 00:05:55,266 --> 00:06:00,000 omega t, one from the E and the other one from the B. 71 00:06:00,000 --> 00:06:05,084 And that is one-half. And so we can conclude then 72 00:06:05,084 --> 00:06:09,851 that the average value of the poynting vector, 73 00:06:09,851 --> 00:06:13,771 time averaged, is one-half that value, 74 00:06:13,771 --> 00:06:19,597 one-half EB divided mu zero for which you can also write 75 00:06:19,597 --> 00:06:25,000 one-half, if you want to kill completely -- 76 00:06:25,000 --> 00:06:27,707 By the way, this is one-half E0B0. 77 00:06:27,707 --> 00:06:32,711 It is important that you have the E0 because the cosine square 78 00:06:32,711 --> 00:06:36,321 average is one-half. Here are the amplitudes. 79 00:06:36,321 --> 00:06:39,603 You can also write for that one-half E02. 80 00:06:39,603 --> 00:06:42,966 And so you write down for BE divided by c, 81 00:06:42,966 --> 00:06:46,904 and then you get downstairs mu zero divided by c. 82 00:06:46,904 --> 00:06:51,662 And the reason why I write it in this form is that it tells 83 00:06:51,662 --> 00:06:56,257 you that if you know what the strength of the E field is, 84 00:06:56,257 --> 00:07:02,000 that alone tells you then what the poynting vector is. 85 00:07:02,000 --> 00:07:06,339 Because B is coupled to E through Maxwell's equations. 86 00:07:06,339 --> 00:07:11,005 That is where B is E over c. And so all that matters then, 87 00:07:11,005 --> 00:07:15,263 if you want to calculate what the poynting vector is, 88 00:07:15,263 --> 00:07:18,210 is E0. Of course, B0 alone would also 89 00:07:18,210 --> 00:07:20,666 be fine. Let's take an example. 90 00:07:20,666 --> 00:07:24,678 Suppose we have an electromagnetic wave whereby E0 91 00:07:24,678 --> 00:07:30,000 were 100 volts per meter, then I can calculate now -- 92 00:07:30,000 --> 00:07:34,632 And it is a traveling electromagnetic wave. 93 00:07:34,632 --> 00:07:40,477 I can calculate what the average value of the poynting 94 00:07:40,477 --> 00:07:44,448 vector is. That would become one-half 95 00:07:44,448 --> 00:07:49,301 times 100 squared divided by mu zero times c. 96 00:07:49,301 --> 00:07:56,029 And, if you do your homework on that, you will find that it is 97 00:07:56,029 --> 00:08:01,886 13 watts per square meter. Now, if you exposed yourself to 98 00:08:01,886 --> 00:08:05,052 13 watts per square meter, visible light and infrared 99 00:08:05,052 --> 00:08:08,947 light, you take all your clothes off and expose yourself to that, 100 00:08:08,947 --> 00:08:12,234 your body will absorb that. It will not go through your 101 00:08:12,234 --> 00:08:14,121 body. X-rays may go through your 102 00:08:14,121 --> 00:08:17,104 body, gamma rays certainly, but infrared radiation 103 00:08:17,104 --> 00:08:19,660 invisible light gets absorbed by your body. 104 00:08:19,660 --> 00:08:22,399 And so the question now is will that harm you? 105 00:08:22,399 --> 00:08:25,565 And then the answer is no. You would hardly notice 13 106 00:08:25,565 --> 00:08:30,595 watts per square meter. Your body itself radiates about 107 00:08:30,595 --> 00:08:35,617 100 joules per second because of the body heat that you have. 108 00:08:35,617 --> 00:08:39,970 And so the 13 watts per square meter that you absorb, 109 00:08:39,970 --> 00:08:45,076 let's say your cross-sectional area is about one square meter, 110 00:08:45,076 --> 00:08:49,095 just to round it off, so that means 13 joules per 111 00:08:49,095 --> 00:08:52,192 second would be absorbed by your body. 112 00:08:52,192 --> 00:08:55,457 And that will not affect you in any way. 113 00:08:55,457 --> 00:09:00,563 But let's now take a situation that we have E0 equals 10 times 114 00:09:00,563 --> 00:09:05,000 higher, which is 1000 volts per meter. 