1 00:00:00,070 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,209 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,209 --> 00:00:17,834 at ocw.mit.edu. 8 00:00:20,830 --> 00:00:22,550 PROFESSOR: So welcome back. 9 00:00:22,550 --> 00:00:27,170 And today, we will solve one type 10 00:00:27,170 --> 00:00:33,890 of problem, a very important problem, 11 00:00:33,890 --> 00:00:35,770 basically, the following. 12 00:00:35,770 --> 00:00:44,060 Suppose all that you have is a single charge, but it can move. 13 00:00:44,060 --> 00:00:47,560 Here is the origin of coordinates. 14 00:00:47,560 --> 00:00:50,700 This is the charge, Q. Its position is 15 00:00:50,700 --> 00:00:54,570 given by some vector, z of t. 16 00:00:54,570 --> 00:00:56,140 And the charge can move. 17 00:00:56,140 --> 00:00:58,140 It can have a velocity. 18 00:00:58,140 --> 00:01:00,210 It can have an acceleration, et cetera. 19 00:01:00,210 --> 00:01:04,250 And the problem is, somewhere in space, 20 00:01:04,250 --> 00:01:08,680 at a point, p, what is the electric field? 21 00:01:08,680 --> 00:01:13,740 What is the magnetic field at all times? 22 00:01:13,740 --> 00:01:15,070 That's the problem. 23 00:01:15,070 --> 00:01:19,600 Now the differences between different particular problems 24 00:01:19,600 --> 00:01:25,070 will be where we are and what the charge is moving et cetera. 25 00:01:25,070 --> 00:01:29,850 In particular, we will consider problems 26 00:01:29,850 --> 00:01:35,110 where this charge is oscillating in a simple harmonic motion. 27 00:01:35,110 --> 00:01:36,860 It's along a line. 28 00:01:36,860 --> 00:01:38,480 And then, in another problem, we'll 29 00:01:38,480 --> 00:01:41,530 consider it's rotating with uniform velocity. 30 00:01:41,530 --> 00:01:46,090 But in principle, we could be doing anything. 31 00:01:46,090 --> 00:01:50,230 So how do you calculate the electric field? 32 00:01:50,230 --> 00:01:54,340 How do you find out what the magnetic field is everywhere? 33 00:01:54,340 --> 00:01:57,600 Also, if there is an electric and magnetic field, 34 00:01:57,600 --> 00:02:02,040 there will be flow of energy and energy flux or pointing vector. 35 00:02:02,040 --> 00:02:04,930 What will that be at that location? 36 00:02:04,930 --> 00:02:09,810 And finally, if a charge does accelerate here and it does 37 00:02:09,810 --> 00:02:15,240 radiate energy, and the question is, 38 00:02:15,240 --> 00:02:20,810 what is the average total power radiated by such a charge? 39 00:02:20,810 --> 00:02:25,920 Now this kind of problem is extremely important. 40 00:02:25,920 --> 00:02:28,680 It's important just from the point 41 00:02:28,680 --> 00:02:31,960 of view of understanding how the world works. 42 00:02:31,960 --> 00:02:34,020 There are charges all over the place. 43 00:02:34,020 --> 00:02:35,730 And they are moving. 44 00:02:35,730 --> 00:02:37,920 And they do produce electric magnetic fields. 45 00:02:43,170 --> 00:02:46,410 A problem like this gives you insight 46 00:02:46,410 --> 00:02:49,590 how the phenomena takes place. 47 00:02:49,590 --> 00:02:53,320 But let me tell you immediately, in general, 48 00:02:53,320 --> 00:02:57,442 this is a horrendous problem. 49 00:02:57,442 --> 00:03:01,950 You know that, if you have charges in space, 50 00:03:01,950 --> 00:03:07,580 and you have electric and magnetic fields, 51 00:03:07,580 --> 00:03:11,140 the charges, the currents, the electric and magnetic fields 52 00:03:11,140 --> 00:03:13,670 have to satisfy Maxwell's equations, those 53 00:03:13,670 --> 00:03:15,360 four Maxwell's equations. 54 00:03:15,360 --> 00:03:18,600 So you can imagine how complicated 55 00:03:18,600 --> 00:03:23,930 it must be to find electric and magnetic fields, in general, 56 00:03:23,930 --> 00:03:26,950 for some arbitrary motion like this. 57 00:03:26,950 --> 00:03:29,780 And the answer is, you would never be asked to do that. 58 00:03:29,780 --> 00:03:34,970 And you would never have to do that, in general, analytically. 59 00:03:34,970 --> 00:03:39,440 If you really had a problem like this that you had to solve, 60 00:03:39,440 --> 00:03:42,776 in general, you would do it on a computer. 61 00:03:42,776 --> 00:03:43,275 All right? 62 00:03:46,050 --> 00:03:49,590 But let me continue. 63 00:03:54,090 --> 00:04:00,350 But it turns out that, under certain conditions, 64 00:04:00,350 --> 00:04:03,680 this problem is not so hard. 65 00:04:03,680 --> 00:04:09,210 And yet, it contains all the physics 66 00:04:09,210 --> 00:04:12,600 and gives you all the insights you need. 67 00:04:12,600 --> 00:04:16,980 And that is the following situation. 68 00:04:16,980 --> 00:04:25,840 If you have the charge moving in a confined region, which 69 00:04:25,840 --> 00:04:29,010 is much smaller, whose dimensions 70 00:04:29,010 --> 00:04:34,930 are much smaller than the distance at which you are 71 00:04:34,930 --> 00:04:38,640 interested to calculate the electric magnetic fields, 72 00:04:38,640 --> 00:04:44,290 suddenly, this problem does not become hard, or in fact, 73 00:04:44,290 --> 00:04:48,480 not hard at all, relatively speaking. 74 00:04:48,480 --> 00:04:52,790 So in some sense-- and this often happens in physics 75 00:04:52,790 --> 00:04:56,630 when you are given the problem-- you 76 00:04:56,630 --> 00:05:02,000 have to have an understanding of the situation 77 00:05:02,000 --> 00:05:09,730 in order to understand what the person is asking you to solve. 78 00:05:09,730 --> 00:05:15,040 So in a situation like this, the buzz words, 79 00:05:15,040 --> 00:05:17,650 or what you would find is, for example, say, 80 00:05:17,650 --> 00:05:18,860 there is a charge. 81 00:05:18,860 --> 00:05:20,180 It's accelerating. 82 00:05:20,180 --> 00:05:25,280 Calculate the electric field at a distance far from the charge, 83 00:05:25,280 --> 00:05:28,040 or the far field, as it's called, 84 00:05:28,040 --> 00:05:29,460 rather than the near field. 85 00:05:33,790 --> 00:05:37,090 And the reason for it is the following, 86 00:05:37,090 --> 00:05:44,240 in this case, that the hard part of this 87 00:05:44,240 --> 00:05:49,440 is calculating the field relatively near the charge, 88 00:05:49,440 --> 00:05:52,180 because there, you have a superposition 89 00:05:52,180 --> 00:05:57,580 of this static field, for example, the Coulomb field. 90 00:05:57,580 --> 00:06:00,780 If a charge is moving, it's like a little current. 91 00:06:00,780 --> 00:06:04,580 And so it produces the magnetic fields 92 00:06:04,580 --> 00:06:08,150 through the Biot-Savart Law, et cetera. 93 00:06:08,150 --> 00:06:14,940 But these fields drop off, like 1/r, from the charge. 94 00:06:26,650 --> 00:06:32,690 And so, if you are far away, then, 95 00:06:32,690 --> 00:06:35,690 in general, in most situations where you're 96 00:06:35,690 --> 00:06:38,350 interested in understanding what's going on, 97 00:06:38,350 --> 00:06:42,490 you can ignore those fields, because the further you 98 00:06:42,490 --> 00:06:44,800 are away, the smaller they are. 99 00:06:44,800 --> 00:06:48,190 And they become negligible. 100 00:06:48,190 --> 00:06:51,440 There is another contribution to the field, 101 00:06:51,440 --> 00:06:55,600 to the electromagnetic field, a very important contribution. 102 00:06:55,600 --> 00:06:59,970 And that is-- and in fact, you know 103 00:06:59,970 --> 00:07:05,880 that, in vacuum, without any charges there, 104 00:07:05,880 --> 00:07:12,920 you can have electric and magnetic fields present 105 00:07:12,920 --> 00:07:17,810 that come from propagating electromagnetic waves. 106 00:07:17,810 --> 00:07:21,460 And it turns out that, if you have a charge which 107 00:07:21,460 --> 00:07:26,490 is accelerating, that does produce 108 00:07:26,490 --> 00:07:30,120 an electromagnetic wave. 109 00:07:30,120 --> 00:07:34,520 And that does not drop off as 1 over r squared, 110 00:07:34,520 --> 00:07:38,430 but as 1/r, much more slowly. 