1 00:00:00 --> 00:00:00,336 2 00:00:00,336 --> 00:00:06,024 I'm going to talk today about energy in electromagnetic waves. 3 00:00:06,024 --> 00:00:11,527 There must be energy in there, because we know that electric 4 00:00:11,527 --> 00:00:16,096 fields contain energy, and magnetic fields contain 5 00:00:16,096 --> 00:00:19,454 energy. And you may remember that the 6 00:00:19,454 --> 00:00:25,422 electric field energy density is one-half epsilon zero E squared, 7 00:00:25,422 --> 00:00:31,111 this is now in joules per cubic meter, and the magnetic energy 8 00:00:31,111 --> 00:00:35,988 field density, which also has come up 9 00:00:35,988 --> 00:00:41,456 earlier in the course, is one over two mu zero times B 10 00:00:41,456 --> 00:00:45,377 squared. Again, joules per cubic meter. 11 00:00:45,377 --> 00:00:49,917 Now, we -- when we deal with traveling waves, 12 00:00:49,917 --> 00:00:55,695 in vacuum, at any moment in time, the magnitude of B is E 13 00:00:55,695 --> 00:01:00,132 divided by C. So this is also one divided by 14 00:01:00,132 --> 00:01:05,909 two mu zero, I can replace this by E squared 15 00:01:05,909 --> 00:01:09,492 divided by C squared. But C squared is one over 16 00:01:09,492 --> 00:01:12,997 epsilon zero my zero. So this is also one-half 17 00:01:12,997 --> 00:01:16,501 epsilon zero E squared. And when you see this, 18 00:01:16,501 --> 00:01:21,174 this is an absolutely amazing result, because what this tells 19 00:01:21,174 --> 00:01:25,535 you is that the energy density in the magnetic field of a 20 00:01:25,535 --> 00:01:30,052 traveling wave is exactly the same as the energy density in 21 00:01:30,052 --> 00:01:33,557 the electric field of an electromagnetic wave. 22 00:01:33,557 --> 00:01:39,806 That's really an amazing thing, the symmetry is absolutely 23 00:01:39,806 --> 00:01:43,907 beautiful. So the total energy density is 24 00:01:43,907 --> 00:01:48,009 the sum of the two, so I double this one, 25 00:01:48,009 --> 00:01:53,956 so that it's epsilon zero E squared joules per cubic meter, 26 00:01:53,956 --> 00:01:58,058 and of course, I can also write for that, 27 00:01:58,058 --> 00:02:02,57 epsilon zero, just one E, and then the other, 28 00:02:02,57 --> 00:02:07,492 I write B times C. So I write for E, 29 00:02:07,492 --> 00:02:10,245 B times C. So this is, again, 30 00:02:10,245 --> 00:02:15,26 in joules per cubic meter. Yes, I'm happy with that, 31 00:02:15,26 --> 00:02:18,603 that's fine. Now, I want to ask the 32 00:02:18,603 --> 00:02:23,028 question, if electromagnetic waves come by me, 33 00:02:23,028 --> 00:02:27,65 how much energy passes through one square meter? 34 00:02:27,65 --> 00:02:32,96 It's like an energy flux. And so I have here one square 35 00:02:32,96 --> 00:02:37,68 meter, and this one square meter is 36 00:02:37,68 --> 00:02:42,219 perpendicular to the direction that the electromagnetic wave is 37 00:02:42,219 --> 00:02:46,685 going, and I want to know how much energy flows through there. 38 00:02:46,685 --> 00:02:49,907 One square meter every second. In one second, 39 00:02:49,907 --> 00:02:54,154 light travels a distance C, which is a horrendous distance, 40 00:02:54,154 --> 00:02:56,57 three hundred thousand kilometers. 41 00:02:56,57 --> 00:03:00,743 This side of this box is what light travels in one second. 42 00:03:00,743 --> 00:03:05,063 And this is one square meter, and I'm going to calculate how 43 00:03:05,063 --> 00:03:10,555 energy now goes through this one square meter in one second. 44 00:03:10,555 --> 00:03:14,658 Of course, I could have chosen this box a billion times 45 00:03:14,658 --> 00:03:18,61 smaller, would have gotten the same answer of course, 46 00:03:18,61 --> 00:03:21,802 but for convenience, I choose this to be C, 47 00:03:21,802 --> 00:03:24,158 and this to be one square meter. 48 00:03:24,158 --> 00:03:27,425 So the volume of that box is C cubic meters. 49 00:03:27,425 --> 00:03:32,137 And all the energy of that box is going to come out here in one 50 00:03:32,137 --> 00:03:36,545 second, because I know that electromagnetic waves move with 51 00:03:36,545 --> 00:03:38,825 the speed of light, which is C. 52 00:03:38,825 --> 00:03:44,113 And so therefore, the energy that comes out here 53 00:03:44,113 --> 00:03:49,614 per square meter per second is that U total there that I have, 54 00:03:49,614 --> 00:03:54,212 which is the amount of energy for every cubic meter, 55 00:03:54,212 --> 00:03:57,007 but I have so many cubic meters. 56 00:03:57,007 --> 00:04:02,417 And so I can use this result, now, here, and I can substitute 57 00:04:02,417 --> 00:04:08,188 that here, so I get epsilon zero times E times B times C squared. 58 00:04:08,188 --> 00:04:11,434 I have to multiply it by this C here. 59 00:04:11,434 --> 00:04:16,843 And this, of course, is also E times B divided 60 00:04:16,843 --> 00:04:23,393 by mu zero, because C squared, in vacuum, is one over epsilon 61 00:04:23,393 --> 00:04:26,449 zero, mu zero. And this, now, 62 00:04:26,449 --> 00:04:30,597 is joules per square meter, per second. 63 00:04:30,597 --> 00:04:35,836 Because in one second, all that energy comes out, 64 00:04:35,836 --> 00:04:40,966 and I have already chosen one square meter area. 65 00:04:40,966 --> 00:04:45,66 Let me see whether I'm happy with that, yes, 66 00:04:45,66 --> 00:04:52,047 I'm happy with that. We call this the pointing 67 00:04:52,047 --> 00:04:54,61 vector. And we write it, 68 00:04:54,61 --> 00:04:58,51 in general as a vector. We write it, 69 00:04:58,51 --> 00:05:03,413 S with a vector, and it's called the pointing 70 00:05:03,413 --> 00:05:08,538 vector with a Y, and we write that as E cross B 71 00:05:08,538 --> 00:05:13,218 divided by mu zero. E B divided by mu zero. 72 00:05:13,218 --> 00:05:19,793 You don't need the cross, really, because E and 73 00:05:19,793 --> 00:05:24,323 B are always perpendicular to each other in a traveling wave. 74 00:05:24,323 --> 00:05:28,854 The advantage of this notation is that S, which is the energy 75 00:05:28,854 --> 00:05:32,857 flux, goes in a certain direction, and the velocity of 76 00:05:32,857 --> 00:05:36,557 the weave is always in the direction of E cross B, 77 00:05:36,557 --> 00:05:40,257 so it also tells you, then, in which direction the 78 00:05:40,257 --> 00:05:42,825 radiation is flowing, whereas here, 79 00:05:42,825 --> 00:05:45,241 you -- you lack that information. 80 00:05:45,241 --> 00:05:47,657 And so this, then, to remind you, 81 00:05:47,657 --> 00:05:50,905 is in these units, which is watts per square 82 00:05:50,905 --> 00:05:54,934 meter. How many joules per second? 83 00:05:54,934 --> 00:05:59,059 One square meter through a plane perpendicular to the 84 00:05:59,059 --> 00:06:03,421 direction of propagation. Now, E and B are changing with 85 00:06:03,421 --> 00:06:07,229 a frequency omega, cosines omega T or sines omega 86 00:06:07,229 --> 00:06:09,688 T. And so S is changing with the 87 00:06:09,688 --> 00:06:14,05 cosine square of omega T. If you are somewhere in space, 88 00:06:14,05 --> 00:06:18,889 and this electromagnetic wave comes by, there are moments that 89 00:06:18,889 --> 00:06:23,966 S is zero, namely, when E and B happen to be zero. 90 00:06:23,966 --> 00:06:27,074 And there are moments that it is at maximum, 91 00:06:27,074 --> 00:06:29,676 when E and B happen to be at maximum. 92 00:06:29,676 --> 00:06:33,29 And so when we deal with electromagnetic radiation, 93 00:06:33,29 --> 00:06:37,194 it's more meaningful to discuss the time-average value. 94 00:06:37,194 --> 00:06:40,953 And the time-average value of the pointing vector is, 95 00:06:40,953 --> 00:06:44,061 first of all, the time-average of the cosine 96 00:06:44,061 --> 00:06:47,097 square omega T, or the sine square omega T, 97 00:06:47,097 --> 00:06:51,145 whatever the case may be. And the average value of cosine 98 00:06:51,145 --> 00:06:53,919 squared is one-half. 99 00:06:53,919 --> 00:06:57,482 And then I can write now, for E, E zero, 100 00:06:57,482 --> 00:07:02,964 and for B, I can write B zero, which is now the maximum value 101 00:07:02,964 --> 00:07:08,081 possible, divided by mu zero. And if you want to write it 102 00:07:08,081 --> 00:07:13,015 differently, if you want to write it only in terms of E 103 00:07:13,015 --> 00:07:18,497 zero, then you can write down one-half E zero squared divided 104 00:07:18,497 --> 00:07:22,883 by mu zero times C. So this is the more practical 105 00:07:22,883 --> 00:07:27,014 equation, that gives you a number which 106 00:07:27,014 --> 00:07:29,675 is time-averaged over the oscillations. 