115 00:09:05,000 --> 00:09:09,761 Now the poynting vector goes up by the square of E, 116 00:09:09,761 --> 00:09:15,285 so now you are going to get that the poynting vector is 1.3 117 00:09:15,285 --> 00:09:19,952 kilowatts per square meter. And that will fry you. 118 00:09:19,952 --> 00:09:24,809 If you walk naked in a field of infrared or optical, 119 00:09:24,809 --> 00:09:29,857 whereby the absorption 1.3 kilowatts per square meter, 120 00:09:29,857 --> 00:09:36,151 that is very dangerous. You get skin cancer and worse. 121 00:09:36,151 --> 00:09:42,035 Now, why do I mention that and why do I focus on that 1.3 122 00:09:42,035 --> 00:09:47,813 kilowatts per square meter? Because that is exactly what 123 00:09:47,813 --> 00:09:50,859 the sun does. Here is the sun, 124 00:09:50,859 --> 00:09:55,377 and the sun is a rather powerful light bulb, 125 00:09:55,377 --> 00:10:00,000 about 3.9 times 10 to the 26 watts. 126 00:10:00,000 --> 00:10:05,612 And this is where you are on earth, and the distance to earth 127 00:10:05,612 --> 00:10:10,664 is 150 million kilometers. I can calculate now how many 128 00:10:10,664 --> 00:10:15,341 joules per second go through one square meter here, 129 00:10:15,341 --> 00:10:20,299 and this one square meter is held perpendicular to the 130 00:10:20,299 --> 00:10:24,135 direction to the sun. That value for S is, 131 00:10:24,135 --> 00:10:29,000 of course, the radiation that leaves here. 132 00:10:29,000 --> 00:10:31,850 That is my 3.9 times 10 to the 26. 133 00:10:31,850 --> 00:10:37,120 And now I have to divide it by the entire surface area of this 134 00:10:37,120 --> 00:10:41,871 sphere that goes in all directions that has this radius. 135 00:10:41,871 --> 00:10:46,795 And that surface area is four pi times that radius square, 136 00:10:46,795 --> 00:10:51,979 so that is 150 times 10 to the 9, because I have to go to MKS 137 00:10:51,979 --> 00:10:55,693 units squared. And that is now the number of 138 00:10:55,693 --> 00:11:02,000 watts per square meter that we receive at earth from the sun. 139 00:11:02,000 --> 00:11:04,956 And that number, a very famous number, 140 00:11:04,956 --> 00:11:07,592 is 1.4 kilowatts per square meter. 141 00:11:07,592 --> 00:11:12,386 And that is why I picked the 1.3 to show you that if you walk 142 00:11:12,386 --> 00:11:16,541 around on the beach and do not take care of yourself, 143 00:11:16,541 --> 00:11:21,574 if you do that too long that is very dangerous because your body 144 00:11:21,574 --> 00:11:23,572 absorbs that. This number, 145 00:11:23,572 --> 00:11:27,487 called the solar constant, has major implications, 146 00:11:27,487 --> 00:11:33,000 of course, for people who want to harvest solar energy. 147 00:11:33,000 --> 00:11:38,370 The maximum that you can ever harvest, for every square meter 148 00:11:38,370 --> 00:11:43,561 that you dedicate to solar energy, you only get 1400 joules 149 00:11:43,561 --> 00:11:46,604 per second. You can never get more. 150 00:11:46,604 --> 00:11:51,796 The electric power capacity of the United States is 700,000 151 00:11:51,796 --> 00:11:55,824 megawatts for which you need 700 power plants. 152 00:11:55,824 --> 00:12:02,000 A full size power plant is about 1000 megawatt power plant. 153 00:12:02,000 --> 00:12:06,377 You need 700 of these power plants to have the capacity for 154 00:12:06,377 --> 00:12:09,320 the United States. We are energy hungry. 