111 00:07:38,430 --> 00:07:41,940 So if you are far away, you are dominated 112 00:07:41,940 --> 00:07:46,450 by the propagating electromagnetic wave. 113 00:07:49,710 --> 00:07:53,480 Now it's not that this is trivial, 114 00:07:53,480 --> 00:07:59,490 but certainly, the propagating electromagnetic wave 115 00:07:59,490 --> 00:08:03,540 is much easier to calculate without the presence 116 00:08:03,540 --> 00:08:05,530 of these other fields. 117 00:08:09,920 --> 00:08:16,330 Now Professor Walter Lewin has done that in his classes. 118 00:08:16,330 --> 00:08:21,810 He has shown how, if you have somewhere a charge which 119 00:08:21,810 --> 00:08:28,780 is accelerating, how it generates an electric field far 120 00:08:28,780 --> 00:08:32,380 away from that charge and a magnetic field. 121 00:08:32,380 --> 00:08:34,630 I won't derive it for you, but I'll 122 00:08:34,630 --> 00:08:40,669 summarize what you have learned. 123 00:08:40,669 --> 00:08:44,290 Now for a second, I'll just want to point out 124 00:08:44,290 --> 00:08:48,660 why the problem of electromagnetic waves 125 00:08:48,660 --> 00:08:52,420 is more difficult than, say, that of a string. 126 00:08:52,420 --> 00:08:57,080 The analogous problem we want to recount for a string 127 00:08:57,080 --> 00:08:57,840 is the following. 128 00:08:57,840 --> 00:09:01,970 Suppose you have a string, you hold it at one end. 129 00:09:01,970 --> 00:09:02,610 Right? 130 00:09:02,610 --> 00:09:08,200 And suppose I move this one end sinusoidally? 131 00:09:08,200 --> 00:09:14,780 You know that a string like this is a line. 132 00:09:14,780 --> 00:09:18,750 And in there are oscillators, infinitely 133 00:09:18,750 --> 00:09:21,420 close to each other, infinite number of them. 134 00:09:21,420 --> 00:09:23,700 All right? 135 00:09:23,700 --> 00:09:29,290 Each one moves along a single line, say, up and down. 136 00:09:29,290 --> 00:09:33,290 So for each one, it's a single number 137 00:09:33,290 --> 00:09:35,600 tells you what the displacement is. 138 00:09:38,540 --> 00:09:43,600 You know that the string satisfies the wave equation. 139 00:09:43,600 --> 00:09:48,460 If you put a boundary at one end, 140 00:09:48,460 --> 00:09:54,140 the oscillator, the first one, is somehow externally driven. 141 00:09:54,140 --> 00:09:56,470 That excites the next one, et cetera. 142 00:09:56,470 --> 00:09:59,580 And you know that the wave propagates. 143 00:09:59,580 --> 00:10:04,280 Now in the case of electromagnetism 144 00:10:04,280 --> 00:10:08,890 of this kind of a problem, the situation is very similar, 145 00:10:08,890 --> 00:10:11,550 but you have this extra complication. 146 00:10:11,550 --> 00:10:16,880 In the string, each oscillator was on the straight line 147 00:10:16,880 --> 00:10:22,050 and only moved up and down in one direction. 148 00:10:22,050 --> 00:10:28,170 For a charge, every point in space you could imagine, 149 00:10:28,170 --> 00:10:30,530 there is an oscillator. 150 00:10:30,530 --> 00:10:33,060 The excitation of that oscillator 151 00:10:33,060 --> 00:10:36,660 is the electric and the magnetic field. 152 00:10:36,660 --> 00:10:39,020 That's the analogous to the displacement. 153 00:10:39,020 --> 00:10:41,310 But notice, already, two quantities, 154 00:10:41,310 --> 00:10:43,830 electric and magnetic. 155 00:10:43,830 --> 00:10:47,220 Here, for the string, you had the displacement, 156 00:10:47,220 --> 00:10:49,870 a single number in one direction. 157 00:10:49,870 --> 00:10:52,860 Here, the electric and the magnetic fields 158 00:10:52,860 --> 00:10:55,340 are vector quantities. 159 00:10:55,340 --> 00:10:59,040 So not only do you have them spread in three dimensions, 160 00:10:59,040 --> 00:11:05,760 instead of along the string, the displacement 161 00:11:05,760 --> 00:11:06,780 is a vector quantity. 162 00:11:06,780 --> 00:11:10,250 So is the directions in three-dimensional space, 163 00:11:10,250 --> 00:11:11,010 et cetera. 164 00:11:11,010 --> 00:11:13,030 So that, in principle, is the same, 165 00:11:13,030 --> 00:11:15,940 but it is much more complicated. 166 00:11:15,940 --> 00:11:18,940 But coming back now, fortunately, 167 00:11:18,940 --> 00:11:26,230 if you analyze what happens, if you take a charge, 168 00:11:26,230 --> 00:11:28,760 if it's stationary, it does not radiate. 169 00:11:28,760 --> 00:11:32,150 It has the Coulomb field, which, as I told you, 170 00:11:32,150 --> 00:11:35,160 drops off like 1 over r squared. 171 00:11:35,160 --> 00:11:37,330 If it's moving with uniform velocity, 172 00:11:37,330 --> 00:11:42,720 for example, it does generate magnetic fields. 173 00:11:42,720 --> 00:11:44,570 But again, they drop off. 174 00:11:44,570 --> 00:11:46,350 It doesn't radiate. 175 00:11:46,350 --> 00:11:53,310 But if it accelerates, then it does produce, far away, 176 00:11:53,310 --> 00:12:00,330 an electron, a field which propagates. 177 00:12:00,330 --> 00:12:04,830 And as Professor Walter Lewin has showed in his lectures, 178 00:12:04,830 --> 00:12:07,500 you find the following. 179 00:12:07,500 --> 00:12:14,780 That if I have a charge which accelerates, 180 00:12:14,780 --> 00:12:18,690 at the time when it's accelerating, 181 00:12:18,690 --> 00:12:22,360 it produces an electric field, which 182 00:12:22,360 --> 00:12:26,519 I can summarize with this formula. 183 00:12:26,519 --> 00:12:27,435 So it's the following. 184 00:12:30,740 --> 00:12:37,350 Suppose, at the origin, somewhere I have a charge. 185 00:12:37,350 --> 00:12:42,710 And it accelerates, say, in some direction, up. 186 00:12:42,710 --> 00:12:48,610 And I am interested what is the electric field that 187 00:12:48,610 --> 00:12:54,570 propagates outwards in some direction, r. 188 00:12:54,570 --> 00:13:04,600 What one can show is that this charge 189 00:13:04,600 --> 00:13:08,175 will propagate in all directions, including 190 00:13:08,175 --> 00:13:08,800 this direction. 191 00:13:12,350 --> 00:13:18,970 But there's only one component of the acceleration 192 00:13:18,970 --> 00:13:23,860 of the charge which gives rise to the production 193 00:13:23,860 --> 00:13:27,280 of the field which propagates. 194 00:13:27,280 --> 00:13:30,370 And that is the component which is 195 00:13:30,370 --> 00:13:34,230 perpendicular to the direction in which I'm 196 00:13:34,230 --> 00:13:38,470 interested to calculate the electric field. 197 00:13:38,470 --> 00:13:43,140 So if the charge, for example, is accelerating upwards, 198 00:13:43,140 --> 00:13:46,900 like this, then it's only the component 199 00:13:46,900 --> 00:13:52,950 of this acceleration which is perpendicular to this line that 200 00:13:52,950 --> 00:13:56,580 gives rise to the electric field. 201 00:13:56,580 --> 00:13:58,660 And one can calculate it. 202 00:13:58,660 --> 00:14:02,230 And its formula is rather straightforward. 203 00:14:02,230 --> 00:14:07,540 You find the electric field at some distance, r, 204 00:14:07,540 --> 00:14:08,605 from the charge. 205 00:14:11,150 --> 00:14:18,600 At time t, is related to, as I say, 206 00:14:18,600 --> 00:14:24,940 this acceleration of the charge, but only the component, which 207 00:14:24,940 --> 00:14:29,750 I call aperp, the component of a perpendicular 208 00:14:29,750 --> 00:14:35,500 to this line, that's this quantity. 209 00:14:35,500 --> 00:14:45,070 And what is interesting is the field here that is produced 210 00:14:45,070 --> 00:14:53,290 is proportional to this perpendicular component 211 00:14:53,290 --> 00:14:55,130 of the acceleration. 212 00:14:55,130 --> 00:15:01,030 But there is a difference in time because, 213 00:15:01,030 --> 00:15:05,630 as this charge accelerates-- it's 214 00:15:05,630 --> 00:15:08,140 analogous to take the string. 215 00:15:08,140 --> 00:15:10,330 When I took the end of the string 216 00:15:10,330 --> 00:15:14,780 and I moved it, at that instant, the kink 217 00:15:14,780 --> 00:15:19,100 got produced which propagated along this string. 218 00:15:19,100 --> 00:15:20,510 Same happens here. 219 00:15:20,510 --> 00:15:23,900 If I have a charge which accelerates, 220 00:15:23,900 --> 00:15:28,760 it produces an electric field which propagates. 