1 00:00:00,060 --> 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 --> 00:00:17,226 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,226 --> 00:00:17,851 at ocw.mit.edu. 8 00:00:46,750 --> 00:00:48,310 PROFESSOR: So welcome back. 9 00:00:48,310 --> 00:00:54,280 And today, I'm going to do some examples of what 10 00:00:54,280 --> 00:00:58,790 happens when you have more than one charge radiating. 11 00:00:58,790 --> 00:01:04,129 This is a very important example, 12 00:01:04,129 --> 00:01:11,720 because whenever you have many charges radiating, if it's 13 00:01:11,720 --> 00:01:16,720 coherent, you get the phenomenon of interference, 14 00:01:16,720 --> 00:01:20,870 and it plays a very important role in many fields of physics, 15 00:01:20,870 --> 00:01:23,250 in particular, in optics. 16 00:01:23,250 --> 00:01:26,330 You may have heard terms like interference patterns, 17 00:01:26,330 --> 00:01:28,230 diffraction, et cetera. 18 00:01:28,230 --> 00:01:36,600 All that is to do with when you have more than one source 19 00:01:36,600 --> 00:01:38,130 radiating. 20 00:01:38,130 --> 00:01:42,990 So the example I'm going to consider is the following. 21 00:01:42,990 --> 00:01:49,090 Suppose you have two charges, both q. 22 00:01:49,090 --> 00:01:50,810 Each one is oscillating. 23 00:01:55,020 --> 00:01:59,190 It's oscillating with an angular frequency 2 pi over lambda. 24 00:01:59,190 --> 00:02:02,340 This tells you the position of the charge 25 00:02:02,340 --> 00:02:04,010 as a function of time. 26 00:02:04,010 --> 00:02:07,380 So it's centered to distance d from here 27 00:02:07,380 --> 00:02:11,380 and oscillating within amplitude a 28 00:02:11,380 --> 00:02:15,730 as a cosine 2 pi over lambda t. 29 00:02:15,730 --> 00:02:18,750 Well, lambda is, of course, the wavelength 30 00:02:18,750 --> 00:02:21,330 of the radiated electromagnetic wave. 31 00:02:29,630 --> 00:02:33,490 So this one is oscillating up and down like that, 32 00:02:33,490 --> 00:02:37,990 and there is another one over here oscillating just for fun 33 00:02:37,990 --> 00:02:39,940 in the other direction. 34 00:02:39,940 --> 00:02:44,350 This one is oscillating in along the x-axis, 35 00:02:44,350 --> 00:02:49,530 and this one is oscillating along the y-axis. 36 00:02:49,530 --> 00:02:52,600 And here is the formula, which tells you 37 00:02:52,600 --> 00:02:57,970 what is the position of this one at the given instant of time. 38 00:02:57,970 --> 00:03:00,940 They're both given in terms of the same time, 39 00:03:00,940 --> 00:03:02,830 with the same time. 40 00:03:02,830 --> 00:03:03,850 t equals 0. 41 00:03:03,850 --> 00:03:08,290 Therefore, these are two coherent sources. 42 00:03:08,290 --> 00:03:12,850 The question is, what will be the resultant field 43 00:03:12,850 --> 00:03:14,800 anywhere in space? 44 00:03:14,800 --> 00:03:20,640 At some point we want to know what is the resultant field. 45 00:03:20,640 --> 00:03:25,770 Now, each one of these-- you know what it does. 46 00:03:25,770 --> 00:03:32,230 It radiates a spherical wave with a certain angular 47 00:03:32,230 --> 00:03:34,060 distribution. 48 00:03:34,060 --> 00:03:34,560 Alright? 49 00:03:37,280 --> 00:03:41,400 And the waves overlap in space. 50 00:03:41,400 --> 00:03:45,210 Now because the electromagnetic waves 51 00:03:45,210 --> 00:03:48,920 are solutions of the electromagnetic wave 52 00:03:48,920 --> 00:03:55,890 equation, which is a linear equation, if we have 53 00:03:55,890 --> 00:04:00,010 a coherent source of radiation at any point, 54 00:04:00,010 --> 00:04:06,300 we simply add vectorially the electromagnetic waves. 55 00:04:06,300 --> 00:04:09,610 The electric fields you add vectorially 56 00:04:09,610 --> 00:04:11,145 all the magnetic field. 57 00:04:14,320 --> 00:04:25,910 Now in principle, we should go-- to first principles. 58 00:04:25,910 --> 00:04:27,920 Consider Maxwell's equations. 59 00:04:27,920 --> 00:04:32,190 Consider what happens when a charge is oscillating. 60 00:04:36,740 --> 00:04:41,110 And in fact, consider both charges, 61 00:04:41,110 --> 00:04:43,260 what they're doing at any instant of time, 62 00:04:43,260 --> 00:04:47,380 and solve in all of space Maxwell's equation 63 00:04:47,380 --> 00:04:50,790 to see what the electric field is everywhere. 64 00:04:50,790 --> 00:04:56,400 We build on our experience, and what we will do 65 00:04:56,400 --> 00:05:02,170 is we will assume that we've solved the problem of what 66 00:05:02,170 --> 00:05:06,710 happens when this single charge is accelerating. 67 00:05:06,710 --> 00:05:09,100 We've done this several times in the [? PAR sensor ?]. 68 00:05:09,100 --> 00:05:11,940 Professor Walter Lewin has done it in the course. 69 00:05:11,940 --> 00:05:18,640 Whenever a charge is accelerating, it radiates. 70 00:05:18,640 --> 00:05:21,210 And this is the formula, which tells 71 00:05:21,210 --> 00:05:26,370 you what is the electric field from the given charge, which 72 00:05:26,370 --> 00:05:31,640 is accelerating at some position, r and t. 73 00:05:31,640 --> 00:05:36,090 I'm just reminding you that what we find 74 00:05:36,090 --> 00:05:44,120 is that the electric field at any position at some time t 75 00:05:44,120 --> 00:05:50,870 is given by, or related to, the perpendicular 76 00:05:50,870 --> 00:05:56,050 component of the acceleration of the charge. 