155 00:12:09,320 --> 00:12:13,245 We consume more than one-quarter of all the energy in 156 00:12:13,245 --> 00:12:16,264 the world. If you want to replace this by 157 00:12:16,264 --> 00:12:20,792 solar energy that is a major problem because you can only get 158 00:12:20,792 --> 00:12:24,415 1.4 kilojoules per second for every square meter. 159 00:12:24,415 --> 00:12:28,716 You can calculate how many hundreds of square miles in the 160 00:12:28,716 --> 00:12:32,339 desert you would have to commit with solar cells, 161 00:12:32,339 --> 00:12:38,000 which are extremely expensive, in order to get electricity. 162 00:12:38,000 --> 00:12:41,303 And the efficiency, of course, is never 100%. 163 00:12:41,303 --> 00:12:44,606 And also, when the sun is low in the horizon, 164 00:12:44,606 --> 00:12:48,885 then you don't have this one square meter perpendicular to 165 00:12:48,885 --> 00:12:52,789 the direction of the sun. All of that has to be taken 166 00:12:52,789 --> 00:12:55,567 into account. Solar energy is not very 167 00:12:55,567 --> 00:12:58,345 important in our lives, unfortunately. 168 00:12:58,345 --> 00:13:03,000 And this is the number that is the ultimate limit. 169 00:13:03,000 --> 00:13:06,812 Now comes the question, is there such a thing as an 170 00:13:06,812 --> 00:13:10,777 electric field over 1000 volts per meter in the solar 171 00:13:10,777 --> 00:13:13,751 radiation? Could we actually measure the 172 00:13:13,751 --> 00:13:17,488 electric field just from the radiation of the sun? 173 00:13:17,488 --> 00:13:21,071 And the answer is no. And the reason is that the 174 00:13:21,071 --> 00:13:25,418 radiation is not really in the form of our idealized plane 175 00:13:25,418 --> 00:13:29,993 waves, but more important than anything else is that there is 176 00:13:29,993 --> 00:13:34,187 no such thing as just one wave from the sun that has the 177 00:13:34,187 --> 00:13:38,000 amplitude of 1000 volts per meter. 178 00:13:38,000 --> 00:13:42,694 In fact, the radiation reaches us in small packages. 179 00:13:42,694 --> 00:13:46,192 It is broken up in pieces, so to speak. 180 00:13:46,192 --> 00:13:51,439 However, since the energy flow is 1.4 kilowatts per square 181 00:13:51,439 --> 00:13:57,146 meter, it is perfectly OK with me that you refer to this number 182 00:13:57,146 --> 00:14:02,745 as the poynting vector. I have no problem with that. 183 00:14:02,745 --> 00:14:08,823 But it is a little bit naive to associate with that an electric 184 00:14:08,823 --> 00:14:14,705 field that can be measured that has an amplitude then of 1000 185 00:14:14,705 --> 00:14:19,019 volts per meter. And so this now is the right 186 00:14:19,019 --> 00:14:24,607 time to take a close look at how electromagnetic waves are 187 00:14:24,607 --> 00:14:28,431 produced. In a nutshell it comes down to 188 00:14:28,431 --> 00:14:32,185 this. You can create electromagnetic 189 00:14:32,185 --> 00:14:36,703 waves if you accelerate charges. Charges that are stationary or 190 00:14:36,703 --> 00:14:41,002 moving at constant velocity are surrounded by a radial field 191 00:14:41,002 --> 00:14:45,082 poynting away or poynting inwards, depending upon whether 192 00:14:45,082 --> 00:14:49,090 the charge is positive or negative, and there is no kink 193 00:14:49,090 --> 00:14:52,733 anywhere in these fields. Whether it has a constant 194 00:14:52,733 --> 00:14:57,178 velocity or whether it stands still they are radially electric 195 00:14:57,178 --> 00:15:00,596 field lines. The moment, however, 196 00:15:00,596 --> 00:15:04,187 that you accelerate it, as you will see today, 197 00:15:04,187 --> 00:15:07,460 you introduce a kink in those field lines. 