221 00:15:28,760 --> 00:15:32,390 And it propagates with the velocity of light, c. 222 00:15:32,390 --> 00:15:40,010 So the field over here is given by the acceleration 223 00:15:40,010 --> 00:15:44,950 of the charge at a time, t prime, which is earlier. 224 00:15:44,950 --> 00:15:49,290 If you're interested in the field here at time t, 225 00:15:49,290 --> 00:15:55,060 then you have to know what was the acceleration of the charge 226 00:15:55,060 --> 00:15:58,160 here at the earlier time, t prime, 227 00:15:58,160 --> 00:16:02,670 which is equal to t minus r/c, which 228 00:16:02,670 --> 00:16:08,350 is the time the electric field took to get to that point. 229 00:16:08,350 --> 00:16:09,430 That's the tricky part. 230 00:16:09,430 --> 00:16:14,780 This is the only tricky part of this equation. 231 00:16:14,780 --> 00:16:20,290 So the electric field at the position r at time t 232 00:16:20,290 --> 00:16:23,560 is related to the acceleration of the charge, 233 00:16:23,560 --> 00:16:25,950 the perpendicular component of it, 234 00:16:25,950 --> 00:16:30,280 at an earlier time times some constants, the charge 235 00:16:30,280 --> 00:16:31,610 and a constant. 236 00:16:31,610 --> 00:16:35,331 And it drops off like 1/r. 237 00:16:35,331 --> 00:16:35,830 OK? 238 00:16:52,020 --> 00:16:55,320 So this is the crucial equation. 239 00:16:55,320 --> 00:17:00,370 It tells us that, as I say-- I'm repeating this, 240 00:17:00,370 --> 00:17:02,960 because it's so important. 241 00:17:02,960 --> 00:17:06,230 Whenever I have a charge which is accelerating, 242 00:17:06,230 --> 00:17:11,079 it, at that instant, produces an electric field 243 00:17:11,079 --> 00:17:16,050 which is moving outwards at the speed of light. 244 00:17:16,050 --> 00:17:21,730 Its amplitude is dropping like 1/r, all right? 245 00:17:21,730 --> 00:17:23,740 And goes on forever. 246 00:17:23,740 --> 00:17:26,400 Now you know from Maxwell's equations, 247 00:17:26,400 --> 00:17:31,890 if you have an electric field which is propagating like that, 248 00:17:31,890 --> 00:17:35,660 associated, there will be a magnetic field. 249 00:17:35,660 --> 00:17:39,940 This is now just the same as you've always 250 00:17:39,940 --> 00:17:42,960 seen for electromagnetic waves. 251 00:17:42,960 --> 00:17:44,990 I like sort of visualizing. 252 00:17:44,990 --> 00:17:47,880 If an electromagnetic wave moves in that direction, 253 00:17:47,880 --> 00:17:51,170 and the electric field is this direction, at that instant, 254 00:17:51,170 --> 00:17:55,120 there will be a magnetic field perpendicular to it. 255 00:17:55,120 --> 00:17:58,380 And the B is proportional to E. And they both 256 00:17:58,380 --> 00:18:00,040 move forward like that. 257 00:18:00,040 --> 00:18:00,790 OK? 258 00:18:00,790 --> 00:18:06,170 So if this is E, then you can get from a B. 259 00:18:06,170 --> 00:18:07,670 They are proportional to each other. 260 00:18:07,670 --> 00:18:10,800 There's just that constant, C, because the units. 261 00:18:10,800 --> 00:18:13,720 And they are perpendicular to each other. 262 00:18:13,720 --> 00:18:14,220 OK. 263 00:18:14,220 --> 00:18:17,660 So if you understand this formula, 264 00:18:17,660 --> 00:18:21,480 all the problems that you would be normally asked 265 00:18:21,480 --> 00:18:26,060 can extremely easily be done, OK? 266 00:18:26,060 --> 00:18:28,040 Once you know E and B, of course, 267 00:18:28,040 --> 00:18:30,350 you can calculate at every location. 268 00:18:30,350 --> 00:18:34,080 The Poynting vector is just a cross product between those. 269 00:18:34,080 --> 00:18:39,260 And I just summarize them on the formula we may need later. 270 00:18:39,260 --> 00:18:41,640 So I'm now going to start from scratch 271 00:18:41,640 --> 00:18:44,090 by taking a concrete example. 272 00:18:44,090 --> 00:18:44,710 All right? 273 00:18:44,710 --> 00:18:49,750 So if this refresher didn't help, maybe the problem 274 00:18:49,750 --> 00:18:51,500 itself will. 275 00:18:51,500 --> 00:18:53,955 So now let's take a concrete example. 276 00:18:57,180 --> 00:19:02,840 I'll take a single charge, Q. OK? 277 00:19:02,840 --> 00:19:05,130 So I have a coordinate system. 278 00:19:05,130 --> 00:19:07,750 Here is my origin of coordinates. 279 00:19:07,750 --> 00:19:11,690 And I'll take a charge here, charge of magnitude, 280 00:19:11,690 --> 00:19:18,670 Q, moving up and down, up and down sinusoidally. 281 00:19:18,670 --> 00:19:22,970 So it's given by the amplitude z0. 282 00:19:22,970 --> 00:19:28,980 It's moving along the z-axis, cosine omega t. 283 00:19:28,980 --> 00:19:32,580 This is the motion. 284 00:19:32,580 --> 00:19:35,630 Of course, something must be moving it. 285 00:19:35,630 --> 00:19:41,780 But the assumption here is it doesn't in any way 286 00:19:41,780 --> 00:19:45,080 affect the electrical and magnetic fields. 287 00:19:45,080 --> 00:19:46,850 So that's the only thing that we have 288 00:19:46,850 --> 00:19:48,480 to consider in the universe. 289 00:19:48,480 --> 00:19:54,150 And the question are, at different places 290 00:19:54,150 --> 00:20:00,660 in space, what is the electric and magnetic field? 291 00:20:00,660 --> 00:20:10,260 So for example, one place is a distance, R, along the z-axis 292 00:20:10,260 --> 00:20:12,150 up. 293 00:20:12,150 --> 00:20:16,790 So it's not along the axis. 294 00:20:16,790 --> 00:20:20,886 So the position where we are interested, where p is, 295 00:20:20,886 --> 00:20:25,530 has coordinates 0, 0, R. In other words, 296 00:20:25,530 --> 00:20:28,590 it's a distance, R, along the z-axis. 297 00:20:28,590 --> 00:20:33,165 And we are all asked, at this time, 298 00:20:33,165 --> 00:20:40,780 R divided by C, what is the electric and magnetic field 299 00:20:40,780 --> 00:20:41,840 at that location? 300 00:20:41,840 --> 00:20:42,570 OK? 301 00:20:42,570 --> 00:20:45,130 That's the first part of the question. 302 00:20:45,130 --> 00:20:49,850 Well, that, I hope you've already done in your head. 303 00:20:49,850 --> 00:20:53,500 If the charge is moving up and down like this-- oh, 304 00:20:53,500 --> 00:20:56,320 and by the way, I have to emphasize, 305 00:20:56,320 --> 00:21:01,300 this problem, in general-- except this one part-- 306 00:21:01,300 --> 00:21:07,080 you are able to do only if we satisfy what I said here, 307 00:21:07,080 --> 00:21:13,860 that the region within which the charge is moving or located 308 00:21:13,860 --> 00:21:18,490 is small, compared to the distance where I'm asking what 309 00:21:18,490 --> 00:21:20,380 the electric and magnetic fields are. 310 00:21:20,380 --> 00:21:26,100 So here, it says for R very much greater than Z0. 311 00:21:26,100 --> 00:21:28,640 So whatever I say from now on, I'm 312 00:21:28,640 --> 00:21:31,720 making the assumption I'm very far away. 313 00:21:31,720 --> 00:21:34,170 I only have the radiated field. 314 00:21:34,170 --> 00:21:40,340 I can ignore the complicated fields like Coulomb's field 315 00:21:40,340 --> 00:21:42,870 and Biot-Savart, et cetera. 316 00:21:42,870 --> 00:21:43,370 OK. 317 00:21:43,370 --> 00:21:46,550 So the first part, even without this, 318 00:21:46,550 --> 00:21:50,150 I can do because imagine we are asking 319 00:21:50,150 --> 00:21:54,050 what is the electric field along the z-axis. 320 00:21:54,050 --> 00:21:56,655 And so suppose I am at that location, 321 00:21:56,655 --> 00:21:59,060 and I'm looking down at the charge. 322 00:21:59,060 --> 00:22:00,380 What do I see? 323 00:22:00,380 --> 00:22:03,870 I see the charge going towards me and away from me, 324 00:22:03,870 --> 00:22:06,570 and towards me and away from me. 325 00:22:06,570 --> 00:22:10,190 So it does have velocity at some stage 326 00:22:10,190 --> 00:22:12,320 and suddenly has acceleration. 327 00:22:12,320 --> 00:22:16,010 But the acceleration is towards me and away from me. 328 00:22:16,010 --> 00:22:19,650 So what is the perpendicular component 329 00:22:19,650 --> 00:22:22,320 of the acceleration in my direction? 330 00:22:22,320 --> 00:22:23,990 0. 331 00:22:23,990 --> 00:22:29,490 At no time do I see the charge accelerating sideways. 