107 00:07:29,675 --> 00:07:33,805 Let me make sure that there is nowhere, a slip of the pen -- 108 00:07:33,805 --> 00:07:37,305 no, there isn't -- I hate to have slips of the pen, 109 00:07:37,305 --> 00:07:41,154 because you can't edit them out later on your videotape. 110 00:07:41,154 --> 00:07:43,675 Slips of the tongue, you can edit it. 111 00:07:43,675 --> 00:07:45,845 The slips of the pen, you can't. 112 00:07:45,845 --> 00:07:49,065 OK, this looks good. So now, we have an average 113 00:07:49,065 --> 00:07:52,845 value for the pointing vector, and so we can calculate, 114 00:07:52,845 --> 00:07:56,589 now, how much energy flows through 115 00:07:56,589 --> 00:08:00,997 one square meter per second. And I can give you an example, 116 00:08:00,997 --> 00:08:04,265 suppose I have a plane electromagnetic wave, 117 00:08:04,265 --> 00:08:07,381 and E zero is one hundred volts per meter. 118 00:08:07,381 --> 00:08:10,194 That's just what it is, how we get it, 119 00:08:10,194 --> 00:08:13,842 that's a different story. And this could be radio 120 00:08:13,842 --> 00:08:17,642 emission, this could infrared, this could be light, 121 00:08:17,642 --> 00:08:20,226 I d- I don't specify the frequency. 122 00:08:20,226 --> 00:08:23,342 I don't have to. Frequency doesn't come up 123 00:08:23,342 --> 00:08:28,351 there. And so the value for -- average 124 00:08:28,351 --> 00:08:34 value for the pointing vector, I can pick this equation. 125 00:08:34 --> 00:08:37,39 E zero is hundred. I know mu zero, 126 00:08:37,39 --> 00:08:41,396 I know C, so I can calculate what it is. 127 00:08:41,396 --> 00:08:45,402 It's one hundred squared divided by two, 128 00:08:45,402 --> 00:08:48,689 divided by mu zero, divided by C. 129 00:08:48,689 --> 00:08:53,517 And when I do that, I find that this is thirteen 130 00:08:53,517 --> 00:08:56,773 watts per square meter. 131 00:08:56,773 --> 00:09:01,298 So imagine that you would stand in this electromagnetic wave 132 00:09:01,298 --> 00:09:04,75 coming to you, we take all our clothes off and 133 00:09:04,75 --> 00:09:08,125 we let it hit us, and suppose it absorbed us. 134 00:09:08,125 --> 00:09:12,727 Suppose it is radiation that absorbs us -- some radiation may 135 00:09:12,727 --> 00:09:15,872 go through you. Gamma rays may go straight 136 00:09:15,872 --> 00:09:19,631 through you -- they are electromagnetic radiation. 137 00:09:19,631 --> 00:09:23,006 But certainly, light will not go through you. 138 00:09:23,006 --> 00:09:26,074 And radio waves, some of them will not go 139 00:09:26,074 --> 00:09:30,352 through you. So you observe them with your 140 00:09:30,352 --> 00:09:32,192 body. Would you notice that? 141 00:09:32,192 --> 00:09:34,781 I doubt it. You probably have a surface 142 00:09:34,781 --> 00:09:37,303 area that's close to one square meter. 143 00:09:37,303 --> 00:09:40,165 Thirteen watts, thirteen joules per second, 144 00:09:40,165 --> 00:09:42,345 not very noticeable. You radiate, 145 00:09:42,345 --> 00:09:45,753 yourself, hundred watts, hundred joules per second, 146 00:09:45,753 --> 00:09:48,274 so I don't think you will notice that. 147 00:09:48,274 --> 00:09:51,204 But imagine, now, that we increase the value 148 00:09:51,204 --> 00:09:54,612 of E zero, and we make it thousand volts per meter. 149 00:09:54,612 --> 00:09:58,292 Now, this goes up by a factor of a 150 00:09:58,292 --> 00:10:01,665 hundred, because if E goes up by a factor of ten, 151 00:10:01,665 --> 00:10:04,827 automatically, B goes up by a factor of ten -- 152 00:10:04,827 --> 00:10:08,903 remember, in electromagnetic waves, they're always coupled. 153 00:10:08,903 --> 00:10:12,416 So the pointing vector, which is the product of the 154 00:10:12,416 --> 00:10:15,579 two, goes up by a factor of a hundred, so now, 155 00:10:15,579 --> 00:10:19,514 you're talking about one point three kilowatts per square 156 00:10:19,514 --> 00:10:22,044 meter. And if you absorb that on your 157 00:10:22,044 --> 00:10:24,573 body, believe me, uh, it may fry you. 158 00:10:24,573 --> 00:10:28,72 Certainly if you do that out on the beach, 159 00:10:28,72 --> 00:10:32,832 you get a -- a deep suntan. And if you do it long enough, 160 00:10:32,832 --> 00:10:35,549 then you can hurt yourself very badly. 161 00:10:35,549 --> 00:10:39,514 So now comes the question, does a light bulb emit plane 162 00:10:39,514 --> 00:10:41,203 waves? Well, not really. 163 00:10:41,203 --> 00:10:44,875 Plane waves have no beginning and they have no end, 164 00:10:44,875 --> 00:10:47,518 they exist at all time and all space. 165 00:10:47,518 --> 00:10:51,117 Look at my plane wave solutions from last lecture. 166 00:10:51,117 --> 00:10:54,495 You can substitute in there any value for X, Y, 167 00:10:54,495 --> 00:11:00,076 and Z, and any moment in time, the year five thousand B.C., 168 00:11:00,076 --> 00:11:02,676 you get an answer. Doesn't specify when, 169 00:11:02,676 --> 00:11:06,078 doesn't specify where. And of course that's not very 170 00:11:06,078 --> 00:11:07,945 realistic. In the real world, 171 00:11:07,945 --> 00:11:12,08 there is a beginning and there is an end to the electromagnetic 172 00:11:12,08 --> 00:11:15,815 radiation, and therefore, they have also a finite length. 173 00:11:15,815 --> 00:11:19,949 Remember the quarter-nanosecond pulses that we sent to the moon 174 00:11:19,949 --> 00:11:23,151 that we discussed last time. They were only seven 175 00:11:23,151 --> 00:11:26,085 centimeters long. That's not very much like a 176 00:11:26,085 --> 00:11:30,086 plane wave. So I want to discuss with you a 177 00:11:30,086 --> 00:11:34,377 little further what these waves look like, and I want to take a 178 00:11:34,377 --> 00:11:37,353 closer look at how electromagnetic waves are 179 00:11:37,353 --> 00:11:40,26 produced by charges that we begin to shake, 180 00:11:40,26 --> 00:11:43,859 that we are accelerating. Key in the whole process is 181 00:11:43,859 --> 00:11:48,011 that you accelerate a charge. If a charge is just moving at a 182 00:11:48,011 --> 00:11:50,572 constant velocity, it will not produce 183 00:11:50,572 --> 00:11:54,24 electromagnetic radiation. But I want to give you some 184 00:11:54,24 --> 00:11:58,462 feeling, at least some classical physical feeling on how these 185 00:11:58,462 --> 00:12:02,052 electromagnetic waves are produced. 186 00:12:02,052 --> 00:12:05,239 It is a picture that has its own limitations, 187 00:12:05,239 --> 00:12:08,932 but it's still useful. It's not a quantum mechanical 188 00:12:08,932 --> 00:12:12,988 treatment, but it is something that you and I can see and 189 00:12:12,988 --> 00:12:15,16 therefore, perhaps, appreciate. 190 00:12:15,16 --> 00:12:18,636 Suppose I have here a charge which is not moving. 191 00:12:18,636 --> 00:12:21,605 Just sitting there. These are field lines, 192 00:12:21,605 --> 00:12:24,068 I just draw only a few field lines. 193 00:12:24,068 --> 00:12:27,978 And if it's a positive charge, the arrows are outwards, 194 00:12:27,978 --> 00:12:33,338 if it's a negative charge, the arrows are inwards. 195 00:12:33,338 --> 00:12:38,156 And I'm going to accelerate this for a time delta T. 196 00:12:38,156 --> 00:12:41,652 Let's accelerate it in this direction. 197 00:12:41,652 --> 00:12:45,809 And then we bring it, again, to a halt state. 198 00:12:45,809 --> 00:12:49,966 I redraw this. This point is the same as this 199 00:12:49,966 --> 00:12:52,517 point here. I accelerate it, 200 00:12:52,517 --> 00:12:57,052 and delta T seconds later, it happens to be here. 201 00:12:57,052 --> 00:13:01,493 So this is T zero, and this is T equals delta T. 202 00:13:01,493 --> 00:13:06,5 And now I'm going to draw a circle -- 203 00:13:06,5 --> 00:13:13,68 it should actually be a sphere, three-dimensionally -- about 204 00:13:13,68 --> 00:13:17,939 this point. And here is that sphere. 