198 00:15:07,460 --> 00:15:11,530 And that kink is responsible for the electromagnetic 199 00:15:11,530 --> 00:15:14,084 radiation. It manifests itself as 200 00:15:14,084 --> 00:15:17,995 electromagnetic radiation. I will follow a classic 201 00:15:17,995 --> 00:15:22,384 derivation that is verbatim given that way in Bekefi and 202 00:15:22,384 --> 00:15:24,779 Barrett. I therefore advise you 203 00:15:24,779 --> 00:15:29,169 strongly, for the next 30 minutes, not to take any notes 204 00:15:29,169 --> 00:15:35,207 but try to follow my arguments. That will help you way more 205 00:15:35,207 --> 00:15:39,786 than if you try to also takes notes, because it is really 206 00:15:39,786 --> 00:15:42,402 verbatim from Bekefi and Barrett. 207 00:15:42,402 --> 00:15:46,408 It is a classical derivation with many simplifying 208 00:15:46,408 --> 00:15:51,232 assumptions, but it gives a very nice result which has great 209 00:15:51,232 --> 00:15:55,566 practical applications. Suppose I have here a charge q 210 00:15:55,566 --> 00:16:00,727 which is located at position A. We have a charge q. 211 00:16:00,727 --> 00:16:05,909 It is at 0 and it is at rest. And I am going to accelerate 212 00:16:05,909 --> 00:16:10,454 that in this direction. I am going to accelerate it 213 00:16:10,454 --> 00:16:14,727 with an acceleration which is in this direction. 214 00:16:14,727 --> 00:16:18,181 I will not put the vector in there now. 215 00:16:18,181 --> 00:16:22,818 I will do that later. Otherwise, the picture becomes 216 00:16:22,818 --> 00:16:26,545 too complicated. And I do that for delta t 217 00:16:26,545 --> 00:16:32,990 seconds, only very brief. And then it ends up at location 218 00:16:32,990 --> 00:16:37,909 0 prime which is here. Now it has a velocity in this 219 00:16:37,909 --> 00:16:43,890 direction, and that velocity u in this direction is now A times 220 00:16:43,890 --> 00:16:45,916 delta t. That is 8.01. 221 00:16:45,916 --> 00:16:50,546 And so it is cruising now with constant velocity, 222 00:16:50,546 --> 00:16:54,115 and we just let it cruise all the way. 223 00:16:54,115 --> 00:16:57,877 It is now, again, a charge with constant 224 00:16:57,877 --> 00:17:01,843 velocity. And I look at it t second 225 00:17:01,843 --> 00:17:04,423 later. And so at time t we will find 226 00:17:04,423 --> 00:17:07,815 it in 0 double prime. And it is still cruising, 227 00:17:07,815 --> 00:17:11,870 and we just let it cruise. We are not going to interfere 228 00:17:11,870 --> 00:17:14,894 with it anymore. And here is then 0 double 229 00:17:14,894 --> 00:17:16,000 prime. 230 00:17:16,000 --> 00:17:22,000 231 00:17:22,000 --> 00:17:28,769 The whole exercise from 0 to 0 double prime took so many 232 00:17:28,769 --> 00:17:32,545 seconds. That means there is a sphere 233 00:17:32,545 --> 00:17:37,636 around point zero and that sphere has a radius which is c 234 00:17:37,636 --> 00:17:42,090 time t plus delta t. Outside that sphere the world 235 00:17:42,090 --> 00:17:47,181 has no knowledge that this charge was accelerated because 236 00:17:47,181 --> 00:17:51,727 that message has to travel with the speed of light. 237 00:17:51,727 --> 00:17:56,909 I am going to draw a circle, but in reality it is a sphere 238 00:17:56,909 --> 00:18:00,454 in all directions, which has a radius c, 239 00:18:00,454 --> 00:18:04,708 t plus delta t. Delta t, by the way, 240 00:18:04,708 --> 00:18:09,493 is way much smaller than t. And outside that sphere there 241 00:18:09,493 --> 00:18:13,167 is no knowledge, the world does not have any 242 00:18:13,167 --> 00:18:18,037 clue about the fact that this object is being accelerated. 