332 00:22:29,490 --> 00:22:32,550 I only see it going towards and away from me. 333 00:22:32,550 --> 00:22:35,720 So the aperp is 0. 334 00:22:35,720 --> 00:22:38,950 And then I go back to my equation here. 335 00:22:38,950 --> 00:22:43,750 If aperp is 0, it doesn't matter what you do with it, 336 00:22:43,750 --> 00:22:44,960 E will be 0. 337 00:22:44,960 --> 00:22:49,360 If E is 0, B is 0, because they are proportional to each other. 338 00:22:49,360 --> 00:22:55,825 So at all times this will be 0, OK? 339 00:22:59,500 --> 00:23:01,280 The next part of the question. 340 00:23:01,280 --> 00:23:04,840 And let we quickly tell you, in the next part of the question 341 00:23:04,840 --> 00:23:09,760 it says, now let's have this charge not move up and down, 342 00:23:09,760 --> 00:23:14,048 but along the y-- sorry. 343 00:23:14,048 --> 00:23:15,024 No. 344 00:23:15,024 --> 00:23:16,000 No. 345 00:23:16,000 --> 00:23:17,310 I misspoke. 346 00:23:17,310 --> 00:23:19,210 Let's leave this moving that way. 347 00:23:19,210 --> 00:23:27,330 But instead of looking along the z-axis, 348 00:23:27,330 --> 00:23:30,840 let's look at the point along the y-axis. 349 00:23:30,840 --> 00:23:34,610 So at the point 0,R,0, or in other words, 350 00:23:34,610 --> 00:23:40,900 at distance R from the origin along the y-axis, again, 351 00:23:40,900 --> 00:23:43,550 at this time. 352 00:23:43,550 --> 00:23:54,790 And then the third one, it asks let's look at it at 30 degrees. 353 00:23:54,790 --> 00:24:00,750 So now, in this case, we look at in the direction 354 00:24:00,750 --> 00:24:05,270 where it's in the y, z plane, but at 30 355 00:24:05,270 --> 00:24:09,170 degrees to the z-axis, so in some direction 356 00:24:09,170 --> 00:24:12,860 there where the coordinate is this. 357 00:24:12,860 --> 00:24:15,250 OK? 358 00:24:15,250 --> 00:24:17,040 And the whole purpose is to just get 359 00:24:17,040 --> 00:24:23,100 you a feel how accelerated charges radiate 360 00:24:23,100 --> 00:24:26,040 in different directions. 361 00:24:26,040 --> 00:24:28,370 So the first one I've already done for you. 362 00:24:31,140 --> 00:24:32,820 OK? 363 00:24:32,820 --> 00:24:38,980 But here, let me just repeat myself or point out. 364 00:24:38,980 --> 00:24:42,480 In each part of this question, we 365 00:24:42,480 --> 00:24:46,056 are a distance, R, from the charge. 366 00:24:46,056 --> 00:24:47,430 Now you can say, well, the charge 367 00:24:47,430 --> 00:24:49,620 is moving a small amount. 368 00:24:49,620 --> 00:24:56,450 But the whole essence of making this a doable problem 369 00:24:56,450 --> 00:25:00,120 and getting a situation which you can solve 370 00:25:00,120 --> 00:25:03,800 is where all the motion of the charges 371 00:25:03,800 --> 00:25:06,640 over distances which is very much 372 00:25:06,640 --> 00:25:08,900 smaller than where you are. 373 00:25:08,900 --> 00:25:12,860 So because R is very much greater than z0, 374 00:25:12,860 --> 00:25:17,820 from the distance of the accelerated charge, 375 00:25:17,820 --> 00:25:21,650 from where you want to calculate the electric field, 376 00:25:21,650 --> 00:25:25,110 you can assume is constant in distance R. 377 00:25:25,110 --> 00:25:27,590 So in all parts of this problem, you 378 00:25:27,590 --> 00:25:31,210 are the same distance away, but at a different angle. 379 00:25:33,900 --> 00:25:38,350 And in all cases, we are asked to find the field 380 00:25:38,350 --> 00:25:44,350 at some time, t, when this is oscillating. 381 00:25:44,350 --> 00:25:46,290 Let's go to the second part. 382 00:25:46,290 --> 00:25:55,425 How about in a direction which is along the y-axis? 383 00:25:55,425 --> 00:25:57,330 All right? 384 00:25:57,330 --> 00:26:00,480 So in the y direction, if something 385 00:26:00,480 --> 00:26:03,680 is accelerating in the z direction, 386 00:26:03,680 --> 00:26:08,160 that is perpendicular to the y-axis. 387 00:26:08,160 --> 00:26:12,140 And so for the second part, the acceleration 388 00:26:12,140 --> 00:26:20,650 of the charge at time t prime, is 389 00:26:20,650 --> 00:26:26,190 simply I have to take this and differentiate it twice, 390 00:26:26,190 --> 00:26:27,380 all right? 391 00:26:27,380 --> 00:26:29,100 And so we've done it. 392 00:26:29,100 --> 00:26:29,800 This is this. 393 00:26:32,410 --> 00:26:38,880 And this acceleration, that has a component in the z direction 394 00:26:38,880 --> 00:26:42,520 which is perpendicular to the y direction. 395 00:26:42,520 --> 00:26:48,990 So in our formula, this is Eperp. 396 00:26:48,990 --> 00:26:51,790 And so we can plug it into that formula. 397 00:26:51,790 --> 00:26:56,690 And we get that the electric field 398 00:26:56,690 --> 00:27:02,490 at position R along the y-axis is given by this. 399 00:27:02,490 --> 00:27:06,400 And I just took that formula-- let me just remind you-- 400 00:27:06,400 --> 00:27:12,170 I have just taken this formula and plugged in the aperp. 401 00:27:12,170 --> 00:27:16,780 And I end up with this equation. 402 00:27:16,780 --> 00:27:21,440 And so I see that, if a charge is oscillating up and down 403 00:27:21,440 --> 00:27:26,550 like that, it does radiate perpendicular to it. 404 00:27:26,550 --> 00:27:30,580 This is the nearest analogy to that string case. 405 00:27:30,580 --> 00:27:32,370 If the string is moving like that, 406 00:27:32,370 --> 00:27:38,680 it propagates a wave along the string. 407 00:27:38,680 --> 00:27:41,970 And here, if a charge is accelerating like this, 408 00:27:41,970 --> 00:27:45,230 it does radiate along that direction. 409 00:27:45,230 --> 00:27:48,290 This is the magnitude. 410 00:27:48,290 --> 00:27:53,360 And as a function, that's what it is. 411 00:27:53,360 --> 00:27:57,100 The actual problem was not in general 412 00:27:57,100 --> 00:28:08,090 what it is, what will it be at the time R/C? 413 00:28:08,090 --> 00:28:15,620 Well, at t R/C, this is R/C minus R/C. This is 0. 414 00:28:15,620 --> 00:28:18,370 So this is cosine of 0, so this is 1. 415 00:28:18,370 --> 00:28:23,290 And so we get, finally, that at that time, 416 00:28:23,290 --> 00:28:28,520 the electric field is equal to this. 417 00:28:28,520 --> 00:28:30,910 It's just that times 1. 418 00:28:30,910 --> 00:28:32,880 This is the electric field. 419 00:28:32,880 --> 00:28:34,700 What is the magnetic field? 420 00:28:34,700 --> 00:28:37,290 Well, as I told you, if you know the electric field, 421 00:28:37,290 --> 00:28:40,630 B is perpendicular to it, proportional to it. 422 00:28:40,630 --> 00:28:45,350 And there's just a 1/C. So they here have it C cubed. 423 00:28:45,350 --> 00:28:51,760 And the magnetic field will be in the x direction. 424 00:28:51,760 --> 00:28:56,530 The electric field is always parallel 425 00:28:56,530 --> 00:28:59,550 to the accelerated charge. 426 00:28:59,550 --> 00:29:04,300 The magnetic field is in the perpendicular direction. 427 00:29:04,300 --> 00:29:10,190 Also, it is important to remember the minus sign. 428 00:29:10,190 --> 00:29:13,550 I come back to this formula. 429 00:29:13,550 --> 00:29:17,400 This minus sign is very important. 430 00:29:17,400 --> 00:29:20,410 What it's saying is, at the instant 431 00:29:20,410 --> 00:29:25,640 when the charge is accelerating in that direction, 432 00:29:25,640 --> 00:29:30,090 the electric field it produces is in the opposite direction. 433 00:29:30,090 --> 00:29:33,540 So when this is accelerating, at the instant this accelerates, 434 00:29:33,540 --> 00:29:36,540 it produces an electric field which goes in this direction. 435 00:29:36,540 --> 00:29:39,880 And that field propagates like that. 436 00:29:39,880 --> 00:29:40,860 OK? 437 00:29:40,860 --> 00:29:43,680 And the magnetic field is perpendicular to it. 438 00:29:43,680 --> 00:29:49,570 And so that is the Part 2. 439 00:29:49,570 --> 00:29:54,700 And now we come to doing the third one-- it's actually 440 00:29:54,700 --> 00:29:58,290 the same problem-- where we are considering 441 00:29:58,290 --> 00:30:04,640 the radiation at the slightly more complicated angle. 