205 00:13:17,939 --> 00:13:22,319 And this sphere has radius C delta T. 206 00:13:22,319 --> 00:13:27,795 This field line, which was the field line that 207 00:13:27,795 --> 00:13:34,245 goes with this charge, is here, and this field line is 208 00:13:34,245 --> 00:13:37,832 here, and this one is here, 209 00:13:37,832 --> 00:13:40,722 I only draw three. This one, this one, 210 00:13:40,722 --> 00:13:43,456 and this one. And the message that I 211 00:13:43,456 --> 00:13:48,065 accelerated this charge could not possibly have reached this 212 00:13:48,065 --> 00:13:51,736 location in space, because that message can only 213 00:13:51,736 --> 00:13:56,423 travel with the speed of light. So the electric field here is 214 00:13:56,423 --> 00:14:00,251 exactly the same as it was when it was still here. 215 00:14:00,251 --> 00:14:04,626 So when the object is here, the electric field must still 216 00:14:04,626 --> 00:14:09,477 be like this here, and it must be like this here, 217 00:14:09,477 --> 00:14:13,648 and it must be like this there, because the message hasn't 218 00:14:13,648 --> 00:14:17,161 reached that point. But now, look at this charge, 219 00:14:17,161 --> 00:14:19,723 which is now here at time D delta T. 220 00:14:19,723 --> 00:14:23,089 Now, the electric field is like so, is like so, 221 00:14:23,089 --> 00:14:26,456 and is like so. So this field line must somehow 222 00:14:26,456 --> 00:14:30,042 meet up with this one, it's one and the same field 223 00:14:30,042 --> 00:14:32,164 line. And what does that mean? 224 00:14:32,164 --> 00:14:35,677 That somewhere there, there must be a kink in the 225 00:14:35,677 --> 00:14:39,922 electric field. And there must be a kink 226 00:14:39,922 --> 00:14:42,635 here. Notice there is no kink here, 227 00:14:42,635 --> 00:14:46,706 which is interesting. And so it is the collection of 228 00:14:46,706 --> 00:14:50,936 these kinks that propagate outwards, with the speed of 229 00:14:50,936 --> 00:14:55,246 light, and they produce an electromagnetic disturbance, 230 00:14:55,246 --> 00:14:58,359 a change. If you were out in space here, 231 00:14:58,359 --> 00:15:02,509 and if, for instance, I were to oscillate this charge 232 00:15:02,509 --> 00:15:06,18 back and forth, you would see these kinks go by 233 00:15:06,18 --> 00:15:10,011 all the time, these breaks in the 234 00:15:10,011 --> 00:15:13,325 electric field, and you would experience that 235 00:15:13,325 --> 00:15:17,692 as an electromagnetic wave, if there is a changing electric 236 00:15:17,692 --> 00:15:22,21 field, according to Maxwell's equations, there has to be also 237 00:15:22,21 --> 00:15:26,351 a changing magnetic field. But the interesting thing is, 238 00:15:26,351 --> 00:15:29,89 even though this is an extremely simple picture, 239 00:15:29,89 --> 00:15:33,429 notice that in this direction, if you were here, 240 00:15:33,429 --> 00:15:37,721 you would not see any kinks. So there's no electromagnetic 241 00:15:37,721 --> 00:15:41,41 radiation going in this direction, 242 00:15:41,41 --> 00:15:44,033 nor is there any going in this direction. 243 00:15:44,033 --> 00:15:46,787 And the maximum is going in this direction, 244 00:15:46,787 --> 00:15:50,132 and something in between is going in that direction. 245 00:15:50,132 --> 00:15:53,148 It's not much of a plane wave, for that matter. 246 00:15:53,148 --> 00:15:56,164 I mean, if anything, it's more like a spherical 247 00:15:56,164 --> 00:15:58 wave. But it's a very special 248 00:15:58 --> 00:16:00,82 spherical wave, not the same strength in all 249 00:16:00,82 --> 00:16:03,377 directions. And so even though this is a 250 00:16:03,377 --> 00:16:05,803 rather classical picture, it helps me, 251 00:16:05,803 --> 00:16:09,541 at least, to see how these changing electric fields -- and 252 00:16:09,541 --> 00:16:13,523 therefore, associated B fields -- are 253 00:16:13,523 --> 00:16:16,222 formed by charges that we accelerate. 254 00:16:16,222 --> 00:16:19,596 And I have a two-minute movie that shows that, 255 00:16:19,596 --> 00:16:23,72 also, in a slightly more detailed way that I was able to 256 00:16:23,72 --> 00:16:25,819 do. And so let's look at that 257 00:16:25,819 --> 00:16:28,294 movie, Marcos, you ready for that? 258 00:16:28,294 --> 00:16:32,717 OK, then you can start that. So this is a computer-generated 259 00:16:32,717 --> 00:16:35,342 movie whereby we accelerate charges. 260 00:16:35,342 --> 00:16:39,316 This is a constant speed, now, and we're going to stop 261 00:16:39,316 --> 00:16:42,914 it. Stopping means there is an 262 00:16:42,914 --> 00:16:43,963 acceleration, right? 263 00:16:43,963 --> 00:16:46,665 You may call it a deceleration, but stopping is an 264 00:16:46,665 --> 00:16:48,761 acceleration. It's going to be stopped. 265 00:16:48,761 --> 00:16:50,637 And you see now, here, these kinks? 266 00:16:50,637 --> 00:16:53,174 And they propagate out with the speed of light, 267 00:16:53,174 --> 00:16:55,546 then, if this was electromagnetic radiation. 268 00:16:55,546 --> 00:16:57,642 You're going to see this several times, 269 00:16:57,642 --> 00:17:00,566 so you get another chance. Your charge is now stopped, 270 00:17:00,566 --> 00:17:03,82 you see the magnetic -- the electric field lines coming out, 271 00:17:03,82 --> 00:17:07,13 and now it's being accelerated. And you see these kinks here? 272 00:17:07,13 --> 00:17:09,005 Moving out with the speed of light. 273 00:17:09,005 --> 00:17:11,046 You're going to see more. There it is, 274 00:17:11,046 --> 00:17:16,334 it's going to be accelerated, and doing the acceleration, 275 00:17:16,334 --> 00:17:21,382 only during the acceleration do you see the formation of the 276 00:17:21,382 --> 00:17:24,718 kinks, when it goes with constant speed, 277 00:17:24,718 --> 00:17:28,397 no longer, it's only during -- now it stops. 278 00:17:28,397 --> 00:17:33,274 Stops means an acceleration. There you see this wave front 279 00:17:33,274 --> 00:17:36,867 moving out. Let's look a little bit more as 280 00:17:36,867 --> 00:17:39,776 you can see the oscillating effect. 281 00:17:39,776 --> 00:17:44,738 So you already have seen this, acceleration, 282 00:17:44,738 --> 00:17:47,363 here is the wave, it stops, that means 283 00:17:47,363 --> 00:17:50,059 deceleration, and here you see the wave 284 00:17:50,059 --> 00:17:52,471 front. And when it's sitting still, 285 00:17:52,471 --> 00:17:55,167 or when it's going with constant speed, 286 00:17:55,167 --> 00:17:58,431 then there is no electromagnetic wave produced. 287 00:17:58,431 --> 00:18:01,766 Now we're going to see some oscillating charges, 288 00:18:01,766 --> 00:18:05,172 which is more realistic when you have an antenna, 289 00:18:05,172 --> 00:18:09,358 and you have current going up and down with frequency omega, 290 00:18:09,358 --> 00:18:13,473 you obviously have stops and starts, it oscillates back and 291 00:18:13,473 --> 00:18:17,305 forth, and that's what you see now. 292 00:18:17,305 --> 00:18:22,216 And look at these beautiful -- there's a wave going out, 293 00:18:22,216 --> 00:18:26,949 here's a wave going out, only during the acceleration. 294 00:18:26,949 --> 00:18:31,236 There's one going out, and there's one going out, 295 00:18:31,236 --> 00:18:35,612 so you accelerated backwards, forwards, backwards, 296 00:18:35,612 --> 0. forwards. 297 0. --> 00:18:36,505 298 00:18:36,505 --> 00:18:39,898 I think that's fine, thank you, Marcos. 299 00:18:39,898 --> 00:18:44,274 So the classic picture, even though it has lots of 300 00:18:44,274 --> 00:18:49,99 limitations, it's not a quantum mechanical 301 00:18:49,99 --> 00:18:52,26 treatment, is still very useful. 302 00:18:52,26 --> 00:18:55,483 For instance, if we think of a -- an antenna, 303 00:18:55,483 --> 00:18:59,439 just a straight wire with a current, going up and down, 304 00:18:59,439 --> 00:19:03,101 with high frequency, it could be seventy megaHertz, 305 00:19:03,101 --> 00:19:06,764 could be gigaHertz, and you produce electromagnetic 306 00:19:06,764 --> 00:19:10,793 radiation, then you are accelerating charges up and down 307 00:19:10,793 --> 00:19:15,041 by having current going like this, then from this classical 308 00:19:15,041 --> 00:19:18,924 picture, we know that no radiation will go out in this 309 00:19:18,924 --> 00:19:22,654 direction. But we also know that in the 310 00:19:22,654 --> 00:19:25,347 direction perpendicular to the acceleration, 311 00:19:25,347 --> 00:19:28,039 that is in this direction -- remember, here, 312 00:19:28,039 --> 00:19:31,67 the acceleration was like so -- and in the direction of the 313 00:19:31,67 --> 00:19:34,174 acceleration, no electromagnetic waves go 314 00:19:34,174 --> 00:19:35,99 out. So nothing goes out here, 315 00:19:35,99 --> 00:19:39,308 nothing goes out there. But in the plane perpendicular 316 00:19:39,308 --> 00:19:42,125 to A, which, in this case, was the blackboard, 317 00:19:42,125 --> 00:19:44,88 is a whole plane like this, and in this case, 318 00:19:44,88 --> 00:19:47,321 of the antenna, is the horizontal plane, 319 00:19:47,321 --> 00:19:50,577 here we have a maximum radiation going out everywhere 320 00:19:50,577 --> 00:19:52,878 in this plane. 