243 00:18:18,037 --> 00:18:23,250 I will mark again to make sure that you can make a connection. 244 00:18:23,250 --> 00:18:28,376 This one has its center at 0 and the radius is c times t plus 245 00:18:28,376 --> 00:18:32,479 delta t. And if you ask me what is the 246 00:18:32,479 --> 00:18:37,865 electric field right there that is radially pointing outwards if 247 00:18:37,865 --> 00:18:41,455 q is positive, this world does not know yet 248 00:18:41,455 --> 00:18:46,585 that there was an acceleration. And if you ask me what is the 249 00:18:46,585 --> 00:18:50,518 electric field here, the field line is line so. 250 00:18:50,518 --> 00:18:55,049 And so the electric field, if it is a positive charge, 251 00:18:55,049 --> 00:19:00,093 is pointing radially outward and has no knowledge that there 252 00:19:00,093 --> 00:19:05,187 was any acceleration. And so I call this my position 253 00:19:05,187 --> 00:19:09,416 vector r, but you can take that r in any direction that you 254 00:19:09,416 --> 00:19:10,000 want. 255 00:19:10,000 --> 00:19:15,000 256 00:19:15,000 --> 00:19:19,531 Now let's look at the world that does know that there was a 257 00:19:19,531 --> 00:19:22,421 change. When the object was at 0 prime 258 00:19:22,421 --> 00:19:26,796 the acceleration stopped, and so it started cruising with 259 00:19:26,796 --> 00:19:30,703 a constant velocity. And so from that moment on the 260 00:19:30,703 --> 00:19:35,000 field lines are again nicely radially outwards. 261 00:19:35,000 --> 00:19:39,692 And so by the time that it reaches point 0 double prime, 262 00:19:39,692 --> 00:19:43,617 I can draw these field lines radially outwards. 263 00:19:43,617 --> 00:19:47,030 And I think, of this original field line, 264 00:19:47,030 --> 00:19:52,235 as a stick that was connected to the charge in that direction. 265 00:19:52,235 --> 00:19:55,392 I could have picked another direction. 266 00:19:55,392 --> 00:20:00,000 And so that field line is now here outwards. 267 00:20:00,000 --> 00:20:05,954 And the world that knows about this is the world which has a 268 00:20:05,954 --> 00:20:10,495 sphere around 0 prime with a radius c times t. 269 00:20:10,495 --> 00:20:14,532 And so I am going to draw another circle, 270 00:20:14,532 --> 00:20:19,376 which in reality is a sphere about point 0 prime. 271 00:20:19,376 --> 00:20:23,211 And this sphere here has now radius ct. 272 00:20:23,211 --> 00:20:26,944 And so this, to remind you what it is, 273 00:20:26,944 --> 00:20:33,000 has the origin at 0 prime. And the radius is ct. 274 00:20:33,000 --> 00:20:38,019 And so everything inside that sphere recognizes that the field 275 00:20:38,019 --> 00:20:42,216 is radially outwards from 0 point 0 double prime and 276 00:20:42,216 --> 00:20:47,154 everything out side here still thinks that the electric field 277 00:20:47,154 --> 00:20:50,528 is like this. But this field line was this 278 00:20:50,528 --> 00:20:55,384 one, the stick that I attached to it, and so there must be a 279 00:20:55,384 --> 00:20:58,099 connection between here and there. 280 00:20:58,099 --> 00:21:03,201 And that connection is only in that very thin shell which has a 281 00:21:03,201 --> 00:21:10,832 thickness c delta t. Let me first make a drawing of 282 00:21:10,832 --> 00:21:17,436 a triangle. That is going to be important. 