77 00:05:56,050 --> 00:06:01,650 It's perpendicular to the direction of propagation, 78 00:06:01,650 --> 00:06:09,010 but at a point t in space, the electric field there 79 00:06:09,010 --> 00:06:15,420 is related to the acceleration at an earlier time. 80 00:06:15,420 --> 00:06:20,320 At the time, t prime, which is earlier than the time when 81 00:06:20,320 --> 00:06:27,390 we are considering by the amount r/c. 82 00:06:27,390 --> 00:06:31,450 All that this does-- it reflects the fact 83 00:06:31,450 --> 00:06:34,710 that if I have some with an accelerated charge, 84 00:06:34,710 --> 00:06:37,610 at that instant, the electromagnetic wave starts 85 00:06:37,610 --> 00:06:46,780 to propagate, and it takes a time r/c to reach a distance r. 86 00:06:46,780 --> 00:06:50,250 That's all this is telling us. 87 00:06:50,250 --> 00:06:53,680 The other thing this is telling us-- the wave front 88 00:06:53,680 --> 00:06:56,260 is a sphere. 89 00:06:56,260 --> 00:07:01,470 The amplitude of the electric field drops off like 1/r. 90 00:07:01,470 --> 00:07:05,920 And it is not uniform in all direction, 91 00:07:05,920 --> 00:07:08,480 as we've seen in the past. 92 00:07:08,480 --> 00:07:09,450 Right? 93 00:07:09,450 --> 00:07:12,450 For example, if I take this charge 94 00:07:12,450 --> 00:07:18,350 as it's oscillating up and down, the maximum amplitude radiated 95 00:07:18,350 --> 00:07:22,400 is in this plane where we are looking, 96 00:07:22,400 --> 00:07:24,710 where the perpendicular component 97 00:07:24,710 --> 00:07:28,190 of the acceleration of this is the maximum. 98 00:07:28,190 --> 00:07:34,000 And its minimum upwards is 0, because this acceleration 99 00:07:34,000 --> 00:07:37,550 has no component of x perpendicular 100 00:07:37,550 --> 00:07:41,000 component to this direction. 101 00:07:41,000 --> 00:07:44,470 And similarly, for this. 102 00:07:44,470 --> 00:07:46,900 The other thing which I wish to point out 103 00:07:46,900 --> 00:07:53,730 is that the polarization of the electromagnetic wave, 104 00:07:53,730 --> 00:07:57,640 the electric vector, is parallel to the perpendicular 105 00:07:57,640 --> 00:07:59,560 component of a. 106 00:07:59,560 --> 00:08:03,630 So from this, the electric vector, 107 00:08:03,630 --> 00:08:06,490 when it gets to here or here, will 108 00:08:06,490 --> 00:08:10,630 be in the direction of x, while from this one, 109 00:08:10,630 --> 00:08:14,540 it'll polarized in the direction y. 110 00:08:14,540 --> 00:08:16,540 The other thing which I want to stress, 111 00:08:16,540 --> 00:08:19,830 that we will in the other examples-- 112 00:08:19,830 --> 00:08:22,520 always consider a situation where 113 00:08:22,520 --> 00:08:27,670 the point where we're looking at the electromagnetic field 114 00:08:27,670 --> 00:08:32,500 is far away from where the charges are. 115 00:08:32,500 --> 00:08:36,720 The formula, which I just showed you here, 116 00:08:36,720 --> 00:08:39,809 applies only in the far field. 117 00:08:39,809 --> 00:08:42,840 Nearby, the electric field and the magnetic field 118 00:08:42,840 --> 00:08:44,580 are very complicated. 119 00:08:44,580 --> 00:08:47,900 They can be calculated, but they cannot be summarized in that 120 00:08:47,900 --> 00:08:48,980 simple form. 121 00:08:48,980 --> 00:08:54,030 But if I'm far away from the accelerating charge then, 122 00:08:54,030 --> 00:08:58,380 it is possible to write it in this simple form. 123 00:08:58,380 --> 00:09:05,330 So our discussion is only valid if the wave length radiated 124 00:09:05,330 --> 00:09:09,060 is much smaller than the distance 125 00:09:09,060 --> 00:09:10,365 from where we're considering. 126 00:09:13,590 --> 00:09:18,700 This distance is also much smaller than that. 127 00:09:18,700 --> 00:09:21,950 And this is to allow us to approximate sine theta 128 00:09:21,950 --> 00:09:23,710 by theta, et cetera. 129 00:09:23,710 --> 00:09:25,730 And that's the case, alright? 130 00:09:28,380 --> 00:09:29,760 OK. 131 00:09:29,760 --> 00:09:34,680 With that, we can now immediately solve the problem. 132 00:09:34,680 --> 00:09:40,040 Simply consider the acceleration of each charge 133 00:09:40,040 --> 00:09:43,630 from our knowledge of where the charge is. 134 00:09:43,630 --> 00:09:46,140 We can differentiate it twice, and we 135 00:09:46,140 --> 00:09:47,570 can get the acceleration. 136 00:09:47,570 --> 00:09:52,785 So we know that the acceleration of the first charge, 137 00:09:52,785 --> 00:09:55,360 which is oscillating along the x-axis, 138 00:09:55,360 --> 00:09:58,290 is given by minus a omega squared 139 00:09:58,290 --> 00:10:01,400 cosine omega in the x-direction. 140 00:10:01,400 --> 00:10:03,480 I want to emphasize, because this 141 00:10:03,480 --> 00:10:09,760 is in the same direction as the perpendicular component of a1. 142 00:10:14,240 --> 00:10:15,820 And, I'm sorry. 143 00:10:15,820 --> 00:10:17,110 I misspoke. 144 00:10:17,110 --> 00:10:18,950 The answer-- what I said is correct 145 00:10:18,950 --> 00:10:20,120 but for the wrong reason. 146 00:10:20,120 --> 00:10:25,760 In this case, by definition, the acceleration of the first one 147 00:10:25,760 --> 00:10:31,450 is this because the displacement of the first one 148 00:10:31,450 --> 00:10:33,260 is in the x-direction. 149 00:10:33,260 --> 00:10:36,310 Similarly, the acceleration on the second one 150 00:10:36,310 --> 00:10:39,390 is in the y-direction, because the displacement 151 00:10:39,390 --> 00:10:41,800 is in the y-direction. 152 00:10:41,800 --> 00:10:42,370 Alright? 153 00:10:42,370 --> 00:10:44,970 So these are the two accelerations. 