442 00:30:04,640 --> 00:30:09,410 So far, we've looked at the charge is moving up and down. 443 00:30:09,410 --> 00:30:13,310 We said there's no electromagnetic field radiated 444 00:30:13,310 --> 00:30:15,315 upwards, or downwards, for that matter. 445 00:30:18,250 --> 00:30:20,230 Along the y-axis it is. 446 00:30:20,230 --> 00:30:24,690 And of course, if we took any other location 447 00:30:24,690 --> 00:30:30,570 in the plane of our x, y, in the horizontal plane, 448 00:30:30,570 --> 00:30:33,810 in all directions, it will radiate 449 00:30:33,810 --> 00:30:37,820 in a similar manner as along the y-axis. 450 00:30:37,820 --> 00:30:40,070 And finally, just for completion, 451 00:30:40,070 --> 00:30:42,480 let's take one more direction. 452 00:30:42,480 --> 00:30:47,480 And what the problem said, let's consider going at 30 degrees 453 00:30:47,480 --> 00:30:48,810 to the z-axis. 454 00:30:48,810 --> 00:30:50,580 That's an angle like that. 455 00:30:50,580 --> 00:30:53,750 What is the electric field that is 456 00:30:53,750 --> 00:30:55,580 generating in that direction? 457 00:30:55,580 --> 00:30:58,630 Again, this is just to give us practice 458 00:30:58,630 --> 00:31:02,600 at calculating vectors, components, et cetera. 459 00:31:02,600 --> 00:31:05,920 So I will do that now over here. 460 00:31:05,920 --> 00:31:09,300 So this is the solving the Part 3. 461 00:31:12,680 --> 00:31:14,450 So what we're interested in, we have 462 00:31:14,450 --> 00:31:18,280 this charge moving up and down. 463 00:31:18,280 --> 00:31:21,520 And we're interested in the radiation a long way away, 464 00:31:21,520 --> 00:31:25,170 a distance R away from this charge, 465 00:31:25,170 --> 00:31:31,350 but at some angle like that, with just 30 degrees 466 00:31:31,350 --> 00:31:34,900 to the z-axis. 467 00:31:34,900 --> 00:31:38,250 So in order to use that formula-- 468 00:31:38,250 --> 00:31:39,790 it's all the time the same thing, 469 00:31:39,790 --> 00:31:41,550 I'm just using the same formula-- 470 00:31:41,550 --> 00:31:47,830 I need to calculate the component of the acceleration 471 00:31:47,830 --> 00:31:50,720 of the charge which is perpendicular 472 00:31:50,720 --> 00:31:53,660 to that direction, OK? 473 00:31:53,660 --> 00:31:55,010 Now there are many ways. 474 00:31:55,010 --> 00:31:57,595 You can do it in your head, if you 475 00:31:57,595 --> 00:32:01,010 are good at manipulating vectors. 476 00:32:01,010 --> 00:32:03,990 Here, I wrote it, in case you wanted a complete formula. 477 00:32:06,625 --> 00:32:10,640 Here is how the perpendicular component 478 00:32:10,640 --> 00:32:16,160 is related to the acceleration, a. 479 00:32:16,160 --> 00:32:19,100 And at the beginning there, I summarized it for you. 480 00:32:19,100 --> 00:32:20,660 But you can do it any way you want. 481 00:32:20,660 --> 00:32:23,080 I did it this way. 482 00:32:23,080 --> 00:32:29,810 And so I am taking this charge and calculating, at some time, 483 00:32:29,810 --> 00:32:35,820 t prime, what is the component of the acceleration 484 00:32:35,820 --> 00:32:42,420 of the charge in a direction perpendicular to the direction 485 00:32:42,420 --> 00:32:46,600 in which I am considering the radiation. 486 00:32:46,600 --> 00:32:51,210 And this is what it comes out. 487 00:32:51,210 --> 00:32:57,130 If you remember, I know what is the motion of the charge. 488 00:32:57,130 --> 00:32:59,400 You've seen it over there. 489 00:32:59,400 --> 00:33:03,670 So I know what z is, a function of time. 490 00:33:03,670 --> 00:33:05,500 Therefore, I can calculate this. 491 00:33:05,500 --> 00:33:09,130 And that's what it comes out as, alright? 492 00:33:09,130 --> 00:33:13,510 Here, I formally use this equation. 493 00:33:13,510 --> 00:33:15,500 So it's as before. 494 00:33:23,030 --> 00:33:29,200 We will be interested in the electric field at the position, 495 00:33:29,200 --> 00:33:34,490 R, so at a time which is later. 496 00:33:34,490 --> 00:33:39,580 In other words, t minus R/C is the time 497 00:33:39,580 --> 00:33:42,940 at which the prime time, t prime, at which I'm 498 00:33:42,940 --> 00:33:45,230 interested in this acceleration. 499 00:33:45,230 --> 00:33:49,910 And that's what it looks like, OK? 500 00:33:49,910 --> 00:33:52,630 Now, I can simplify it. 501 00:33:52,630 --> 00:33:54,220 I'm not going to waste your time. 502 00:33:54,220 --> 00:33:57,380 You can go through this algebra. 503 00:33:57,380 --> 00:33:58,880 Simplify it here. 504 00:33:58,880 --> 00:34:02,300 And then I take this acceleration 505 00:34:02,300 --> 00:34:06,300 and plug it into that equation, which tells me, 506 00:34:06,300 --> 00:34:09,830 if I know the perpendicular component of acceleration 507 00:34:09,830 --> 00:34:13,270 at this earlier time, I know what the electric field 508 00:34:13,270 --> 00:34:14,860 is at the time t. 509 00:34:14,860 --> 00:34:16,560 And this is what it comes out at. 510 00:34:16,560 --> 00:34:22,690 So the electric field in a direction 30 511 00:34:22,690 --> 00:34:29,150 degrees to the z-axis, or in other words, 512 00:34:29,150 --> 00:34:32,630 at a distance, capital R, from the origin. 513 00:34:32,630 --> 00:34:37,350 Or in other words, at the coordinates 0, 514 00:34:37,350 --> 00:34:44,050 R/2, R root 3 over-- and this is the y-coordinate, x-coordinate, 515 00:34:44,050 --> 00:34:48,679 y, and z-- at this time, I plug this into that equation. 516 00:34:48,679 --> 00:34:52,060 And I get this and the direction. 517 00:34:52,060 --> 00:34:55,310 If you do it formally, you immediately get the direction. 518 00:34:55,310 --> 00:34:57,400 It's not very hard. 519 00:34:57,400 --> 00:35:03,320 It'll be, in fact, it's 60 degrees to the z-axis. 520 00:35:03,320 --> 00:35:04,370 All right. 521 00:35:04,370 --> 00:35:06,980 So that's what the electric field will be. 522 00:35:06,980 --> 00:35:09,610 Then, knowing the electric field, 523 00:35:09,610 --> 00:35:14,850 again, I know that E and B are perpendicular to each other. 524 00:35:14,850 --> 00:35:18,610 They are proportional to each other, except for the 1/C. 525 00:35:18,610 --> 00:35:19,320 All right? 526 00:35:19,320 --> 00:35:23,540 And so, at the same location, I can calculate B. 527 00:35:23,540 --> 00:35:25,590 And I get this. 528 00:35:25,590 --> 00:35:30,730 The magnitude here and here are the same, 529 00:35:30,730 --> 00:35:34,010 except for one power of C here, C cubed. 530 00:35:34,010 --> 00:35:35,320 The direction is different. 531 00:35:38,000 --> 00:35:40,470 B will be the x direction. 532 00:35:40,470 --> 00:35:45,480 And that, you could do in your head. 533 00:35:45,480 --> 00:35:53,819 If the charge is oscillating like this, at all angles, 534 00:35:53,819 --> 00:35:55,360 you'll still have the magnetic field, 535 00:35:55,360 --> 00:35:58,690 which will perpendicular in that direction. 536 00:35:58,690 --> 00:35:59,440 All right? 537 00:35:59,440 --> 00:36:01,630 And you can check for yourself. 538 00:36:01,630 --> 00:36:09,450 This half is simply the sine of 30 degrees. 539 00:36:09,450 --> 00:36:10,090 OK. 540 00:36:10,090 --> 00:36:14,500 So I've gone very slowly, intentionally, 541 00:36:14,500 --> 00:36:21,265 and taken the simple situation where 542 00:36:21,265 --> 00:36:24,020 a single charge is oscillating up and down 543 00:36:24,020 --> 00:36:27,580 and I've calculated for you the electric field, 544 00:36:27,580 --> 00:36:31,780 the radiated electric field, up and down, at 0. 545 00:36:31,780 --> 00:36:36,640 In a horizontal plane, I've done it effectively, all locations. 546 00:36:36,640 --> 00:36:39,540 And I've done it at an angle, theta, 547 00:36:39,540 --> 00:36:41,660 where theta was 30 degrees. 548 00:36:41,660 --> 00:36:45,900 I could repeat this for every angle, theta, every location. 549 00:36:45,900 --> 00:36:50,870 So in fact, just taking more and more cases, 550 00:36:50,870 --> 00:36:54,700 we have calculated the electric and propagating 551 00:36:54,700 --> 00:36:59,440 electric and magnetic field in all directions, all right? 552 00:36:59,440 --> 00:37:04,546 And we calculated the magnitude at a distance from the origin. 