283 00:21:17,436 --> 00:21:25,651 It is this triangle. This length here has a length u 284 00:21:25,651 --> 00:21:33,307 perpendicular times t. And you can easily see that why 285 00:21:33,307 --> 00:21:38,536 that is the case because this distance here is ut. 286 00:21:38,536 --> 00:21:43,125 Now, you may say, well, it wasn't going with 287 00:21:43,125 --> 00:21:46,646 velocity u here. I grant you that. 288 00:21:46,646 --> 00:21:51,661 But, keep in mind, in whatever follows that u is 289 00:21:51,661 --> 00:21:55,823 way, way smaller than c, delta t is way, 290 00:21:55,823 --> 00:22:01,051 way smaller than t, and so r which is c times t is 291 00:22:01,051 --> 00:22:06,816 way, way larger than ut. When you see a little 292 00:22:06,816 --> 00:22:10,882 distortion in this picture, I cannot, of course, 293 00:22:10,882 --> 00:22:13,996 make delta t way, way smaller than t. 294 00:22:13,996 --> 00:22:17,024 You wouldn't see any decent picture. 295 00:22:17,024 --> 00:22:20,138 Let me first go to u perpendicular t. 296 00:22:20,138 --> 00:22:22,993 There is here a velocity vector u. 297 00:22:22,993 --> 00:22:28,010 And the component perpendicular to r is a component in this 298 00:22:28,010 --> 00:22:32,300 direction. It cruised for t seconds, 299 00:22:32,300 --> 00:22:35,521 so this length is u perpendicular t, 300 00:22:35,521 --> 00:22:40,674 to a very good approximation. Forgetting about this teeny 301 00:22:40,674 --> 00:22:44,723 weenie little distance. We know what this is. 302 00:22:44,723 --> 00:22:49,049 U perpendicular is the component of the vector u 303 00:22:49,049 --> 00:22:52,453 perpendicular to my position vector r. 304 00:22:52,453 --> 00:22:56,503 And this little section here must, of course, 305 00:22:56,503 --> 00:23:02,933 have thickness c delta t. That is the difference in this 306 00:23:02,933 --> 00:23:06,160 radius and this radius, c delta t. 307 00:23:06,160 --> 00:23:11,050 Look at this green triangle. Now comes a key thing. 308 00:23:11,050 --> 00:23:17,015 This electric field line must connect with that electric field 309 00:23:17,015 --> 00:23:19,753 line. It was one in the same. 310 00:23:19,753 --> 00:23:24,545 It has been broken now. And the connection must go 311 00:23:24,545 --> 00:23:31,000 through this thin layer. I will draw here a connection. 312 00:23:31,000 --> 00:23:34,869 And then here you have to round it off a little. 313 00:23:34,869 --> 00:23:38,820 It cannot be sharp. And this rounds off a little. 314 00:23:38,820 --> 00:23:43,760 This point here represents the beginning of the acceleration, 315 00:23:43,760 --> 00:23:47,629 and this represents the end of the acceleration. 316 00:23:47,629 --> 00:23:50,757 And so then the field line goes swoosh. 317 00:23:50,757 --> 00:23:54,874 And now comes my task, and that is to calculate the 318 00:23:54,874 --> 00:23:58,578 electric field inside that shell because that, 319 00:23:58,578 --> 00:24:01,542 and only that, is responsible for the 320 00:24:01,542 --> 00:24:07,683 electromagnetic radiation. And so the field line goes like 321 00:24:07,683 --> 00:24:10,540 this. There must be a component of 322 00:24:10,540 --> 00:24:15,907 the electric field which is in this direction and there must be 323 00:24:15,907 --> 00:24:19,197 a component which is in this direction. 324 00:24:19,197 --> 00:24:23,611 I just decompose it. And I call this one E parallel. 325 00:24:23,611 --> 00:24:27,680 My notation parallel always means parallel to r, 326 00:24:27,680 --> 00:24:34,000 and my notation perpendicular always means perpendicular to r. 327 00:24:34,000 --> 00:24:41,172 And so this then would be called E perpendicular. 328 00:24:41,172 --> 00:24:49,689 And this triangle that you see here is congruent with this 329 00:24:49,689 --> 00:24:55,068 green one. That is why I gave you the 330 00:24:55,068 --> 00:25:01,941 dimension of the green one. And so the question now is what 331 00:25:01,941 --> 00:25:04,008 is this component E perpendicular? 