321 00:19:52,878 --> 00:19:56,944 And in between somewhere, it's not zero and it is not the 322 00:19:56,944 --> 00:20:00,067 maximum value. So now I would like to return 323 00:20:00,067 --> 00:20:02,535 to the idea of the pointing vector. 324 00:20:02,535 --> 00:20:06,021 Because we talked only about the pointing vector, 325 00:20:06,021 --> 00:20:09,433 we derive that in terms of plane wave solutions. 326 00:20:09,433 --> 00:20:13,718 And we now know that plane wave solutions are really not too 327 00:20:13,718 --> 00:20:16,549 realistic. Suppose we have the sun here. 328 00:20:16,549 --> 00:20:19,672 And the sun is a really powerful light bulb, 329 00:20:19,672 --> 00:20:22,649 three times ten to the six watts radiates, 330 00:20:22,649 --> 00:20:27,798 largely invisible light, and in infrared radiation, 331 00:20:27,798 --> 00:20:30,354 which is electromagnetic radiation. 332 00:20:30,354 --> 00:20:33,737 And we are a distance of hundred fifty million 333 00:20:33,737 --> 00:20:37,496 kilometers, here somewhere on Earth, and therefore, 334 00:20:37,496 --> 00:20:41,931 through every square meter in the direction perpendicular to 335 00:20:41,931 --> 00:20:46,141 the line of sight to the sun, through every square meter, 336 00:20:46,141 --> 00:20:50,577 there is about one kilowatt flowing, one kilowatt per square 337 00:20:50,577 --> 00:20:53,208 meter. And you can calculate that at 338 00:20:53,208 --> 00:20:57,794 home to verify that, that's easy calculation, 339 00:20:57,794 --> 00:21:01,597 you know that it is three times ten to the six joules per 340 00:21:01,597 --> 00:21:05,332 second, and if the distance is hundred and fifty million 341 00:21:05,332 --> 00:21:09,272 kilometers, you can calculate how much of that energy flows 342 00:21:09,272 --> 00:21:12,871 through one square meter. And that's very close to one 343 00:21:12,871 --> 00:21:14,977 kilowatt. And we will call that, 344 00:21:14,977 --> 00:21:18,712 nevertheless -- even though it's not really a plane wave 345 00:21:18,712 --> 00:21:21,633 solution in that sense -- we will call that, 346 00:21:21,633 --> 00:21:23,874 nevertheless, the pointing vector. 347 00:21:23,874 --> 00:21:27,405 And I really don't care whether you 348 00:21:27,405 --> 00:21:31,138 think of this as being a plane electromagnetic wave, 349 00:21:31,138 --> 00:21:35,164 or whether you think of it as more like spherical waves, 350 00:21:35,164 --> 00:21:38,677 if you see there, or even if you want to think of 351 00:21:38,677 --> 00:21:42,63 them as individual photons, and I will come back to the 352 00:21:42,63 --> 00:21:44,972 photons. As far as I'm concerned, 353 00:21:44,972 --> 00:21:48,851 it's all of the above. But the result is that there is 354 00:21:48,851 --> 00:21:51,34 an energy flux, every square meter, 355 00:21:51,34 --> 00:21:55,512 aimed in the direction of the sun, one thousand joules per 356 00:21:55,512 --> 00:22:00,049 second will flow through that. If you want to 357 00:22:00,049 --> 00:22:04,116 hang on to the plane wave solution, and if you want to put 358 00:22:04,116 --> 00:22:08,254 any value at the plane wave solution, then you have to draw 359 00:22:08,254 --> 00:22:11,393 the consequence that you can calculate, then, 360 00:22:11,393 --> 00:22:15,46 the E zero of that plane wave. And I really don't know how 361 00:22:15,46 --> 00:22:19,027 meaningful that is, but then you would then get the 362 00:22:19,027 --> 00:22:23,022 equation that one thousand, that is, the average value of 363 00:22:23,022 --> 00:22:25,519 the pointing factor, that's a given, 364 00:22:25,519 --> 00:22:28,872 is then E zero squared divided by two mu zero C. 365 00:22:28,872 --> 00:22:33,509 We derived that there. And so you can calculate, 366 00:22:33,509 --> 00:22:37,11 now, what E zero is, and you'll find that E zero is 367 00:22:37,11 --> 00:22:41,07 approximately eight hundred and seventy volts per meter. 368 00:22:41,07 --> 00:22:44,67 To be frank with you, I don't quite know what to do 369 00:22:44,67 --> 00:22:48,054 with that number. It is some equivalent electric 370 00:22:48,054 --> 00:22:50,719 field. But it's really not an electric 371 00:22:50,719 --> 00:22:53,599 field, of course, that you could measure, 372 00:22:53,599 --> 00:22:57,631 because it's not really a plane wave in the sense that we 373 00:22:57,631 --> 00:23:01,663 derived our pointing vector for the plane wave solutions. 374 00:23:01,663 --> 00:23:07,165 So the bizarre thing is that in physics, you can think of 375 00:23:07,165 --> 00:23:10,111 light, for instance, as a plane wave, 376 00:23:10,111 --> 00:23:13,22 you can think of it as spherical waves, 377 00:23:13,22 --> 00:23:18,13 you can think of it as photons, and you can choose which ever 378 00:23:18,13 --> 00:23:21,812 fits you best to explain a certain phenomenon. 379 00:23:21,812 --> 00:23:26,476 And it is only in quantum mechanics where the wave and the 380 00:23:26,476 --> 00:23:30,567 photon picture merge. Let's talk about photons now. 381 00:23:30,567 --> 00:23:34,412 That is one aspect of the electromagnetic waves, 382 00:23:34,412 --> 00:23:38,586 then. Photons are really individual 383 00:23:38,586 --> 00:23:40,49 packages. Wave trains. 384 00:23:40,49 --> 00:23:43,392 You can think of them as bullets. 385 00:23:43,392 --> 00:23:48,379 Well-defined in time and space. [Wssshhht], there goes a 386 00:23:48,379 --> 00:23:51,281 photon. That photon has a certain 387 00:23:51,281 --> 00:23:56,631 energy which it carries with it. That means it has a certain 388 00:23:56,631 --> 00:24:00,077 momentum. And the momentum of a photon, 389 00:24:00,077 --> 00:24:05,7 which comes from one of -- the great work by Albert Einstein -- 390 00:24:05,7 --> 00:24:11,231 the momentum of a photon P is the energy of one photon divided 391 00:24:11,231 --> 00:24:15,799 by C. You just have to take my word 392 00:24:15,799 --> 00:24:18,725 for that. If you ever take special 393 00:24:18,725 --> 00:24:23,249 relativity you will understand this a little better. 394 00:24:23,249 --> 00:24:27,151 But give it now -- take it now at face value. 395 00:24:27,151 --> 00:24:31,054 And now, if I shoot many photons at a target, 396 00:24:31,054 --> 00:24:35,134 and these photons, if they are absorbed by that 397 00:24:35,134 --> 00:24:39,302 target, then the target will experience a force. 398 00:24:39,302 --> 00:24:45,069 Because if the photons carry momentum, and if the momentum 399 00:24:45,069 --> 00:24:47,393 -- photon, photon, photon, photon, 400 00:24:47,393 --> 00:24:51,336 photon -- I feel a force. You remember from eight oh one, 401 00:24:51,336 --> 00:24:54,645 a force is D P D T, is the transfer of momentum. 402 00:24:54,645 --> 00:24:58,658 If I throw rotten tomatoes at you, then when you get these 403 00:24:58,658 --> 00:25:02,39 tomatoes on your face, they hit you like this and then 404 00:25:02,39 --> 00:25:05,98 they go [pffft] like this, so they come in a certain 405 00:25:05,98 --> 00:25:09,501 direction, and they lose all their momentum in that 406 00:25:09,501 --> 00:25:12,036 direction. That gives a force on you, 407 00:25:12,036 --> 00:25:15,486 in this direction. Because that 408 00:25:15,486 --> 00:25:20,412 momentum in this direction is destroyed, but momentum is 409 00:25:20,412 --> 00:25:23,816 conserved and so you get that momentum. 