553 00:37:04,546 --> 00:37:05,045 OK? 554 00:37:08,410 --> 00:37:11,880 It drops off like 1/R we saw. 555 00:37:11,880 --> 00:37:17,200 And like the other field, we drop off like 1 over r squared. 556 00:37:17,200 --> 00:37:24,280 And so this charge, which is oscillating up and down, 557 00:37:24,280 --> 00:37:29,870 radiates in all directions, but with different magnitudes, 558 00:37:29,870 --> 00:37:33,580 biggest in the plane perpendicular 559 00:37:33,580 --> 00:37:37,710 to the vector describing the acceleration. 560 00:37:40,900 --> 00:37:42,010 OK. 561 00:37:42,010 --> 00:37:47,045 Now the next thing that I will do now is the following. 562 00:37:54,410 --> 00:38:00,020 Let's take the same situation and try 563 00:38:00,020 --> 00:38:07,960 to calculate the total power radiated by the charge. 564 00:38:07,960 --> 00:38:12,430 So I've taken this same problem. 565 00:38:12,430 --> 00:38:17,160 There is the charge oscillating up and down 566 00:38:17,160 --> 00:38:24,280 with amplitude z0 cosine omega t prime. 567 00:38:24,280 --> 00:38:30,980 And what we see-- well, we know that it radiates energy. 568 00:38:30,980 --> 00:38:35,570 If it radiates energy, of course, normally, 569 00:38:35,570 --> 00:38:40,950 it would have to finally stop. 570 00:38:40,950 --> 00:38:45,730 But I'm assuming there is some mechanism which uniformly 571 00:38:45,730 --> 00:38:52,420 maintains the motion of these charges in a certain motion. 572 00:38:52,420 --> 00:38:54,710 Energy will be radiated. 573 00:38:54,710 --> 00:39:01,410 And I want to calculate what is the total energy radiated 574 00:39:01,410 --> 00:39:04,990 per second by this charge. 575 00:39:04,990 --> 00:39:09,957 And the way we are going to do this-- so the question now 576 00:39:09,957 --> 00:39:16,040 is, for this particular charge, oscillating like we discussed, 577 00:39:16,040 --> 00:39:20,120 what is the total energy per second-- in other words, 578 00:39:20,120 --> 00:39:24,100 power-- radiated at the time t prime? 579 00:39:27,660 --> 00:39:32,280 Clearly, if it's radiating energy outwards, 580 00:39:32,280 --> 00:39:34,910 by conservation of energy, the amount 581 00:39:34,910 --> 00:39:38,250 of energy coming out of this will 582 00:39:38,250 --> 00:39:43,090 be the same as the amount of energy crossing 583 00:39:43,090 --> 00:39:50,930 a sphere of radius, R, at a later time. 584 00:39:50,930 --> 00:39:56,580 So if I calculate the energy crossing a sphere at the time 585 00:39:56,580 --> 00:40:01,220 t, where t is this t prime plus R/C, 586 00:40:01,220 --> 00:40:06,240 at a later time, that will be the total energy 587 00:40:06,240 --> 00:40:11,440 per second radiated by this charge at the time, t prime. 588 00:40:11,440 --> 00:40:15,900 So I will now calculate the energy 589 00:40:15,900 --> 00:40:22,500 per second which is crossing the surface of this sphere 590 00:40:22,500 --> 00:40:25,720 at the time, t. 591 00:40:25,720 --> 00:40:28,450 Well, how much energy is crossing the surface? 592 00:40:28,450 --> 00:40:34,340 We know that, at the surface of this sphere, 593 00:40:34,340 --> 00:40:41,240 there is, at every location, an electric field which 594 00:40:41,240 --> 00:40:44,140 is parallel to the surface, because this 595 00:40:44,140 --> 00:40:47,280 is perpendicular to R. The electric field is 596 00:40:47,280 --> 00:40:49,080 perpendicular to this. 597 00:40:49,080 --> 00:40:53,310 So at this location, for example, the electric field 598 00:40:53,310 --> 00:40:57,530 will be like this, crossing it, part 599 00:40:57,530 --> 00:41:00,080 of the surface of this sphere. 600 00:41:00,080 --> 00:41:04,000 There will be a magnetic field perpendicular to it. 601 00:41:04,000 --> 00:41:07,470 And so, at this location, there will 602 00:41:07,470 --> 00:41:11,720 be a energy flux, the so-called Poynting vector, 603 00:41:11,720 --> 00:41:16,800 the energy per second crossing a unit [INAUDIBLE]. 604 00:41:16,800 --> 00:41:20,050 And that's this vector, s, the Poynting vector, 605 00:41:20,050 --> 00:41:23,030 which, as you know from Professor Walter 606 00:41:23,030 --> 00:41:27,740 Lewin's lectures, is E cross B. I forgot my vector sign 607 00:41:27,740 --> 00:41:35,550 on top of the B. These are vectors, of course, E cross B. 608 00:41:35,550 --> 00:41:40,700 Now we know, for an electromagnetic wave in vacuum, 609 00:41:40,700 --> 00:41:45,280 that the E and B are perpendicular and proportional 610 00:41:45,280 --> 00:41:48,060 to each other, just the fact the C. So this will 611 00:41:48,060 --> 00:41:54,430 be the magnitude of E squared divided by this mu 0 times 612 00:41:54,430 --> 00:41:57,400 C, because B is E/C. 613 00:41:57,400 --> 00:41:59,310 This is a scalar quant-- these were vectors. 614 00:41:59,310 --> 00:42:00,970 This is a scalar. 615 00:42:00,970 --> 00:42:09,000 So this is the energy flowing per unit area, per unit time 616 00:42:09,000 --> 00:42:10,455 at the surface of this. 617 00:42:13,330 --> 00:42:20,000 And the E will depend on the location on this sphere. 618 00:42:20,000 --> 00:42:24,060 So what is the total energy that's 619 00:42:24,060 --> 00:42:28,700 leaving the charge at time t prime? 620 00:42:28,700 --> 00:42:34,330 It will be the sum of all the energies 621 00:42:34,330 --> 00:42:36,410 that are coming out per unit area 622 00:42:36,410 --> 00:42:39,940 along this surface at time t. 623 00:42:39,940 --> 00:42:46,140 So it's going to be the integral of SdA. 624 00:42:46,140 --> 00:42:50,410 What I'm doing, basically, calculating at every location 625 00:42:50,410 --> 00:42:54,870 S multiplied by the area there-- so that gives me 626 00:42:54,870 --> 00:42:57,010 how much energy goes through that area-- 627 00:42:57,010 --> 00:43:00,470 and adding them across the surface 628 00:43:00,470 --> 00:43:03,600 everywhere around the complete sphere. 629 00:43:03,600 --> 00:43:07,350 That will be the total energy leaving here. 630 00:43:07,350 --> 00:43:10,170 So imagine this is oscillating up and down, 631 00:43:10,170 --> 00:43:14,460 radiating these spherical waves outwards. 632 00:43:14,460 --> 00:43:16,670 And they cross this surface. 633 00:43:16,670 --> 00:43:21,190 And at every place, there is energy flowing, the Poynting 634 00:43:21,190 --> 00:43:21,790 vector. 635 00:43:21,790 --> 00:43:23,850 And here, I'm adding them all. 636 00:43:23,850 --> 00:43:28,790 This will be the total power radiated 637 00:43:28,790 --> 00:43:32,521 by this charge at that earlier time, t prime. 638 00:43:32,521 --> 00:43:33,020 OK. 639 00:43:33,020 --> 00:43:37,660 So I now have to just log through and calculate this. 640 00:43:37,660 --> 00:43:42,615 S, I have to calculate from this. 641 00:43:45,460 --> 00:43:51,190 And then I have to calculate the piece of area. 642 00:43:51,190 --> 00:43:55,120 Well, let me tell you what I will do. 643 00:43:55,120 --> 00:44:01,480 From what we've learned from the earlier part, 644 00:44:01,480 --> 00:44:11,150 the magnitude of E only depended on this angle, theta, 645 00:44:11,150 --> 00:44:13,220 and this distance. 646 00:44:13,220 --> 00:44:18,080 So imagine, along the surface, I take a slice 647 00:44:18,080 --> 00:44:26,470 this surface, like this, where every point along this surface 648 00:44:26,470 --> 00:44:32,890 is at an angle, theta, to the z-axis, OK? 649 00:44:36,400 --> 00:44:39,040 And it's a thin slice. 650 00:44:39,040 --> 00:44:41,890 So what will be this distance? 651 00:44:41,890 --> 00:44:44,830 I'll take this to be d theta. 652 00:44:44,830 --> 00:44:49,450 This distance is R, so that distance is already Rd theta. 653 00:44:49,450 --> 00:44:55,050 If I multiply that by 2 pi r sine theta, 654 00:44:55,050 --> 00:45:02,020 r sine theta is the radius of this circle. 655 00:45:02,020 --> 00:45:07,850 So this total area going all the way around this sphere 656 00:45:07,850 --> 00:45:14,790 is 2 pi r sine theta multiplied by Rd theta. 