332 00:25:04,008 --> 00:25:07,327 You can just feel in your stomach that that is the one 333 00:25:07,327 --> 00:25:10,396 that is responsible, that is electric field in the 334 00:25:10,396 --> 00:25:13,653 traveling wave that moves outwards because this whole 335 00:25:13,653 --> 00:25:15,845 shell moves outwards with a speed c. 336 00:25:15,845 --> 00:25:19,102 And notice this one is perpendicular to the direction 337 00:25:19,102 --> 00:25:22,171 that I have chosen of propagation you are watching 338 00:25:22,171 --> 00:25:24,363 here. And this shell comes over you. 339 00:25:24,363 --> 00:25:27,933 And it is always the E field that is perpendicular to your 340 00:25:27,933 --> 00:25:33,021 line of sight. That is the E field in the 341 00:25:33,021 --> 00:25:38,017 traveling wave. Our task is now to calculate 342 00:25:38,017 --> 00:25:45,105 the E vector perpendicular here. Well, if you look at the fact 343 00:25:45,105 --> 00:25:52,309 that this triangle is congruent with this one then you see that 344 00:25:52,309 --> 00:25:59,281 E perpendicular divided by E parallel must be u perpendicular 345 00:25:59,281 --> 00:26:04,495 t divided by c delta t. U perpendicular t, 346 00:26:04,495 --> 00:26:08,150 we know that u is at, so u perpendicular is a 347 00:26:08,150 --> 00:26:12,055 perpendicular time t. A perpendicular is now the 348 00:26:12,055 --> 00:26:16,043 component of the acceleration perpendicular to r. 349 00:26:16,043 --> 00:26:20,363 I don't want to put it in here because it becomes too 350 00:26:20,363 --> 00:26:23,270 cluttered. I will make a new drawing 351 00:26:23,270 --> 00:26:26,095 shortly. I will put the a in there. 352 00:26:26,095 --> 00:26:30,000 Acceleration is in this direction. 353 00:26:30,000 --> 00:26:33,593 And so a perpendicular is like this. 354 00:26:33,593 --> 00:26:37,495 And so I can replace u perpendicular t. 355 00:26:37,495 --> 00:26:41,089 I can replace it by a perpendicular. 356 00:26:41,089 --> 00:26:46,428 Sorry, this is delta t. You guys should have screamed 357 00:26:46,428 --> 00:26:53,000 because the acceleration only lasts for delta t seconds. 358 00:26:53,000 --> 00:27:00,000 359 00:27:00,000 --> 00:27:06,704 Now I get a perpendicular times delta t times t divided by c 360 00:27:06,704 --> 00:27:11,022 delta t. And so that is a perpendicular 361 00:27:11,022 --> 00:27:16,704 times t divided by c. If now I can calculate what E 362 00:27:16,704 --> 00:27:23,750 parallel is, I am done because then I know what E perpendicular 363 00:27:23,750 --> 00:27:27,500 is. I put E parallel here and I am 364 00:27:27,500 --> 00:27:31,961 in business. Before I do that, 365 00:27:31,961 --> 00:27:37,615 I want to eliminate t. And I am going to write for 366 00:27:37,615 --> 00:27:43,269 this t, r divided by c. It is also a perpendicular 367 00:27:43,269 --> 00:27:48,923 times r divided by c2. And so I can write down now 368 00:27:48,923 --> 00:27:54,000 that E perpendicular is this time E parallel. 369 00:27:54,000 --> 00:28:00,000 How do we find E parallel? 8.02 Gauss' Law. 370 00:28:00,000 --> 00:28:04,481 I make a pill box, and the pill box is going to be 371 00:28:04,481 --> 00:28:07,957 like this. I will make a drawing of it. 372 00:28:07,957 --> 00:28:13,719 And this surface is here in the world that doesn't know yet what 373 00:28:13,719 --> 00:28:17,286 happens. And this surface here is in the 374 00:28:17,286 --> 00:28:21,859 world which is in turmoil, which is the transition. 