154 00:10:44,970 --> 00:10:49,170 This is just some-- occasionally I'll use omega, 155 00:10:49,170 --> 00:10:52,580 occasionally lambda just to remind you of that. 156 00:10:52,580 --> 00:10:53,080 OK. 157 00:10:53,080 --> 00:10:59,736 So all I now have to do for each of the charges 158 00:10:59,736 --> 00:11:05,170 is calculate this quantity at the position 159 00:11:05,170 --> 00:11:08,720 where I want to calculate the electric and the magnetic 160 00:11:08,720 --> 00:11:10,040 field. 161 00:11:10,040 --> 00:11:13,040 So, first of all in this problem, 162 00:11:13,040 --> 00:11:15,170 we're asked to do it in two places. 163 00:11:15,170 --> 00:11:18,120 We're asked to do it at position p1 164 00:11:18,120 --> 00:11:21,290 and at p2. p1 is along the z-axis, 165 00:11:21,290 --> 00:11:24,450 and p2 is off the z-axis. 166 00:11:24,450 --> 00:11:27,570 First, let's do it for p1. 167 00:11:27,570 --> 00:11:33,740 So, the electric field, due to the first charge 168 00:11:33,740 --> 00:11:40,780 at position p1 at time t, is given by this formula 169 00:11:40,780 --> 00:11:49,610 if I insert in this for the distance r, which is L. OK? 170 00:11:49,610 --> 00:11:54,050 And I put in the acceleration of this one 171 00:11:54,050 --> 00:11:59,400 and the perpendicular component of the acceleration 172 00:11:59,400 --> 00:12:04,870 perpendicular to the line drawing my two charges. 173 00:12:04,870 --> 00:12:08,300 And the position p1 is-- in fact, 174 00:12:08,300 --> 00:12:10,250 that line is along the z-axis. 175 00:12:10,250 --> 00:12:13,790 So the perpendicular direction is, in fact, 176 00:12:13,790 --> 00:12:15,650 in the x-direction. 177 00:12:15,650 --> 00:12:16,150 OK. 178 00:12:16,150 --> 00:12:21,670 So I don't even have to take the-- the actual acceleration 179 00:12:21,670 --> 00:12:23,870 is the perpendicular component. 180 00:12:23,870 --> 00:12:24,580 OK. 181 00:12:24,580 --> 00:12:30,570 And I have to calculate it at a time which 182 00:12:30,570 --> 00:12:40,230 is t minus the distance p1 is from in my charges, which is L. 183 00:12:40,230 --> 00:12:43,730 So it's minus omega L over c. 184 00:12:43,730 --> 00:12:47,660 I calculate t prime in here. 185 00:12:47,660 --> 00:12:51,620 Notice, I have not tried to calculate 186 00:12:51,620 --> 00:12:56,110 the distance between the charge and p1 exactly, 187 00:12:56,110 --> 00:13:00,060 because I told you that the distance d is very much 188 00:13:00,060 --> 00:13:05,900 smaller than L. And so the overall distance is negligently 189 00:13:05,900 --> 00:13:11,250 different from being just L. So that's 190 00:13:11,250 --> 00:13:13,540 the electric field due to the first one. 191 00:13:13,540 --> 00:13:17,250 Similarly, I can do the same for the second one. 192 00:13:17,250 --> 00:13:19,150 Everything is the same. 193 00:13:19,150 --> 00:13:22,650 The only difference in that case is 194 00:13:22,650 --> 00:13:27,720 that now the electric vector-- the acceleration 195 00:13:27,720 --> 00:13:32,470 is along the y-axis and therefore, the electric field 196 00:13:32,470 --> 00:13:35,710 is polarized along the y-direction. 197 00:13:35,710 --> 00:13:42,290 So at the point p1, the two electric fields are the same. 198 00:13:42,290 --> 00:13:44,830 They're both oscillating in phase 199 00:13:44,830 --> 00:13:47,260 with the same frequency, et cetera. 200 00:13:47,260 --> 00:13:50,270 But this one is pointing in the x-direction. 201 00:13:50,270 --> 00:13:52,940 This in the y-direction. 202 00:13:52,940 --> 00:13:56,870 Now, as I mentioned at the beginning, if at any point 203 00:13:56,870 --> 00:14:01,450 you have electric field due to two sources, 204 00:14:01,450 --> 00:14:05,780 they simply add because the system is a linear system. 205 00:14:05,780 --> 00:14:08,800 So we simply have to add the electric fields. 206 00:14:08,800 --> 00:14:11,060 Electric fields are vectors, therefore 207 00:14:11,060 --> 00:14:16,390 we don't add them algebraic as scalars, 208 00:14:16,390 --> 00:14:19,750 but we have to add them vectorially. 209 00:14:19,750 --> 00:14:24,220 And so, we take this vector and that vector and add it. 210 00:14:24,220 --> 00:14:28,910 And that gives us the total electric field at point p1. 211 00:14:28,910 --> 00:14:32,390 And simply from here, I get d0 like that, 212 00:14:32,390 --> 00:14:38,840 which I can rewrite in terms of a unit vector. 213 00:14:38,840 --> 00:14:39,750 Alright? 214 00:14:39,750 --> 00:14:45,980 This is a unit vector, which is at 45 degrees. 215 00:14:45,980 --> 00:14:51,680 In other words, if that's the z-direction x and y, 216 00:14:51,680 --> 00:14:56,180 it's at 45 degrees to both the x and y-axes. 217 00:14:56,180 --> 00:14:57,830 That's the unit vector. 218 00:14:57,830 --> 00:14:59,780 This is the amplitude of it. 219 00:14:59,780 --> 00:15:05,280 And this is d-- tell us what is the oscillating frequency 220 00:15:05,280 --> 00:15:08,730 and what is the phase. 221 00:15:08,730 --> 00:15:13,670 Notice, although each one of these sources 222 00:15:13,670 --> 00:15:19,430 radiated there an electric field of magnitude E0, 223 00:15:19,430 --> 00:15:25,080 when I add them, I did not get to 2E0, because as I mentioned, 224 00:15:25,080 --> 00:15:26,980 electric fields are vectors, and we 225 00:15:26,980 --> 00:15:28,550 have to add them vectorially. 226 00:15:28,550 --> 00:15:31,770 And that's why you get here the root 2. 227 00:15:31,770 --> 00:15:36,940 So this is the first example, which I've done. 228 00:15:36,940 --> 00:15:40,870 Now I want to take the same geometry 229 00:15:40,870 --> 00:15:46,050 but calculate what is the electric field. 