410 00:25:23,816 --> 00:25:27,937 If, for instance, I threw one kilogram tomatoes 411 00:25:27,937 --> 00:25:32,864 at you each second -- so one after another -- on average 412 00:25:32,864 --> 00:25:38,417 time, average one kilogram per second, and if each tomato had a 413 00:25:38,417 --> 00:25:43,255 speed of five meters per second -- and I cal that the X 414 00:25:43,255 --> 00:25:47,733 direction -- and here is your face -- 415 00:25:47,733 --> 00:25:51,591 they hit your face, and then they go [plllt] down. 416 00:25:51,591 --> 00:25:55,606 So all the momentum in the X direction is destroyed, 417 00:25:55,606 --> 00:25:59,857 then you will experience a force in that X direction -- 418 00:25:59,857 --> 00:26:03,794 it's in the X direction -- which is one times five, 419 00:26:03,794 --> 00:26:07,573 which is five newtons. It's a time-average force. 420 00:26:07,573 --> 00:26:11,273 If, for some reason, the pot- the tomatoes would 421 00:26:11,273 --> 00:26:14,737 bounce back with a perfect elastic collision, 422 00:26:14,737 --> 00:26:20,448 like tennis balls would do, then the momentum transfer is 423 00:26:20,448 --> 00:26:24,178 twice, comes in with momentum, and comes back. 424 00:26:24,178 --> 00:26:28,241 So now the momentum transfer is twice this number, 425 00:26:28,241 --> 00:26:31,722 it's not just destroyed, no, it's reversed. 426 00:26:31,722 --> 00:26:35,619 So in that case, the force on your case would be 427 00:26:35,619 --> 00:26:38,355 double that, would be ten newtons. 428 00:26:38,355 --> 00:26:43,246 And so we can now carry this further, to our electromagnetic 429 00:26:43,246 --> 00:26:47,474 radiation, and we can return to our pointing vector. 430 00:26:47,474 --> 00:26:54,38 Here, I have one square meter. And radiation is coming in. 431 00:26:54,38 --> 00:27:00,107 And I know exactly how much radiation is coming in, 432 00:27:00,107 --> 00:27:05,262 for every square meter, there is this value S. 433 00:27:05,262 --> 00:27:10,073 So S, which is an energy, per square meter, 434 00:27:10,073 --> 00:27:14,196 per second. Remember, that's -- did I 435 00:27:14,196 --> 00:27:21,642 mention, of the pointing vector? If I divide this by C, 436 00:27:21,642 --> 00:27:25,833 and I divide this by C, then look what I have here. 437 00:27:25,833 --> 00:27:29,355 I have energy divided by C. But the energy, 438 00:27:29,355 --> 00:27:32,876 in electromagnetic radiation, divided by C, 439 00:27:32,876 --> 00:27:35,726 according to Einstein, is momentum. 440 00:27:35,726 --> 00:27:40,002 And I don't care whether this is hundred billions of 441 00:27:40,002 --> 00:27:43,104 individual photons, that's OK with me. 442 00:27:43,104 --> 00:27:47,045 That is a momentum. And so I have now a momentum 443 00:27:47,045 --> 00:27:51,32 per unit time that makes it a force. 444 00:27:51,32 --> 00:27:55,505 But a force per square meter makes it a pressure. 445 00:27:55,505 --> 00:27:59,863 And so this is pressure. And we call that radiation 446 00:27:59,863 --> 00:28:02,74 pressure. It means that if you are 447 00:28:02,74 --> 00:28:07,186 exposed to bombardment of electromagnetic radiation, 448 00:28:07,186 --> 00:28:10,934 and you absorb it, that you feel a pressure. 449 00:28:10,934 --> 00:28:15,293 The radiation comes to you, and you're being pushed 450 00:28:15,293 --> 00:28:18,693 backwards. And this is the pressure that 451 00:28:18,693 --> 00:28:22,964 you will experience. It makes a difference whether 452 00:28:22,964 --> 00:28:26,456 the radiation is absorbed, 453 00:28:26,456 --> 00:28:30,827 or whether you are capable of reflecting that radiation, 454 00:28:30,827 --> 00:28:35,357 just like with the tomatoes. And so if I'm trying to be as 455 00:28:35,357 --> 00:28:38,536 general as I can -- let me, first of all, 456 00:28:38,536 --> 00:28:42,669 specify the mean value of S, that's the only thing in 457 00:28:42,669 --> 00:28:45,689 electromagnetic radiation was mattered. 458 00:28:45,689 --> 00:28:49,107 It's the mean value. So the mean value then, 459 00:28:49,107 --> 00:28:54,272 if I divide that by C, and I multiply that by an -- 460 00:28:54,272 --> 00:28:58,005 a vector alpha, U will come up shortly and tell 461 00:28:58,005 --> 00:29:02,468 you what alpha is -- that is now the radiation pressure. 462 00:29:02,468 --> 00:29:06,85 An alpha, if alpha is one, then I have full absorption. 463 00:29:06,85 --> 00:29:11,395 All the radiation is absorbed. If there is no absorption, 464 00:29:11,395 --> 00:29:16,263 if it's completely transparent -- and there is some radiation 465 00:29:16,263 --> 00:29:20,483 that goes straight through you -- then alpha is zero. 466 00:29:20,483 --> 00:29:25,19 But if there is hundred percent reflection -- 467 00:29:25,19 --> 00:29:28,51 which can happen, you can reflect radio waves 468 00:29:28,51 --> 00:29:33,264 from metals with almost hundred percent reflection -- then alpha 469 00:29:33,264 --> 00:29:35,679 is two. So that you get twice the 470 00:29:35,679 --> 00:29:37,867 pressure. If you take the sun, 471 00:29:37,867 --> 00:29:41,188 whereby the mean value for S is one thousand, 472 00:29:41,188 --> 00:29:45,036 you can calculate now, if you absorb that radiation, 473 00:29:45,036 --> 00:29:49,036 over one square meter, what the pressure would be that 474 00:29:49,036 --> 00:29:51,601 you would experience. So your body, 475 00:29:51,601 --> 00:29:54,771 one square meter, you expose it to the sun, 476 00:29:54,771 --> 00:30:00,32 in the direction of the sun. That force is insignificantly 477 00:30:00,32 --> 00:30:02,53 small. The pressure would only be 478 00:30:02,53 --> 00:30:06,466 three times ten to the minus six newtons per square meter, 479 00:30:06,466 --> 00:30:10,61 because you have to multiply that number thousand -- you have 480 00:30:10,61 --> 00:30:12,958 to -- not multiply, divide it by C. 481 00:30:12,958 --> 00:30:16,963 There's not much left over. And if you hold your hand up in 482 00:30:16,963 --> 00:30:20,623 the direction of the sun, your hand would only have an 483 00:30:20,623 --> 00:30:23,731 area maybe of one-hundredth of a square meter, 484 00:30:23,731 --> 00:30:27,115 that force is unnoticeable. So in our daily lives, 485 00:30:27,115 --> 00:30:31,839 we don't experience radiation pressure very much. 486 00:30:31,839 --> 00:30:36,664 Regardless of whether you think of electromagnetic radiation as 487 00:30:36,664 --> 00:30:40,789 plane waves, or you want to think of them as spherical 488 00:30:40,789 --> 00:30:45,225 waves, or you want to think of them as photons -- I really 489 00:30:45,225 --> 00:30:49,972 don't care -- but what matters as far as radiation pressure is 490 00:30:49,972 --> 00:30:54,563 concerned, is how much energy flow through one square meter, 491 00:30:54,563 --> 00:30:58,065 and which fraction of that energy is absorbed, 492 00:30:58,065 --> 00:31:00,789 or maybe, some of it even reflected. 493 00:31:00,789 --> 00:31:04,525 So the radiation pressure is not 494 00:31:04,525 --> 00:31:08,623 that important in our daily lives, but is important in 495 00:31:08,623 --> 00:31:10,324 astronomy. And in fact, 496 00:31:10,324 --> 00:31:13,881 you can see it. Some -- some of you can see it. 497 00:31:13,881 --> 00:31:17,824 A comet has two tails, and one of those tails is the 498 00:31:17,824 --> 00:31:22 result of radiation pressure. A comet is made of carbon 499 00:31:22 --> 00:31:26,021 dioxide, and it has also dust, it's about the size of 500 00:31:26,021 --> 00:31:29,036 Manhattan. And when it gets close to the 501 00:31:29,036 --> 00:31:33,831 sun, the radiation pressure of the sun, which is this radiation 502 00:31:33,831 --> 00:31:37,852 pressure that we just talked about, 503 00:31:37,852 --> 00:31:41,049 pushes onto the dust particles, forming the tail, 504 00:31:41,049 --> 00:31:44,447 and this tail is white light, white-yellowish light, 505 00:31:44,447 --> 00:31:47,911 which is reflected sunlight off these dust particles. 506 00:31:47,911 --> 00:31:51,042 And then there is a second tail which is bluish, 507 00:31:51,042 --> 00:31:55,039 which is hard to see with your naked eye, and that one is the 508 00:31:55,039 --> 00:31:58,57 result of the solar winds. We discussed the solar wind 509 00:31:58,57 --> 00:32:02,301 earlier, causes the aurora in the upper atmosphere of the 510 00:32:02,301 --> 00:32:04,766 Earth. The solar winds are protons and 511 00:32:04,766 --> 00:32:10,096 electrons which are emitted by the sun in a rather erratic way, 512 00:32:10,096 --> 00:32:12,539 sometimes a lot, sometimes not so much, 513 00:32:12,539 --> 00:32:16,141 and they move with a speed of two hundred fifty miles per 514 00:32:16,141 --> 00:32:18,649 second or so, and they ionize the C O 2. 