375 00:28:21,859 --> 00:28:26,158 And so I am going to make that pill box for you. 376 00:28:26,158 --> 00:28:31,700 And so I draw here again this. This is a line that goes 377 00:28:31,700 --> 00:28:34,039 somewhere through this point here. 378 00:28:34,039 --> 00:28:36,944 It is not this one. It is somewhere there. 379 00:28:36,944 --> 00:28:39,000 And here is my pill box. 380 00:28:39,000 --> 00:28:47,000 381 00:28:47,000 --> 00:28:50,979 That is my pill box. And so, in that pill box, 382 00:28:50,979 --> 00:28:56,020 I have in the outside world, the outside world here I have 383 00:28:56,020 --> 00:29:00,000 that E field. I will put that in as an E. 384 00:29:00,000 --> 00:29:04,963 It is in the world that doesn't know yet what happens. 385 00:29:04,963 --> 00:29:08,240 That is radially go through point 0. 386 00:29:08,240 --> 00:29:12,361 Inside the box, I have this E parallel coming 387 00:29:12,361 --> 00:29:16,481 in like this. And then going straight through 388 00:29:16,481 --> 00:29:20,976 the sides of the box is this one E perpendicular. 389 00:29:20,976 --> 00:29:26,501 This comes in E perpendicular, and that E perpendicular goes 390 00:29:26,501 --> 00:29:30,297 out. Now, since this is in vacuum, 391 00:29:30,297 --> 00:29:33,623 there is no charge density inside this box. 392 00:29:33,623 --> 00:29:36,158 The divergence of E must be zero. 393 00:29:36,158 --> 00:29:40,594 And that means that the E vector here must be exactly the 394 00:29:40,594 --> 00:29:45,029 same as the E vector here, because the contribution these 395 00:29:45,029 --> 00:29:47,960 two is zero so this must also be zero. 396 00:29:47,960 --> 00:29:50,574 There is no charge inside the box. 397 00:29:50,574 --> 00:29:53,267 But I do know what this E field is. 398 00:29:53,267 --> 00:29:56,435 That is 8.02. If someone tells you that I 399 00:29:56,435 --> 00:30:02,438 have a charge sitting there. What is the electric field at a 400 00:30:02,438 --> 00:30:04,955 distance r that is Coulomb's Law? 401 00:30:04,955 --> 00:30:08,022 This is simple. It is not Coulomb's Law. 402 00:30:08,022 --> 00:30:10,775 It is Gauss's Law. But, in any case, 403 00:30:10,775 --> 00:30:15,494 you will find very easy that his E vector is q divided by 4pi 404 00:30:15,494 --> 00:30:18,876 epsilon zero r2. It falls of as one over r2. 405 00:30:18,876 --> 00:30:22,101 You people should remember that from 8.02. 406 00:30:22,101 --> 00:30:24,539 And so now we have accomplished, 407 00:30:24,539 --> 00:30:30,327 the 13 minutes are almost up. E perpendicular now is, 408 00:30:30,327 --> 00:30:35,893 therefore, A perpendicular times r divided by c2 times q 409 00:30:35,893 --> 00:30:39,537 divided by 4pi epsilon zero times r2. 410 00:30:39,537 --> 00:30:44,091 And you lose one r. And so this is the classic 411 00:30:44,091 --> 00:30:50,366 derivation also already known in the late 19th century which is 412 00:30:50,366 --> 00:30:55,426 now the electric vector, the strength of the vector 413 00:30:55,426 --> 00:31:01,599 which clearly is responsible for the electromagnetic traveling 414 00:31:01,599 --> 00:31:07,773 wave because it is perpendicular to my direction r that I have 415 00:31:07,773 --> 00:31:13,173 chosen. It is inversely proportional to 416 00:31:13,173 --> 00:31:15,937 r. I will come back to that. 417 00:31:15,937 --> 00:31:21,977 That is a natural consequence of the conservation of energy. 418 00:31:21,977 --> 00:31:27,300 And this field is very different from static electric 419 00:31:27,300 --> 00:31:30,985 fields which fall off as one over r2. 420 00:31:30,985 --> 00:31:37,332 This is a travel wave that has an E field that falls off as one 421 00:31:37,332 over r, not as one over r2.