230 00:15:46,050 --> 00:15:49,170 What we've just done, we've calculated here. 231 00:15:49,170 --> 00:15:51,750 Now I want to do the same situation-- 232 00:15:51,750 --> 00:15:57,530 two sources oscillating coherently. 233 00:15:57,530 --> 00:16:01,240 But I want to look at what is the electric field at the point 234 00:16:01,240 --> 00:16:02,560 p2. 235 00:16:02,560 --> 00:16:09,620 And I've taken p2 to be in that the xz-plane, 236 00:16:09,620 --> 00:16:14,986 but in a direction that's an angle lambda over a to d. 237 00:16:14,986 --> 00:16:16,900 d is this distance. 238 00:16:16,900 --> 00:16:22,490 Lambda is the wavelength of the radiated electromagnetic wave. 239 00:16:22,490 --> 00:16:25,760 And I've taken this crazy number simply 240 00:16:25,760 --> 00:16:29,690 to make the arithmetic come out easier at the end. 241 00:16:29,690 --> 00:16:34,020 And so we now want to calculate the electric field here. 242 00:16:34,020 --> 00:16:37,320 And we do exactly the same except we go a little faster. 243 00:16:37,320 --> 00:16:41,330 What I will do is I will calculate the electric field 244 00:16:41,330 --> 00:16:42,310 here. 245 00:16:42,310 --> 00:16:45,610 Due to this charge oscillating, I'll 246 00:16:45,610 --> 00:16:47,990 then calculate the electric field here. 247 00:16:47,990 --> 00:16:51,120 Due to this charge, this one is oscillating like that. 248 00:16:51,120 --> 00:16:54,210 I'm reminding you this one is oscillating like that. 249 00:16:54,210 --> 00:16:57,740 And we will vectorially add the two. 250 00:16:57,740 --> 00:17:02,790 Now, from the point of view of the amplitude, 251 00:17:02,790 --> 00:17:10,630 the difference in distance between this and here or this 252 00:17:10,630 --> 00:17:15,940 and here or this and here is insignificant. 253 00:17:15,940 --> 00:17:19,270 And so I have ignored that in the previous calculations, 254 00:17:19,270 --> 00:17:21,599 and I'll do the same here. 255 00:17:21,599 --> 00:17:28,470 But when you add to the two, we have 256 00:17:28,470 --> 00:17:30,990 to worry about the relative phase 257 00:17:30,990 --> 00:17:34,050 of the alternating the electric fields. 258 00:17:34,050 --> 00:17:38,370 And the difference of this distance compared 259 00:17:38,370 --> 00:17:44,360 to that distance is no longer insignificant 260 00:17:44,360 --> 00:17:50,700 when it comes to the relative phase of the two radiations. 261 00:17:50,700 --> 00:17:58,490 So, in this formula when I'm calculating the electric field, 262 00:17:58,490 --> 00:18:03,630 this r-- it doesn't matter whether I 263 00:18:03,630 --> 00:18:07,390 call this L or L plus a little bit or less. 264 00:18:07,390 --> 00:18:09,730 This doesn't change very much. 265 00:18:09,730 --> 00:18:19,330 But in calculating the phase of the radiation, 266 00:18:19,330 --> 00:18:21,990 I cannot ignore that. 267 00:18:21,990 --> 00:18:25,350 In the previous case, the phase was the same 268 00:18:25,350 --> 00:18:28,890 because this distance and that distance 269 00:18:28,890 --> 00:18:32,050 are exactly the same by symmetry. 270 00:18:32,050 --> 00:18:37,660 For this position, that distance and that distance 271 00:18:37,660 --> 00:18:39,380 are not the same. 272 00:18:39,380 --> 00:18:45,670 And so I what I will do is I'll use as reference that distance, 273 00:18:45,670 --> 00:18:48,420 and when I'm calculating this distance, 274 00:18:48,420 --> 00:18:52,870 I will calculate by how much I have to subtract from here 275 00:18:52,870 --> 00:18:57,690 to end up with this length and for this one, by how much 276 00:18:57,690 --> 00:19:01,750 I've add to it to get to here. 277 00:19:01,750 --> 00:19:05,590 Pay attention to that when I'm doing that in a second. 278 00:19:05,590 --> 00:19:11,410 And the rest is very straightforward and similar 279 00:19:11,410 --> 00:19:12,740 to what I've just done. 280 00:19:17,960 --> 00:19:21,650 So now, as I say, I want to calculate the electric field 281 00:19:21,650 --> 00:19:25,510 at the point p2, which is this. 282 00:19:25,510 --> 00:19:28,350 This is the x-coordinate, y, and z-coordinate. 283 00:19:28,350 --> 00:19:31,950 And I pointed it out to you a second ago. 284 00:19:31,950 --> 00:19:32,450 Alright. 285 00:19:36,050 --> 00:19:41,210 So I don't have to rewrite all those q's and omegas, 286 00:19:41,210 --> 00:19:44,290 et cetera, by analogy, what we've 287 00:19:44,290 --> 00:19:52,290 done-- the electric field due to the first charge at position p2 288 00:19:52,290 --> 00:19:55,845 will be that the amplitude that we've got 289 00:19:55,845 --> 00:19:59,530 is the same for the previous case of P1. 290 00:19:59,530 --> 00:20:02,340 So it's-- E0 is the amplitude. 291 00:20:02,340 --> 00:20:07,800 It's pointing in the x-direction as before. 292 00:20:07,800 --> 00:20:18,780 And we know that the description of the electric field 293 00:20:18,780 --> 00:20:28,745 will be given by cosine omega t1 prime where t is-- so far 294 00:20:28,745 --> 00:20:30,660 I've always talked about t prime, 295 00:20:30,660 --> 00:20:34,740 because I only had one when I was considering just one 296 00:20:34,740 --> 00:20:35,490 charge. 297 00:20:35,490 --> 00:20:42,680 But here the t1 is different to t2. 298 00:20:42,680 --> 00:20:51,920 So quickly, this term comes from the perpendicular component 299 00:20:51,920 --> 00:20:55,630 of a right as before. 