515 00:32:18,649 --> 00:32:22,058 So they excite the molecules when the [unintelligible] 516 00:32:22,058 --> 00:32:25,21 excitation of the molecules, they emit blue light. 517 00:32:25,21 --> 00:32:28,361 So you get two tails, and we'll show you these two 518 00:32:28,361 --> 00:32:30,419 tails. The tails can be a hundred 519 00:32:30,419 --> 00:32:33,249 million kilometers in size, they can be huge. 520 00:32:33,249 --> 00:32:36,272 As some of you may remember, or should remember, 521 00:32:36,272 --> 00:32:39,809 in nineteen ninety seven, we had a fabulous example of a 522 00:32:39,809 --> 00:32:42,99 comet, which was Hale-Bop. 523 00:32:42,99 --> 00:32:46,183 I watched it every night for months on end. 524 00:32:46,183 --> 00:32:50,668 This is a picture that you're going to see of Hale-Bop which 525 00:32:50,668 --> 00:32:54,393 is a time-exposure, it's not what you can see with 526 00:32:54,393 --> 00:32:56,522 your naked eye. And, in fact, 527 00:32:56,522 --> 00:32:59,79 frankly speaking, I have never seen the blue 528 00:32:59,79 --> 00:33:03,439 tail, but the white tail, which is the one due to 529 00:33:03,439 --> 00:33:06,176 radiation pressure, I have seen many, 530 00:33:06,176 --> 00:33:08,761 many times. So let's take a look at 531 00:33:08,761 --> 00:33:13,17 Hale-Bop, and there you see it. And you see clearly the two 532 00:33:13,17 --> 00:33:17,368 tails. You see the blue tail here, 533 00:33:17,368 --> 00:33:22,141 which is the result of the interaction with the solar wind, 534 00:33:22,141 --> 00:33:27,16 and then here you see the dust which is the result -- the tail 535 00:33:27,16 --> 00:33:30,04 is the result of radiation pressure. 536 00:33:30,04 --> 00:33:34,401 The sunlight is absorbed, fully absorbed by these dust 537 00:33:34,401 --> 00:33:38,021 particles, and is pushed away. I just wonder, 538 00:33:38,021 --> 00:33:41,148 who has seen Hale-Bop, three years ago? 539 00:33:41,148 --> 00:33:43,452 Yes? Just hold your hands up, 540 00:33:43,452 --> 00:33:46,907 I'm just curious, it was a 541 00:33:46,907 --> 00:33:50,492 spectacular sight, and it was easy to see for 542 00:33:50,492 --> 00:33:53,017 months on end, you could see it. 543 00:33:53,017 --> 00:33:56,846 You don't see very bright comets all that often. 544 00:33:56,846 --> 00:34:00,756 So without being very qualitative in terms of the 545 00:34:00,756 --> 00:34:05,237 electric and the magnetic field strengths, it is easy to 546 00:34:05,237 --> 00:34:09,88 determine the direction of the oscillating electric field, 547 00:34:09,88 --> 00:34:13,953 if we know in what direction the charges were being 548 00:34:13,953 --> 00:34:18,353 oscillated. And that has to do with the 549 00:34:18,353 --> 00:34:24,874 polarization of the radiation, and so I would like to send the 550 00:34:24,874 --> 00:34:31,074 remaining time today on that. And let's stick to the center 551 00:34:31,074 --> 00:34:34,282 board. So we have a oscillating 552 00:34:34,282 --> 00:34:40,268 charge, being accelerated, and I will choose a coordinate 553 00:34:40,268 --> 00:34:45,079 system like so, and let's suppose we oscillate 554 00:34:45,079 --> 00:34:49,03 the charge in this direction. 555 00:34:49,03 --> 00:34:53,881 Acceleration [wssshhht] all the time, and the frequency is 556 00:34:53,881 --> 00:34:56,944 omega. And we are somewhere in space, 557 00:34:56,944 --> 00:35:00,859 here, and we receive electromagnetic radiation, 558 00:35:00,859 --> 00:35:04,518 we are at point P -- like to make you see it 559 00:35:04,518 --> 00:35:07,752 three-dimensionally. And at this angle, 560 00:35:07,752 --> 00:35:11,326 B theta, and this is the position vector R, 561 00:35:11,326 --> 00:35:15,836 from the origin of the oscillating charge to where you 562 00:35:15,836 --> 00:35:21,219 are, you are at point P. And when you are at P, 563 00:35:21,219 --> 00:35:26,709 I'm going to give you some very simple rules which will always 564 00:35:26,709 --> 00:35:31,928 allow you to determine the direction of the electric field. 565 00:35:31,928 --> 00:35:37,148 The electric field is always perpendicular to the direction 566 00:35:37,148 --> 00:35:41,198 position vector, the direction of propagation. 567 00:35:41,198 --> 00:35:45,607 So you're looking in this direction to the charge, 568 00:35:45,607 --> 00:35:50,557 the electric field that you experience, if the charge is 569 00:35:50,557 --> 00:35:55,269 doing this, is always perpendicular to R. 570 00:35:55,269 --> 00:35:58,749 And A and R and E are always in one plane. 571 00:35:58,749 --> 00:36:03,249 And notice that's the way I drew it in the blackboard. 572 00:36:03,249 --> 00:36:08,004 E and A and R are in one plane. So it's extremely simple. 573 00:36:08,004 --> 00:36:12,588 So if this is oscillating with angular frequency omega, 574 00:36:12,588 --> 00:36:17,512 then this electric field will also be oscillating with that 575 00:36:17,512 --> 00:36:21,672 angular frequency omega. If you double the charge, 576 00:36:21,672 --> 00:36:25,748 the electric field will double. If 577 00:36:25,748 --> 00:36:29,946 you double the acceleration, the electric field will also 578 00:36:29,946 --> 00:36:32,571 double. That's reasonably intuitive. 579 00:36:32,571 --> 00:36:37,144 So the electric field strength is proportional with the charge 580 00:36:37,144 --> 00:36:40,218 that you oscillate, with the acceleration. 581 00:36:40,218 --> 00:36:44,492 And then, there comes this effect that we discussed there, 582 00:36:44,492 --> 00:36:48,841 that nothing goes out in the direction of the acceleration, 583 00:36:48,841 --> 00:36:53,114 no energy goes out in this direction, so no electric field 584 00:36:53,114 --> 00:36:58,91 is produced in this direction, and the maximum is produced in 585 00:36:58,91 --> 00:37:02,541 this direction, and somewhere in between here, 586 00:37:02,541 --> 00:37:06,334 and that is then reflected by the sine of theta. 587 00:37:06,334 --> 00:37:09,078 If theta is zero, nothing goes out. 588 00:37:09,078 --> 00:37:12,709 If theta is ninety degrees, you get a maximum. 589 00:37:12,709 --> 00:37:16,26 That's the whole plane, perpendicular to this 590 00:37:16,26 --> 00:37:19,166 oscillating charge, this whole plane, 591 00:37:19,166 --> 00:37:23,523 maximum electric field. And then -- and that may not be 592 00:37:23,523 --> 00:37:28,365 so obvious to you now -- it's also inversely 593 00:37:28,365 --> 00:37:32,254 proportional to the distance R. If you double R, 594 00:37:32,254 --> 00:37:36,887 the electric field strength goes down by a factor of two. 595 00:37:36,887 --> 00:37:41,437 And the pointing vector, which is the product of E and B 596 00:37:41,437 --> 00:37:46,567 -- but E is always proportional to B -- so the pointing vector, 597 00:37:46,567 --> 00:37:50,869 then, is proportional to Q squared, proportional to A 598 00:37:50,869 --> 00:37:54,923 squared, proportional to the sine square of theta, 599 00:37:54,923 --> 00:37:58,232 and inversely proportional with R square. 600 00:37:58,232 --> 00:38:00,753 And that is obvious, 601 00:38:00,753 --> 00:38:04,299 that it has to be inversely proportional with R squared. 602 00:38:04,299 --> 00:38:06,943 Because if you go -- if you have a sphere, 603 00:38:06,943 --> 00:38:10,296 and radiation goes out, and you're twice as far away, 604 00:38:10,296 --> 00:38:14,293 then you know that the area of the sphere is four times larger, 605 00:38:14,293 --> 00:38:18,291 and so the amount of energy per square meter must be four times 606 00:38:18,291 --> 00:38:20,418 lower. So the pointing vector must 607 00:38:20,418 --> 00:38:24,094 fall off as one over R squared, that's the conservation of 608 00:38:24,094 --> 00:38:26,544 energy. And if you accept the fact that 609 00:38:26,544 --> 00:38:30,025 the pointing vector falls off as one 610 00:38:30,025 --> 00:38:34,885 over R squared, then the E vector must fall off 611 00:38:34,885 --> 00:38:39,534 as one over R, because the pointing vector is 612 00:38:39,534 --> 00:38:45,028 the product of E and B. E and B must both fall off as 613 00:38:45,028 --> 00:38:49,148 one over R. I want to show you a picture 614 00:38:49,148 --> 00:38:53,903 that may help, to see you -- how the radiation 615 00:38:53,903 --> 00:38:59,925 -- how the electric field is oriented relative to a charge 616 00:38:59,925 --> 00:39:04,42 that we are accelerating. 