657 00:45:14,790 --> 00:45:16,050 OK? 658 00:45:16,050 --> 00:45:22,720 So now along this, everywhere, S is the same. 659 00:45:22,720 --> 00:45:25,970 So if S is at that angle, theta, and I've 660 00:45:25,970 --> 00:45:31,250 multiplied by 2 pi R sine theta Rd theta, 661 00:45:31,250 --> 00:45:36,120 this is the piece of area, if I multiply by that, 662 00:45:36,120 --> 00:45:43,370 this gives me the total energy flux or power 663 00:45:43,370 --> 00:45:52,490 that is leaving the sphere along this part, all 664 00:45:52,490 --> 00:45:57,070 the way behind there, all the way around. 665 00:45:57,070 --> 00:45:59,960 OK? 666 00:45:59,960 --> 00:46:03,920 Now what is the total energy per second leaving 667 00:46:03,920 --> 00:46:06,950 this complete sphere is the addition 668 00:46:06,950 --> 00:46:11,010 of pieces like that for the complete sphere. 669 00:46:11,010 --> 00:46:15,730 So it would be the integral of this 670 00:46:15,730 --> 00:46:18,390 for theta going from 0 to pi. 671 00:46:21,070 --> 00:46:23,410 And then I've covered the compete sphere. 672 00:46:23,410 --> 00:46:26,770 So that will be this integral, which 673 00:46:26,770 --> 00:46:29,870 is the power radiated by the charge. 674 00:46:29,870 --> 00:46:32,540 OK, if you've understood that, then it's 675 00:46:32,540 --> 00:46:36,550 just pure algebra now. 676 00:46:36,550 --> 00:46:39,470 S, I can calculate, because I know 677 00:46:39,470 --> 00:46:44,180 it's the magnitude of E squared over C mu 0. 678 00:46:44,180 --> 00:46:47,550 But that, we've already done. 679 00:46:47,550 --> 00:46:49,070 We did it for 30 degrees. 680 00:46:49,070 --> 00:46:50,310 And now do it for theta. 681 00:46:50,310 --> 00:46:54,430 So here, instead of having sine 30, which was 1/2, 682 00:46:54,430 --> 00:46:57,870 I have sine theta there. 683 00:46:57,870 --> 00:47:03,215 So that squared gives me this divided by C mu 0 that. 684 00:47:09,200 --> 00:47:15,120 And of course, it depends-- so this is what E is. 685 00:47:15,120 --> 00:47:22,880 But it's oscillating as cosine omega times this. 686 00:47:22,880 --> 00:47:26,080 And we have to square it, because this is E squared. 687 00:47:26,080 --> 00:47:33,760 So this is what the Poynting vector is at, 688 00:47:33,760 --> 00:47:39,370 the angle theta, distance R away at time t, OK? 689 00:47:39,370 --> 00:47:41,830 And therefore, the total power is 690 00:47:41,830 --> 00:47:48,060 this multiplied by that integrated from 0 to pi. 691 00:47:48,060 --> 00:47:51,980 I've just rewritten it, this quantity, 692 00:47:51,980 --> 00:47:53,450 plugging that into here. 693 00:47:53,450 --> 00:47:55,470 I end up with this. 694 00:47:55,470 --> 00:47:58,090 Everything does not depend on theta, 695 00:47:58,090 --> 00:48:03,320 except sine theta, which you have a squared from here 696 00:48:03,320 --> 00:48:04,950 and another sign from there. 697 00:48:04,950 --> 00:48:07,980 So I get sine cubed theta d theta. 698 00:48:07,980 --> 00:48:10,150 OK, we're home. 699 00:48:10,150 --> 00:48:12,600 So that's the answer. 700 00:48:12,600 --> 00:48:17,400 Because this you can do, I won't waste your time. 701 00:48:17,400 --> 00:48:21,310 The way to do it is you can take sine theta d theta to be 702 00:48:21,310 --> 00:48:25,170 d cosine theta, and then integrating sine 703 00:48:25,170 --> 00:48:28,080 squared theta d cosine theta. 704 00:48:28,080 --> 00:48:31,600 And if you do that, you get 4/3. 705 00:48:31,600 --> 00:48:34,250 So I plug 4/3 in for that integral. 706 00:48:34,250 --> 00:48:38,790 This is the answer, OK? 707 00:48:38,790 --> 00:48:46,810 So what we see now, through any-- 708 00:48:46,810 --> 00:48:51,390 and it's independent of R, which I hope did not surprise you. 709 00:48:51,390 --> 00:48:53,880 It's a statement of conservation of energy, 710 00:48:53,880 --> 00:48:56,520 because you have this charge oscillating 711 00:48:56,520 --> 00:48:59,000 up and down, radiating. 712 00:48:59,000 --> 00:49:04,370 Through every sphere of radius R, the same amount of energy 713 00:49:04,370 --> 00:49:06,345 has to go through, or otherwise, you' 714 00:49:06,345 --> 00:49:08,350 d be gaining or losing energy. 715 00:49:08,350 --> 00:49:10,057 So that's a check. 716 00:49:10,057 --> 00:49:10,640 You can check. 717 00:49:10,640 --> 00:49:14,130 If R came into this formula, you would have made a mistake. 718 00:49:14,130 --> 00:49:16,360 And the only place where it does, 719 00:49:16,360 --> 00:49:21,010 is only-- it oscillates, so at any position, R, 720 00:49:21,010 --> 00:49:23,890 the energy going through oscillates, because it's 721 00:49:23,890 --> 00:49:25,840 a wave going out, OK? 722 00:49:25,840 --> 00:49:29,550 So this changes the phase, but not the magnitude. 723 00:49:29,550 --> 00:49:33,600 In the actual question, it asked for what 724 00:49:33,600 --> 00:49:38,920 is the time average that's radiated outwards. 725 00:49:38,920 --> 00:49:42,760 So we want to know the time average of this. 726 00:49:42,760 --> 00:49:49,375 And that's the time average of that is here. 727 00:49:52,070 --> 00:49:58,980 You know that the time average of cosine some function of t 728 00:49:58,980 --> 00:49:59,980 is 1/2. 729 00:49:59,980 --> 00:50:04,670 If you take cosine-- sorry, cosine squared, 730 00:50:04,670 --> 00:50:05,940 which was in here, cosine. 731 00:50:05,940 --> 00:50:09,200 So the time average of cosine squared is 1/2. 732 00:50:09,200 --> 00:50:12,790 So finally, we get that the average, time average, 733 00:50:12,790 --> 00:50:17,070 power leaving this oscillating charge 734 00:50:17,070 --> 00:50:18,434 is given by this formula. 735 00:50:18,434 --> 00:50:19,725 It's actually a famous formula. 736 00:50:19,725 --> 00:50:24,310 It's called [? Lambor ?] formula. 737 00:50:24,310 --> 00:50:25,340 OK? 738 00:50:25,340 --> 00:50:28,260 So that's the total energy radiated. 739 00:50:28,260 --> 00:50:29,670 Fine. 740 00:50:29,670 --> 00:50:32,050 We have a few minutes left, and so I'll 741 00:50:32,050 --> 00:50:37,420 do one more quick problem. 742 00:50:45,470 --> 00:50:56,160 OK, just for variety-- so far, we've 743 00:50:56,160 --> 00:50:59,520 considered a charge which is accelerating up and down. 744 00:50:59,520 --> 00:51:02,230 And be considered all permutations and combinations 745 00:51:02,230 --> 00:51:05,940 about what happens to the electromagnetic field going 746 00:51:05,940 --> 00:51:06,860 outwards. 747 00:51:06,860 --> 00:51:08,490 Now let's consider something else, 748 00:51:08,490 --> 00:51:11,140 a charge which is going in a circle. 749 00:51:11,140 --> 00:51:21,180 Let's consider that a charge, Q, which is rotating in the z, 750 00:51:21,180 --> 00:51:28,170 y plane with a uniform velocity, v, 751 00:51:28,170 --> 00:51:33,520 given by omega z0 where omega is the angular velocity of this. 752 00:51:33,520 --> 00:51:37,470 So we have a charge going like this, uniform velocity. 753 00:51:41,700 --> 00:51:45,160 What is the radiated field? 754 00:51:45,160 --> 00:51:50,030 What radiated electric and magnetic fields are produced? 755 00:51:50,030 --> 00:51:56,060 Well, go back to where I started. 756 00:51:56,060 --> 00:52:00,980 The charge is moving with uniform velocity. 757 00:52:00,980 --> 00:52:08,240 Static charges don't radiate, but accelerated charges do. 758 00:52:08,240 --> 00:52:10,290 Does this charge accelerate? 759 00:52:10,290 --> 00:52:14,920 It's moving with a uniform speed in a circle, 760 00:52:14,920 --> 00:52:16,170 but does it radiate? 761 00:52:16,170 --> 00:52:19,270 Well, you know that, if an object is 762 00:52:19,270 --> 00:52:25,160 going with a uniform speed, V, around a circle, 763 00:52:25,160 --> 00:52:28,220 it's all the time accelerating, right? 764 00:52:28,220 --> 00:52:31,480 It's accelerating inwards all the time. 765 00:52:34,070 --> 00:52:38,540 And so this charge is accelerating inwards. 766 00:52:38,540 --> 00:52:41,740 So at every instant of time, it's 767 00:52:41,740 --> 00:52:45,570 accelerating like this, so as this moves more. 768 00:52:45,570 --> 00:52:49,560 So all the time, it has a constant magnitude 769 00:52:49,560 --> 00:52:55,550 of acceleration, but the direction changes all the time. 770 00:52:55,550 --> 00:52:59,995 So this charge, at every instant of time, will radiate. 