300 00:20:55,630 --> 00:21:00,770 And t1 prime is equal to the time 301 00:21:00,770 --> 00:21:11,120 when I'm looking at the electric field minus the time it takes 302 00:21:11,120 --> 00:21:17,030 for this signal from the charge to get to the point 303 00:21:17,030 --> 00:21:20,850 where I'm looking at the electric field to point p2. 304 00:21:20,850 --> 00:21:26,320 So, it will be-- as I told you a second ago, 305 00:21:26,320 --> 00:21:29,440 I'll take L as the reference. 306 00:21:29,440 --> 00:21:35,940 And I'm subtracting from it that distance d sine lambda over 8d. 307 00:21:35,940 --> 00:21:38,380 For a second, let me go back to that picture 308 00:21:38,380 --> 00:21:42,080 because you can easily get confused. 309 00:21:42,080 --> 00:21:44,790 So I'm coming back here. 310 00:21:44,790 --> 00:21:53,160 What we're trying to calculate-- the distance from here to here. 311 00:21:53,160 --> 00:21:58,400 So I'm taking this distance, which is L, 312 00:21:58,400 --> 00:22:04,680 and I'm subtracting from it this distance, which 313 00:22:04,680 --> 00:22:11,500 is d sine this angle here, which is the same as this angle. 314 00:22:11,500 --> 00:22:15,150 So that's what I'm calculating there. 315 00:22:15,150 --> 00:22:21,840 So this is L minus d sine lambda over 8d divided by c. 316 00:22:21,840 --> 00:22:26,570 Then the electric field is then immediately followed 317 00:22:26,570 --> 00:22:33,540 from this d0 in the x-direction, cosine omega t minus this. 318 00:22:33,540 --> 00:22:37,990 And if you assume that this angle is small, 319 00:22:37,990 --> 00:22:43,470 and I told you that that's given in the problem, 320 00:22:43,470 --> 00:22:46,150 the sine of an angle is equal to an angle. 321 00:22:46,150 --> 00:22:48,030 Approximately, they're small. 322 00:22:48,030 --> 00:22:52,500 And if I multiply this out, I simply get an angle here. 323 00:22:52,500 --> 00:22:54,320 That's pi/4. 324 00:22:54,320 --> 00:22:59,680 So I find that the electric field at point p2 325 00:22:59,680 --> 00:23:03,620 is pointing, due to the first charge, is pointing 326 00:23:03,620 --> 00:23:05,630 in the x-direction, as you'd expect, 327 00:23:05,630 --> 00:23:08,350 because of the direction which the charge is moving 328 00:23:08,350 --> 00:23:14,085 times E0 times cosine omega t minus omega L over c 329 00:23:14,085 --> 00:23:20,120 plus a phase like that. 330 00:23:20,120 --> 00:23:22,320 How about from the other charge? 331 00:23:22,320 --> 00:23:25,800 If I take the other charge, everything 332 00:23:25,800 --> 00:23:30,720 will be the same except now this charge is oscillating 333 00:23:30,720 --> 00:23:34,670 in the y-direction, so the electric field will 334 00:23:34,670 --> 00:23:36,960 be in the y-direction. 335 00:23:36,960 --> 00:23:42,070 And this time, it's omega t2 prime 336 00:23:42,070 --> 00:23:46,890 where t2 will be very similar to t1 337 00:23:46,890 --> 00:23:52,180 but now the distance is greater by this amount. 338 00:23:52,180 --> 00:23:55,350 For a second, let's go back to this picture 339 00:23:55,350 --> 00:23:57,810 so you see what I'm talking about. 340 00:23:57,810 --> 00:24:02,360 If I calculate this distance, it's 341 00:24:02,360 --> 00:24:12,400 the same as that distance plus d sine this angle, which 342 00:24:12,400 --> 00:24:17,340 is this lambda over 8d, and I'm subtracting. 343 00:24:17,340 --> 00:24:20,390 And I want to emphasize that what 344 00:24:20,390 --> 00:24:23,740 I'm focusing on here, what is important, 345 00:24:23,740 --> 00:24:28,640 is the difference between this and that 346 00:24:28,640 --> 00:24:31,040 and not the absolute value. 347 00:24:31,040 --> 00:24:35,640 So when I'm approximating-- if you 348 00:24:35,640 --> 00:24:37,160 do this calculation for yourself, 349 00:24:37,160 --> 00:24:39,540 you'll see I'm doing a slight approximation 350 00:24:39,540 --> 00:24:40,750 to calculate that. 351 00:24:40,750 --> 00:24:44,530 But this and this is the same quantity. 352 00:24:44,530 --> 00:24:47,610 But the important thing is the difference of this phase 353 00:24:47,610 --> 00:24:49,320 here is plus pi/4. 354 00:24:49,320 --> 00:24:52,330 Here is minus pi/4, because here it's 355 00:24:52,330 --> 00:24:54,460 minus that little bit of a distance, 356 00:24:54,460 --> 00:24:58,960 and here it's plus a little bit of the distance. 357 00:24:58,960 --> 00:25:00,470 So we finished. 358 00:25:00,470 --> 00:25:05,820 The total electric field at p2 and time t 359 00:25:05,820 --> 00:25:10,700 is the sum of the electric field due to the first charge plus 360 00:25:10,700 --> 00:25:12,000 due to the second challenge. 361 00:25:15,170 --> 00:25:19,810 And as I mentioned a second ago, electric fields are vectors. 362 00:25:19,810 --> 00:25:23,230 Thus, we have to add these two vectorially. 363 00:25:23,230 --> 00:25:26,140 This is in the x-direction. 364 00:25:26,140 --> 00:25:28,350 This is in the y-direction. 365 00:25:28,350 --> 00:25:30,620 And so what we have at that point 366 00:25:30,620 --> 00:25:34,850 is two electric oscillating electric fields. 367 00:25:34,850 --> 00:25:39,430 They're oscillating coherently but out of phase. 368 00:25:39,430 --> 00:25:41,700 This is a plus pi/4 phase. 369 00:25:41,700 --> 00:25:44,600 This is a minus pi/4 phase. 370 00:25:44,600 --> 00:25:48,900 So the difference of phase between those two is pi/2. 371 00:25:48,900 --> 00:25:50,680 It's 90 degrees. 372 00:25:50,680 --> 00:25:55,860 So this is out of phase with, one with respect to the other, 373 00:25:55,860 --> 00:25:57,280 by 90 degrees. 374 00:25:57,280 --> 00:26:01,740 They're coherent, but 90 degrees out of phase. 