617 00:39:04,42 --> 00:39:08,02 In the middle here, we accelerate a charge, 618 00:39:08,02 --> 00:39:10,42 up and down. Frequency omega. 619 00:39:10,42 --> 00:39:14,62 Whatever radiation you want to make, fine with me. 620 00:39:14,62 --> 00:39:18,219 You can make omega as large as you want to. 621 00:39:18,219 --> 00:39:21,819 For one thing, notice, these -- these waves 622 00:39:21,819 --> 00:39:25,591 here, these snakes, represent electromagnetic 623 00:39:25,591 --> 00:39:28,762 radiation. And nothing is going out in 624 00:39:28,762 --> 00:39:33,391 this direction. And we understand why. 625 00:39:33,391 --> 00:39:38,146 The maximum is going out in a plane perpendicular to this 626 00:39:38,146 --> 00:39:40,1 direction. That is this, 627 00:39:40,1 --> 00:39:44,176 this, this and this. And notice that the E-vector 628 00:39:44,176 --> 00:39:49,017 that you receive when you're here is perpendicular to R -- 629 00:39:49,017 --> 00:39:52,074 this is R -- and it is perpendicular. 630 00:39:52,074 --> 00:39:56,49 And notice what I said, that A and R and E are in one 631 00:39:56,49 --> 00:39:59,462 plane. A, which is the acceleration, 632 00:39:59,462 --> 00:40:02,605 R and E are in one plane. A, R, and E, 633 00:40:02,605 --> 00:40:06,256 are in one plane. A, R, E, are in one plane. 634 00:40:06,256 --> 00:40:10,181 So you can always determine the 635 00:40:10,181 --> 00:40:13,409 direction of the oscillating electric field. 636 00:40:13,409 --> 00:40:17,612 And if you go at an angle theta, which is neither zero -- 637 00:40:17,612 --> 00:40:21,966 which it is here -- nor ninety degrees -- which it is here, 638 00:40:21,966 --> 00:40:26,02 here, and here -- but if you have something in between, 639 00:40:26,02 --> 00:40:30,523 then notice that the E vector here is drawn a little smaller. 640 00:40:30,523 --> 00:40:32,926 And that is that sine theta here. 641 00:40:32,926 --> 00:40:36,228 And consequently, the pointing vector in that 642 00:40:36,228 --> 00:40:39,907 direction will be smaller than in 643 00:40:39,907 --> 00:40:42,866 this direction, and in this direction, 644 00:40:42,866 --> 00:40:45,425 the pointing vector will be zero. 645 00:40:45,425 --> 00:40:49,025 We call this radiation -- whether you're here, 646 00:40:49,025 --> 00:40:52,464 or there, or there, or there -- we call that 647 00:40:52,464 --> 00:40:57,183 linearly polarized radiation, for the simple reason that the 648 00:40:57,183 --> 00:41:00,862 electric field is oscillating in one direction. 649 00:41:00,862 --> 00:41:04,062 It's linear. And I'm going to produce for 650 00:41:04,062 --> 00:41:07,821 you linearly polarized radiation, and I have two 651 00:41:07,821 --> 00:41:11,13 demonstrations for that. 652 00:41:11,13 --> 00:41:17,242 So I'll get the lights back on again, and let's discuss these 653 00:41:17,242 --> 00:41:20,094 demonstrations. I have, here, 654 00:41:20,094 --> 00:41:24,78 a transmitter which transmits at ten gigaHertz. 655 00:41:24,78 --> 00:41:28,141 That is, uh, a wavelength of three 656 00:41:28,141 --> 00:41:32,317 centimeters. You wouldn't call that radio, 657 00:41:32,317 --> 00:41:37,716 you would call that radar, but that's just a matter of 658 00:41:37,716 --> 00:41:42,708 names. And we have also a receiver. 659 00:41:42,708 --> 00:41:46,04 We have a transmitter and we have a receiver. 660 00:41:46,04 --> 00:41:49,75 The transmitter is here, and the receiver is here. 661 00:41:49,75 --> 00:41:53,007 If I give you a three-dimensional picture -- 662 00:41:53,007 --> 00:41:57,475 this is my coordinate system, this is coming straight out of 663 00:41:57,475 --> 00:42:01,412 the blackboard to you, that's what I meant by this -- 664 00:42:01,412 --> 00:42:05,426 then the transmitter is aimed like so, just like this. 665 00:42:05,426 --> 00:42:08,758 Current is going to be oscillating like this. 666 00:42:08,758 --> 00:42:13,378 And the receiver -- think of this as my radio, 667 00:42:13,378 --> 00:42:18,246 which is receiving -- is here. And the antenna of the radio is 668 00:42:18,246 --> 00:42:22,156 also in this direction. And so, the electric field 669 00:42:22,156 --> 00:42:25,428 which you see here, which comes from these 670 00:42:25,428 --> 00:42:29,418 oscillating charges -- which oscillate, by the way, 671 00:42:29,418 --> 00:42:33,967 at a horrendous frequency, ten billion times per second -- 672 00:42:33,967 --> 00:42:37,319 their electric field is perpendicular to R, 673 00:42:37,319 --> 00:42:42,027 this is R, and the electric fields R and A are in one plane. 674 00:42:42,027 --> 00:42:46,018 Ah, so that means that the electric 675 00:42:46,018 --> 00:42:48,99 field that arrives here is oscillating like so. 676 00:42:48,99 --> 00:42:50,734 And therefore, this antenna, 677 00:42:50,734 --> 00:42:52,543 this receiver, is very happy. 678 00:42:52,543 --> 00:42:55,58 The radiation comes in exactly in the right way, 679 00:42:55,58 --> 00:42:59,134 and so it will receive it. We have modulated this signal 680 00:42:59,134 --> 00:43:01,847 with an audio signal, amplitude modulation, 681 00:43:01,847 --> 00:43:05,66 as we discussed that earlier, that you also use with radios. 682 00:43:05,66 --> 00:43:08,761 And we modulated with approximately one kiloHertz 683 00:43:08,761 --> 00:43:11,41 audio signal. And we'll make you listen to 684 00:43:11,41 --> 00:43:13,672 that audio signal. And you can tell, 685 00:43:13,672 --> 00:43:18,234 then, that this receiver is, indeed, receiving the 686 00:43:18,234 --> 00:43:21,16 three-centimeter radar from this transmitter. 687 00:43:21,16 --> 00:43:24,685 That's what we will first do. So I'll just turn on the 688 00:43:24,685 --> 00:43:27,811 transmitter [tone], and here you hear [tone] one 689 00:43:27,811 --> 00:43:30,671 [tone] kiloHertz, [tone] which is the [tone] 690 00:43:30,671 --> 00:43:33,597 modulated signal, [tone] and it's received by 691 00:43:33,597 --> 00:43:36,523 this antenna, [tone] which is a straight wire 692 00:43:36,523 --> 00:43:39,25 like this, [tone] and this is the emitter. 693 00:43:39,25 --> 00:43:43,041 [tone] And to demonstrate, to you that it really goes from 694 00:43:43,041 --> 00:43:46,5 here to here, if I put my hands in 695 00:43:46,5 --> 00:43:49,15 between, my hands absorb three centimeters. 696 00:43:49,15 --> 00:43:51,8 It's not there anymore. [tone] There it is. 697 00:43:51,8 --> 00:43:55,523 [tone] It's not there anymore. But now, [tone] now I'm going 698 00:43:55,523 --> 00:43:58,488 to [tone] rotate this antenna by ninety degrees, 699 00:43:58,488 --> 00:44:01,895 [tone] so this one is going to be put in this position. 700 00:44:01,895 --> 00:44:05,555 [tone] So now the electric fields -- [tone] so I'm going to 701 00:44:05,555 --> 00:44:09,278 put it now in this position [tone] -- so the electric field, 702 00:44:09,278 --> 00:44:11,802 now, [tone] is going up and down like so. 703 00:44:11,802 --> 00:44:15,651 [tone] But the receiver doesn't like that, because the antenna 704 00:44:15,651 --> 00:44:20,473 of the receiver is like this. [tone] So it cannot [tone] 705 00:44:20,473 --> 00:44:23,783 drive any current. And so you won't have anything 706 00:44:23,783 --> 00:44:26,472 anymore. [tone] So what it means is that 707 00:44:26,472 --> 00:44:30,748 I have changed the direction of polarization by ninety degrees, 708 00:44:30,748 --> 00:44:33,437 [tone] and the receiver cannot see that. 709 00:44:33,437 --> 00:44:36,748 [tone] I'm going to rotate it now ninety degrees. 710 00:44:36,748 --> 00:44:39,782 And it stops. So now, the electric field goes 711 00:44:39,782 --> 00:44:42,127 like this, [shhhht]. The wire here, 712 00:44:42,127 --> 00:44:43,851 the antenna, is like this, 713 00:44:43,851 --> 00:44:47,161 and the wire says, "Sorry, I can't hear you." But 714 00:44:47,161 --> 00:44:52,008 all I have to do now is rotate the receiver ninety 715 00:44:52,008 --> 00:44:55,68 degrees, and so I can rotate this antenna and put it 716 00:44:55,68 --> 00:44:59,064 vertical, and then, of course, it's happy again. 