771 00:53:02,640 --> 00:53:09,380 And it will radiate in all directions, like before. 772 00:53:09,380 --> 00:53:14,700 For simplicity, so we don't get overburdened by mathematics, 773 00:53:14,700 --> 00:53:18,730 let me just think for a second what a-- let's just 774 00:53:18,730 --> 00:53:20,123 talk qualitatively. 775 00:53:25,970 --> 00:53:32,080 I can describe the motion of a charge going like this by, 776 00:53:32,080 --> 00:53:36,440 at any instant of time, it has a position. 777 00:53:36,440 --> 00:53:42,670 This is Z0 vector is the position 778 00:53:42,670 --> 00:53:46,020 of this charge at time t prime. 779 00:53:46,020 --> 00:53:52,780 I could write it as Z0 cosine theta y plus Z0 sine theta Z, 780 00:53:52,780 --> 00:53:54,460 all right? 781 00:53:54,460 --> 00:53:56,160 That's the y-coordinate. 782 00:53:56,160 --> 00:53:58,210 That's the z-coordinate. 783 00:53:58,210 --> 00:54:05,790 If that is the position, I can rewrite this like this, 784 00:54:05,790 --> 00:54:16,490 because theta, of course, is given by omega t prime. 785 00:54:16,490 --> 00:54:20,770 It's rotating, right? 786 00:54:20,770 --> 00:54:22,160 What is the acceleration? 787 00:54:22,160 --> 00:54:23,645 I differentiate this twice. 788 00:54:27,490 --> 00:54:29,073 So here is the acceleration. 789 00:54:31,680 --> 00:54:35,200 So if we consider the two components, 790 00:54:35,200 --> 00:54:43,540 what I see is that this charge has a component in the y 791 00:54:43,540 --> 00:54:46,590 direction and in the z direction, like this. 792 00:54:46,590 --> 00:54:52,650 And so it will, at every instant of time, radiate. 793 00:54:52,650 --> 00:54:58,390 I can decompose the radiation, using this formula, 794 00:54:58,390 --> 00:55:01,360 into the two components. 795 00:55:01,360 --> 00:55:06,770 And what we see is that the two components are out of phase 796 00:55:06,770 --> 00:55:09,465 from each other by 90 degrees. 797 00:55:13,380 --> 00:55:17,350 So for example, if I am straight ahead looking at it, 798 00:55:17,350 --> 00:55:23,210 I will see a charge which is accelerating up and down, 799 00:55:23,210 --> 00:55:26,480 like that, and out of phase, like this. 800 00:55:26,480 --> 00:55:31,770 So it will radiate straightforward a component 801 00:55:31,770 --> 00:55:37,320 like this, oscillating, and out of phase by 90, like that. 802 00:55:37,320 --> 00:55:38,930 And what is that? 803 00:55:38,930 --> 00:55:39,430 You know. 804 00:55:39,430 --> 00:55:41,465 That's called circularly polarized light. 805 00:55:44,630 --> 00:55:46,190 There are many ways to look at it. 806 00:55:46,190 --> 00:55:48,340 You can either look at the components, 807 00:55:48,340 --> 00:55:51,770 or you could look at the rotation 808 00:55:51,770 --> 00:55:55,290 of the electric vector. 809 00:55:55,290 --> 00:55:57,160 At the moment, I am looking at this 810 00:55:57,160 --> 00:55:59,240 from the point of view of components. 811 00:55:59,240 --> 00:56:03,680 So if I look at this, straight at it, what I will see, 812 00:56:03,680 --> 00:56:10,260 an electromagnetic wave with the electric vector 813 00:56:10,260 --> 00:56:13,360 polarized vertically, and another one 814 00:56:13,360 --> 00:56:18,850 polarized horizontally, out of phase by 90 degrees. 815 00:56:18,850 --> 00:56:21,550 And so, at any location in space, 816 00:56:21,550 --> 00:56:25,510 I'll see an electric vector which is rotating like this. 817 00:56:25,510 --> 00:56:28,460 Of course, there's a magnetic vector perpendicular to it. 818 00:56:28,460 --> 00:56:32,150 And that we call, circularly polarized light. 819 00:56:32,150 --> 00:56:38,630 So this will radiate in that direction circularly polarized 820 00:56:38,630 --> 00:56:39,840 light. 821 00:56:39,840 --> 00:56:42,610 If I look from above, what do I see? 822 00:56:42,610 --> 00:56:45,410 The charge is going like this, so I just 823 00:56:45,410 --> 00:56:48,250 see this component from above-- you 824 00:56:48,250 --> 00:56:51,470 can look at the two components-- like that. 825 00:56:51,470 --> 00:56:55,470 So straight up, I'll see linearly polarized light. 826 00:56:55,470 --> 00:56:59,460 If I look from the side, I see it's going up and down. 827 00:56:59,460 --> 00:57:04,090 I will see also linearly polarized, 828 00:57:04,090 --> 00:57:06,310 but in a different direction. 829 00:57:06,310 --> 00:57:10,050 So a charge like this, as before, 830 00:57:10,050 --> 00:57:12,400 will radiate in all directions. 831 00:57:12,400 --> 00:57:16,640 But the extra complication now is, in different directions, 832 00:57:16,640 --> 00:57:18,830 it'll be different polarization. 833 00:57:18,830 --> 00:57:20,990 For example, if I'm looking at this from here, 834 00:57:20,990 --> 00:57:24,180 this crazy angle, what I will see 835 00:57:24,180 --> 00:57:28,380 is I will see a component from the acceleration like this, 836 00:57:28,380 --> 00:57:31,930 and from that, but they have different amplitude. 837 00:57:31,930 --> 00:57:36,179 And so I'll get the two polarizations which 838 00:57:36,179 --> 00:57:37,720 are different amplitudes, corresponds 839 00:57:37,720 --> 00:57:39,770 to elliptically polarized light. 840 00:57:39,770 --> 00:57:43,100 So in this direction, I'll see an elliptically polarized 841 00:57:43,100 --> 00:57:44,700 light. 842 00:57:44,700 --> 00:57:48,680 And using this formula any place in space, 843 00:57:48,680 --> 00:57:52,430 you can calculate the electric and magnetic field, 844 00:57:52,430 --> 00:57:56,320 just doing the same as we did before, but take 845 00:57:56,320 --> 00:58:02,530 the full vector description of the perpendicular 846 00:58:02,530 --> 00:58:05,820 component of the acceleration of the charge. 847 00:58:05,820 --> 00:58:09,010 So let me stop there and just sort of summarize. 848 00:58:09,010 --> 00:58:12,230 In principle, we've done just very little. 849 00:58:12,230 --> 00:58:16,680 What we showed today is, in general, 850 00:58:16,680 --> 00:58:21,480 if I have a charge which moving with some velocity, 851 00:58:21,480 --> 00:58:23,440 some acceleration, et cetera, it'll 852 00:58:23,440 --> 00:58:26,150 be surrounded by a very complicated field. 853 00:58:26,150 --> 00:58:28,470 There will be the Coulomb field, the Biot-Savart field, 854 00:58:28,470 --> 00:58:31,120 the radiated field, et cetera. 855 00:58:31,120 --> 00:58:34,139 If you need to solve the complete thing properly, 856 00:58:34,139 --> 00:58:35,180 you will need a computer. 857 00:58:35,180 --> 00:58:36,730 You can't do it. 858 00:58:36,730 --> 00:58:42,540 But if you are only interested in the radiated field which 859 00:58:42,540 --> 00:58:47,240 is far away from the accelerated charge, 860 00:58:47,240 --> 00:58:50,570 it turns out the situation is sufficiently simple, 861 00:58:50,570 --> 00:58:54,310 you can do it almost on the back of an envelope. 862 00:58:54,310 --> 00:58:57,620 All you need to know, if you are interested in the field 863 00:58:57,620 --> 00:59:01,460 in any location, is you ask yourself, 864 00:59:01,460 --> 00:59:07,950 at an earlier time, what was the charge doing? 865 00:59:07,950 --> 00:59:13,680 And by earlier, I mean at time that light 866 00:59:13,680 --> 00:59:17,650 had time to come from the charge to me. 867 00:59:17,650 --> 00:59:22,130 So I take that distance, R/C, and at that earlier time, 868 00:59:22,130 --> 00:59:25,030 I need to know what the charge was doing. 869 00:59:25,030 --> 00:59:28,870 I ask myself, in which direction it was accelerating? 870 00:59:28,870 --> 00:59:31,590 I take the magnitude of acceleration, 871 00:59:31,590 --> 00:59:35,740 which is perpendicular to the direction in which I'm looking, 872 00:59:35,740 --> 00:59:40,710 and I calculate that and multiply by some constants 873 00:59:40,710 --> 00:59:43,640 that we've done over and over again, 874 00:59:43,640 --> 00:59:45,960 and that will tell me what the electric field is. 875 00:59:45,960 --> 00:59:47,900 From that, I can get the magnetic field, 876 00:59:47,900 --> 00:59:49,360 and I get the full radiation. 877 00:59:49,360 --> 00:59:51,210 Thank you.