375 00:26:01,740 --> 00:26:04,510 The magnitudes are the same, but this one 376 00:26:04,510 --> 00:26:07,830 is pointing in the x-direction and this in the y-direction, 377 00:26:07,830 --> 00:26:10,900 and you know what that corresponds to. 378 00:26:10,900 --> 00:26:15,480 If I have at the location two electric fields, one 379 00:26:15,480 --> 00:26:20,410 doing that and the other one doing this, 380 00:26:20,410 --> 00:26:27,960 if they're out of phase by 90 degrees, and I add the two, 381 00:26:27,960 --> 00:26:29,010 what do I get? 382 00:26:29,010 --> 00:26:34,010 I get a constant size of a radius, 383 00:26:34,010 --> 00:26:35,900 and the thing is rotating. 384 00:26:35,900 --> 00:26:42,100 That is the description of a rotating 385 00:26:42,100 --> 00:26:45,340 vector of magnitude E0. 386 00:26:45,340 --> 00:26:53,310 In the other case, at P1, there was not this phase difference, 387 00:26:53,310 --> 00:26:59,280 so the two were-- one was oscillating like this. 388 00:26:59,280 --> 00:27:02,140 The other one was oscillating like that. 389 00:27:02,140 --> 00:27:05,330 But they were oscillating at the same phase. 390 00:27:05,330 --> 00:27:07,230 So this is what they were doing. 391 00:27:07,230 --> 00:27:12,200 And if you add those up, clearly you get a line diagonally. 392 00:27:12,200 --> 00:27:17,700 And that's why before, the result was a linearly polarized 393 00:27:17,700 --> 00:27:21,990 electric field there at 45 degrees to the x and y-axis. 394 00:27:21,990 --> 00:27:28,740 In this case, these two are out of phase by 90 degrees. 395 00:27:28,740 --> 00:27:32,440 So when one is doing this, the other one 396 00:27:32,440 --> 00:27:33,790 is also but out of phase. 397 00:27:33,790 --> 00:27:35,420 And so when this one's at the maximum, 398 00:27:35,420 --> 00:27:38,670 this one's at the minimum, and so that's what's happening. 399 00:27:38,670 --> 00:27:39,810 So they're out of phase. 400 00:27:39,810 --> 00:27:42,030 And then if you add the two up, you 401 00:27:42,030 --> 00:27:43,710 get something doing this, which we 402 00:27:43,710 --> 00:27:46,000 called circularly polarized light. 403 00:27:46,000 --> 00:27:48,600 And that's why I chose that crazy angle, 404 00:27:48,600 --> 00:27:51,920 because it came out like this. 405 00:27:51,920 --> 00:27:53,910 So that's the end of that problem. 406 00:27:53,910 --> 00:27:57,840 I will now do another one, which to emphasize 407 00:27:57,840 --> 00:28:04,110 some technique-- a different technique of doing it. 408 00:28:04,110 --> 00:28:04,850 OK. 409 00:28:04,850 --> 00:28:06,170 Let's move. 410 00:28:06,170 --> 00:28:12,300 So the next problem is the following. 411 00:28:12,300 --> 00:28:20,670 We're again dealing with several charges oscillating coherently. 412 00:28:20,670 --> 00:28:22,790 Their cohering sources. 413 00:28:22,790 --> 00:28:26,440 And we're asking, what are the electric fields? 414 00:28:26,440 --> 00:28:27,920 Somewhere in space? 415 00:28:27,920 --> 00:28:29,880 And if we found the electric field, of course, 416 00:28:29,880 --> 00:28:33,990 we could always calculate the magnetic field 417 00:28:33,990 --> 00:28:37,420 at that location using Maxwell's equations or our knowledge 418 00:28:37,420 --> 00:28:39,960 of the relation between electromagnetic field 419 00:28:39,960 --> 00:28:43,790 in a progressive electromagnetic way. 420 00:28:43,790 --> 00:28:47,540 Now I'll take three charges. 421 00:28:47,540 --> 00:28:50,630 But that's not the hope. 422 00:28:50,630 --> 00:28:55,780 It's more different, but what this problem adds 423 00:28:55,780 --> 00:29:03,060 is I'll use it to show a technique that is often used, 424 00:29:03,060 --> 00:29:05,910 which helps in the solution of such problems. 425 00:29:05,910 --> 00:29:09,190 So the problem now is the following. 426 00:29:09,190 --> 00:29:11,010 I have three charges. 427 00:29:11,010 --> 00:29:13,540 Each of the same magnitude. 428 00:29:13,540 --> 00:29:17,482 They're located along the x-axis at position 0 429 00:29:17,482 --> 00:29:20,300 minus d and minus 2d. 430 00:29:20,300 --> 00:29:25,100 So these distances are the same-- each distance d. 431 00:29:25,100 --> 00:29:31,260 And what the problem is-- up to t equals 0, 432 00:29:31,260 --> 00:29:37,080 these three charges are displaced from equilibrium. 433 00:29:37,080 --> 00:29:39,000 I put y is 0. 434 00:29:39,000 --> 00:29:40,715 So, at the height, y0 here. 435 00:29:44,060 --> 00:29:49,740 Then, at t0-- don't ask me how. 436 00:29:49,740 --> 00:29:56,190 Magically, I get these three charges to start oscillating. 437 00:29:56,190 --> 00:30:00,720 Such that at time yt, the displacement 438 00:30:00,720 --> 00:30:04,600 is y0 cosine omega t. 439 00:30:04,600 --> 00:30:09,590 And they continue-- this continues like that forever. 440 00:30:09,590 --> 00:30:16,910 The question is, as a result of this oscillation of charges, 441 00:30:16,910 --> 00:30:23,180 what will be the electric field a long way away from here 442 00:30:23,180 --> 00:30:28,530 at position L along the x-axis, so we call that position p. 443 00:30:28,530 --> 00:30:35,720 So the problem is, calculate E at position p for all times. 444 00:30:35,720 --> 00:30:38,730 Once we've calculated, we could calculate the magnetic field, 445 00:30:38,730 --> 00:30:43,060 But to save you time--I mean, you could do it for yourself. 