717 00:44:59,064 --> 00:45:03,168 So if I rotate this ninety degrees, [tone] there comes the 718 00:45:03,168 --> 00:45:06,048 signal back. [tone] So now the E field is 719 00:45:06,048 --> 00:45:09,72 like so, [tone] and the antenna receives it like so. 720 00:45:09,72 --> 00:45:13,896 [tone] Now, I have one [tone] -- I have one brainteaser for 721 00:45:13,896 --> 00:45:16,704 you, and I want you to think about that. 722 00:45:16,704 --> 00:45:20,449 We can actually have a vote on that. 723 00:45:20,449 --> 00:45:23,837 I have here a wooden frame with metal bars. 724 00:45:23,837 --> 00:45:25,45 No trick. Metal bars. 725 00:45:25,45 --> 00:45:28,758 And I'm going to put that in between here. 726 00:45:28,758 --> 00:45:32,63 So we have the situation that we are transmitting 727 00:45:32,63 --> 00:45:36,987 electromagnetic waves with the E field going like this, 728 00:45:36,987 --> 00:45:39,81 and the receiver is perfectly happy. 729 00:45:39,81 --> 00:45:42,876 The receiver is also aligned like this. 730 00:45:42,876 --> 00:45:46,748 There you have it. I can put this grid in like so 731 00:45:46,748 --> 00:45:51,67 [wssshhht], so that the bars are in the same 732 00:45:51,67 --> 00:45:55,999 direction as E, but I can also put it in like 733 00:45:55,999 --> 00:46:00,231 so, so that the bars are perpendicular to E. 734 00:46:00,231 --> 00:46:05,052 If I make you a drawing, if you would look in this 735 00:46:05,052 --> 00:46:09,185 direction, then I can do one of two things. 736 00:46:09,185 --> 00:46:14,302 Either the bars are like so, or the bars are like so, 737 00:46:14,302 --> 00:46:19,124 but in both cases, is the electric field coming in 738 00:46:19,124 --> 00:46:22,962 like this? Who thinks that if I have the 739 00:46:22,962 --> 00:46:26,215 bars vertical, that the 740 00:46:26,215 --> 00:46:30,903 electromagnetic radiation will go straight through without any 741 00:46:30,903 --> 00:46:33,747 problem? Who thinks that maybe it will 742 00:46:33,747 --> 00:46:37,359 not get through? Who thinks it will get through? 743 00:46:37,359 --> 00:46:39,972 Who thinks it will not get through? 744 00:46:39,972 --> 00:46:42,508 Now I do this. Who thinks that the 745 00:46:42,508 --> 00:46:46,351 electromagnetic radiation, the oscillating E-field, 746 00:46:46,351 --> 00:46:50,424 can easily go through this? Who thinks it will not get 747 00:46:50,424 --> 00:46:53,037 through? It's every evenly divided, 748 00:46:53,037 --> 00:46:57,47 we have about twenty five, twenty five, 749 00:46:57,47 --> 00:46:59,573 twenty five, twenty five. 750 00:46:59,573 --> 00:47:02,64 OK, we'll see. [tone] One kiloHertz. 751 00:47:02,64 --> 00:47:07,546 [tone] E field is like this, [tone] receiver is receiving 752 00:47:07,546 --> 00:47:10,35 it. [tone] What do you want me to 753 00:47:10,35 --> 00:47:14,731 do first, this or this? [tone] Let's do this first. 754 00:47:14,731 --> 00:47:18,236 [tone] Nothing. [tone] It's not absorbed. 755 00:47:18,236 --> 00:47:21,128 [tone] Goes straight through here. 756 00:47:21,128 --> 00:47:25,071 [tone] Now I rotate it. [tone] Ninety degrees. 757 00:47:25,071 --> 00:47:28,293 [tone] And I kill it. 758 00:47:28,293 --> 00:47:30,773 [tone] So in this configuration, 759 00:47:30,773 --> 00:47:35,333 [tone] the electromagnetic radiation does not get through. 760 00:47:35,333 --> 00:47:39,813 [tone] I want you to think about it, and I -- if you have 761 00:47:39,813 --> 00:47:43,493 some sleepless nights about it, that's OK, too, 762 00:47:43,493 --> 00:47:47,492 that's very healthy. I have a second demonstration, 763 00:47:47,492 --> 00:47:51,252 and that has also to do with the polarization of 764 00:47:51,252 --> 00:47:55,252 electromagnetic radiation. I have here seventy five 765 00:47:55,252 --> 00:47:59,891 megaHertz transmitter, so that's more like 766 00:47:59,891 --> 00:48:02,763 radio stuff, that is a four-meter 767 00:48:02,763 --> 00:48:05,904 wavelength. And this is the antenna, 768 00:48:05,904 --> 00:48:10,84 this is the transmitter. So let me give you the numbers. 769 00:48:10,84 --> 00:48:15,327 So this is seventy five megaHertz, and that is four 770 00:48:15,327 --> 00:48:19,365 meters wavelength. And I'm going to turn it on 771 00:48:19,365 --> 00:48:22,326 like now. You don't hear anything, 772 00:48:22,326 --> 00:48:25,736 it's not connected to any receiver yet. 773 00:48:25,736 --> 00:48:30,851 Oscillating. Making electromagnetic waves, 774 00:48:30,851 --> 00:48:32,714 going like this, going up. 775 00:48:32,714 --> 00:48:35,323 Not much is going in this direction. 776 00:48:35,323 --> 00:48:39,048 I have a very special receiver. A very special one. 777 00:48:39,048 --> 00:48:41,88 And that very special receiver is this. 778 00:48:41,88 --> 00:48:45,606 It's just a straight copper wire cut in the middle, 779 00:48:45,606 --> 00:48:50,077 and the left side and the right side I connected with a light 780 00:48:50,077 --> 00:48:52,759 bulb. So if there is a strong current 781 00:48:52,759 --> 00:48:57,379 going through here -- that means if it receives a strong signal 782 00:48:57,379 --> 00:49:00,435 -- then the light bulb will indicate that. 783 00:49:00,435 --> 00:49:04,309 I'm going to make it dark so that 784 00:49:04,309 --> 00:49:07,106 we can, perhaps, see that light bulb. 785 00:49:07,106 --> 00:49:11,3 So that is transmitting. And I want to show you several 786 00:49:11,3 --> 00:49:14,173 things. The first thing I want to show 787 00:49:14,173 --> 00:49:18,057 you is that, indeed, if I hold this parallel to the 788 00:49:18,057 --> 00:49:20,698 transmitter, that I can receive it. 789 00:49:20,698 --> 00:49:25,591 The electric field is now going like this, and so this receiving 790 00:49:25,591 --> 00:49:29,474 antenna is very happy. But now look a what happens, 791 00:49:29,474 --> 00:49:32,969 it's same thing I did with the radar receiver, 792 00:49:32,969 --> 00:49:38,359 I rotate it ninety degrees. And now the E-field that comes 793 00:49:38,359 --> 00:49:41,121 in this cannot cause any current like this. 794 00:49:41,121 --> 00:49:44,212 So the light bulb goes out. I can also show you, 795 00:49:44,212 --> 00:49:47,105 which is obvious, that if you go farther away 796 00:49:47,105 --> 00:49:50,525 from the transmitter, that the strength of the signal 797 00:49:50,525 --> 00:49:52,958 is less. Remember, the pointing vector 798 00:49:52,958 --> 00:49:56,706 goes as one over R squared. So the light in the light bulb 799 00:49:56,706 --> 00:49:59,074 should go down by one over R squared. 800 00:49:59,074 --> 00:50:01,901 So I go a little closer -- cannot too close, 801 00:50:01,901 --> 00:50:05,715 because I can really burn out the bulb if I come too close. 802 00:50:05,715 --> 00:50:09,266 But you see it's quite bright now, 803 00:50:09,266 --> 00:50:12,281 and when I walk away, by holding it still in the 804 00:50:12,281 --> 00:50:14,204 right direction, you see, that, 805 00:50:14,204 --> 00:50:16,449 indeed, the light bulb gets fainter. 806 00:50:16,449 --> 00:50:20,04 One over R square relationship. What is also interesting, 807 00:50:20,04 --> 00:50:23,375 we can explore how much radiation is sent out by this 808 00:50:23,375 --> 00:50:26,966 transmitter in the direction of the antenna, which is the 809 00:50:26,966 --> 00:50:30,814 direction in which we argued that no radiation should go out. 810 00:50:30,814 --> 00:50:34,662 So let me walk over here -- or let me walk to the other side, 811 00:50:34,662 --> 00:50:37,163 because it's darker here. So right here, 812 00:50:37,163 --> 00:50:40,498 I am now standing such that theta 813 00:50:40,498 --> 00:50:42,298 is zero. No matter what I do, 814 00:50:42,298 --> 00:50:45,449 whether I hold my antenna like this, or like this, 815 00:50:45,449 --> 00:50:47,893 or like this, there's nothing I can do. 816 00:50:47,893 --> 00:50:51,044 I can even come a little closer, you see no light, 817 00:50:51,044 --> 00:50:54,388 because there is no energy flowing in this direction, 818 00:50:54,388 --> 00:50:57,86 no electromagnetic wave. Largely going out in the plane 819 00:50:57,86 --> 00:51:00,626 perpendicular to the antenna, but of course, 820 00:51:00,626 --> 00:51:02,683 if I'm here, which is not exactly 821 00:51:02,683 --> 00:51:04,548 perpendicular, then of course, 822 00:51:04,548 --> 00:51:06,671 I do pick up also, some radiation. 823 00:51:06,671 --> 51:12 OK, see you Friday.