446 00:30:43,060 --> 00:30:45,220 You know if you calculate the electric field, 447 00:30:45,220 --> 00:30:47,390 there is a progressive wave over here. 448 00:30:47,390 --> 00:30:49,790 if E is like this, then b will be 449 00:30:49,790 --> 00:30:52,250 perpendicular to it of the amplitude which 450 00:30:52,250 --> 00:30:54,300 is just a dE over c. 451 00:30:54,300 --> 00:30:57,350 So that will be straightforward, so I'm not asking it. 452 00:31:00,030 --> 00:31:02,280 Now, let's just think for a second 453 00:31:02,280 --> 00:31:04,070 what goes on in this problem. 454 00:31:04,070 --> 00:31:07,820 Initially, the charges are stationary, 455 00:31:07,820 --> 00:31:10,290 so there will be a Coulomb field around. 456 00:31:10,290 --> 00:31:14,210 But we are specifically asked to ignore static fields. 457 00:31:14,210 --> 00:31:18,440 So in the last problem too, I ignored static fields. 458 00:31:18,440 --> 00:31:22,590 We were only considering the time-dependent fields. 459 00:31:25,410 --> 00:31:29,200 You can always have superimposed on the time-dependent field 460 00:31:29,200 --> 00:31:33,690 some static field that doesn't add anything. 461 00:31:33,690 --> 00:31:39,620 So what we have here is at time up to time t equals 0, 462 00:31:39,620 --> 00:31:41,030 the charges are here. 463 00:31:41,030 --> 00:31:43,450 So let's take one of the charges here. 464 00:31:43,450 --> 00:31:47,920 Then it starts moving. 465 00:31:47,920 --> 00:31:51,880 It will have an acceleration, which 466 00:31:51,880 --> 00:31:54,750 is perpendicular to the direction in which I'm 467 00:31:54,750 --> 00:31:58,990 interested the propagation of the electromagnetic wave. 468 00:31:58,990 --> 00:32:02,840 So it will certainly radiate in this direction. 469 00:32:02,840 --> 00:32:08,860 So this charge, which was oscillating, will radiate. 470 00:32:08,860 --> 00:32:12,470 Over here at the point p, the electric field 471 00:32:12,470 --> 00:32:14,690 will be initially 0. 472 00:32:14,690 --> 00:32:23,380 And it will continue being 0 until the electric field, which 473 00:32:23,380 --> 00:32:28,530 is generated here, propagates that distance. 474 00:32:28,530 --> 00:32:42,970 So, only after a time, L/c, will the electric vector get here? 475 00:32:42,970 --> 00:32:49,440 So up to the time L/c, there will be no electric field here. 476 00:32:53,670 --> 00:32:55,230 How about this charge? 477 00:32:55,230 --> 00:32:59,520 This charge is initially is displaced at y0. 478 00:32:59,520 --> 00:33:02,990 Then it starts oscillating the same. 479 00:33:02,990 --> 00:33:05,670 Initially, it's stationary. 480 00:33:05,670 --> 00:33:08,450 It'll produce a Coulomb field, which is a static field. 481 00:33:08,450 --> 00:33:10,330 We're not interested in it. 482 00:33:10,330 --> 00:33:16,010 But once it starts accelerating, it starts radiating. 483 00:33:16,010 --> 00:33:18,800 And that radiating progresses. 484 00:33:18,800 --> 00:33:23,410 So that will also produce a field over here. 485 00:33:23,410 --> 00:33:29,660 But since these two distances are different, 486 00:33:29,660 --> 00:33:32,700 the radiation from here, first of all, will get there a little 487 00:33:32,700 --> 00:33:33,970 later. 488 00:33:33,970 --> 00:33:39,240 But also, once it gets here, it will have a different phase 489 00:33:39,240 --> 00:33:43,780 because the radiation traveled a different distance. 490 00:33:43,780 --> 00:33:45,420 And same for the last one. 491 00:33:48,190 --> 00:33:50,520 So now, how do we do this? 492 00:33:54,030 --> 00:33:58,350 I almost sound like a broken record. 493 00:33:58,350 --> 00:34:02,240 As always, I can calculate the electric field 494 00:34:02,240 --> 00:34:06,940 from each charge, and it'll be given by this formula, which 495 00:34:06,940 --> 00:34:11,620 we've seen over and over again where t prime is 496 00:34:11,620 --> 00:34:17,469 the distance from the charge to the point p. 497 00:34:17,469 --> 00:34:21,880 Furthermore, I know the perpendicular acceleration 498 00:34:21,880 --> 00:34:24,719 and also the perpendicular component of it, 499 00:34:24,719 --> 00:34:27,560 because in this problem, the perpendicular component 500 00:34:27,560 --> 00:34:29,404 is the same as the actual acceleration. 501 00:34:32,429 --> 00:34:37,380 After time t equals 0, the acceleration 502 00:34:37,380 --> 00:34:42,630 of every one of these charges a1, a2, a3, is the same. 503 00:34:42,630 --> 00:34:44,179 It's the same direction. 504 00:34:44,179 --> 00:34:46,980 And it's given by that simply because we 505 00:34:46,980 --> 00:34:50,620 know what is the displacement y of t. 506 00:34:50,620 --> 00:34:54,389 If I differentiate it twice, I get the acceleration. 507 00:34:54,389 --> 00:34:58,460 If I plug this into here for each charge, 508 00:34:58,460 --> 00:35:02,400 I will know what is the electric fields from each charge. 509 00:35:02,400 --> 00:35:06,550 I can then add the electric fields from each charge, 510 00:35:06,550 --> 00:35:09,730 and I'll get the total electric field. 511 00:35:09,730 --> 00:35:12,380 You have to add them vectorially. 512 00:35:12,380 --> 00:35:18,900 So, once again, for t less than L/c, 513 00:35:18,900 --> 00:35:23,880 the electric field at position p will be 0. 514 00:35:23,880 --> 00:35:27,820 There was no earlier accelerated charge 515 00:35:27,820 --> 00:35:30,830 which radiated an electric field which 516 00:35:30,830 --> 00:35:35,110 got here before this time. 517 00:35:35,110 --> 00:35:37,330 So that's nice and easy. 518 00:35:37,330 --> 00:35:45,110 How about for a later period, a period between L/c or just 519 00:35:45,110 --> 00:35:47,010 after L/c. 520 00:35:47,010 --> 00:35:53,320 For t just after L/c, the acceleration 521 00:35:53,320 --> 00:36:01,870 of the charge at x equals 0 would produce an electric field 522 00:36:01,870 --> 00:36:07,720 which propagated and would have got to my point p. 523 00:36:07,720 --> 00:36:14,220 So between that time L/c and the time when 524 00:36:14,220 --> 00:36:20,420 the radiation from the second charge got to the point p, 525 00:36:20,420 --> 00:36:24,050 I will have an electric field but only 526 00:36:24,050 --> 00:36:28,470 from the first radiation due to the first charge. 527 00:36:28,470 --> 00:36:32,630 I can forget the second and third charge at minus d 528 00:36:32,630 --> 00:36:38,530 and a minus 2d but not the charge at x equals 0. 529 00:36:38,530 --> 00:36:40,980 So that will be I use this formula. 530 00:36:40,980 --> 00:36:41,830 I plug it in. 531 00:36:41,830 --> 00:36:43,880 I use the acceleration of a1. 532 00:36:43,880 --> 00:36:50,771 I put it in here, and I know that t prime is t minus r/c. 533 00:36:50,771 --> 00:36:55,300 r in this case is L. So this formula 534 00:36:55,300 --> 00:37:00,180 immediately tells me that's what it is and lets me save-- 535 00:37:00,180 --> 00:37:02,640 I don't want to write this over and over again. 536 00:37:02,640 --> 00:37:04,400 This-- I'll call that E0. 537 00:37:04,400 --> 00:37:08,310 So from the first charge, this is 538 00:37:08,310 --> 00:37:12,600 what the electric field-- it's an oscillating electric field-- 539 00:37:12,600 --> 00:37:15,440 what it looks like at position p. 540 00:37:15,440 --> 00:37:24,350 But I'm reminding you, this is only true for this time window 541 00:37:24,350 --> 00:37:27,410 when the radiation from the charge at q 542 00:37:27,410 --> 00:37:33,700 equals 0 has got to point p, but from the other one's not. 543 00:37:33,700 --> 00:37:36,880 How about from a little later? 544 00:37:36,880 --> 00:37:42,040 From the time L/c plus d over c-- in other words, 545 00:37:42,040 --> 00:37:49,390 now radiation over a distance L plus d 546 00:37:49,390 --> 00:37:52,570 has had time to reach my point p. 547 00:37:52,570 --> 00:38:00,190 So now, the second charge, the one that minused d, 548 00:38:00,190 --> 00:38:05,190 has had enough time to reach my point p. 549 00:38:05,190 --> 00:38:11,370 But if I limit to the window, this and that, in other words, 550 00:38:11,370 --> 00:38:16,950 there's still a difference of d/c in time between those. 551 00:38:16,950 --> 00:38:21,630 I will get at point p the radiation as 552 00:38:21,630 --> 00:38:23,970 from the first charge. 553 00:38:23,970 --> 00:38:27,010 That's exactly the same as this. 554 00:38:27,010 --> 00:38:28,830 That's obviously got there. 555 00:38:28,830 --> 00:38:32,850 But now from the second charge, it has got there. 556 00:38:32,850 --> 00:38:35,110 And they're very similar. 557 00:38:35,110 --> 00:38:41,800 They are both-- the charges are accelerating 558 00:38:41,800 --> 00:38:50,060 along y-direction in phase. 559 00:38:50,060 --> 00:38:56,150 So they have the same frequency, and the amplitude 560 00:38:56,150 --> 00:39:00,790 is going to be the same, because in this formula, 561 00:39:00,790 --> 00:39:05,790 this r is the total distance between the charge 562 00:39:05,790 --> 00:39:07,570 and the point p. 563 00:39:07,570 --> 00:39:11,210 Now, there is a tiny difference in that distance 564 00:39:11,210 --> 00:39:14,130 for the first and second charge. 565 00:39:14,130 --> 00:39:17,420 But if I take them here a tiny difference, 566 00:39:17,420 --> 00:39:19,465 this will not change very much. 567 00:39:19,465 --> 00:39:21,680 I'm ignoring that difference. 568 00:39:21,680 --> 00:39:24,790 So I'm calling both of them E0 and ignoring 569 00:39:24,790 --> 00:39:26,210 that tiny difference. 570 00:39:26,210 --> 00:39:31,140 But I cannot ignore the difference on the phase. 571 00:39:31,140 --> 00:39:36,920 So the t prime into two cases is different. 572 00:39:36,920 --> 00:39:40,580 In one case, it's t minus L/c. 573 00:39:40,580 --> 00:39:44,640 And the other case, it's minus L/c 574 00:39:44,640 --> 00:39:49,510 minus this little extra time where the radiation 575 00:39:49,510 --> 00:39:54,070 takes from the second to the first charge. 576 00:39:54,070 --> 00:39:57,140 And the third one, the radiation could 577 00:39:57,140 --> 00:39:59,360 not to reach the point p yet. 578 00:39:59,360 --> 00:40:04,805 So that is now the total electric field from the two. 579 00:40:08,400 --> 00:40:11,380 And finally-- and I have to do this. 580 00:40:11,380 --> 00:40:12,870 I haven't done the addition. 581 00:40:12,870 --> 00:40:20,030 And finally, if we take a time which is greater than this, 582 00:40:20,030 --> 00:40:26,140 then we are in a situation where the oscillation 583 00:40:26,140 --> 00:40:29,900 on the last charge has had enough time 584 00:40:29,900 --> 00:40:31,880 to get to the point p. 585 00:40:31,880 --> 00:40:37,650 Let me go back to this picture and just repeat this 586 00:40:37,650 --> 00:40:39,030 so that you don't get lost. 587 00:40:39,030 --> 00:40:45,480 So these charges have started to move all at the same time. 588 00:40:45,480 --> 00:40:51,460 After a time L/c, the radiation from this one 589 00:40:51,460 --> 00:40:52,580 will have got to here. 590 00:40:55,220 --> 00:41:00,740 If I add to the time d/c, that's how long 591 00:41:00,740 --> 00:41:04,380 the electromagnetic radiation takes to get to from here 592 00:41:04,380 --> 00:41:05,060 to here. 593 00:41:05,060 --> 00:41:07,970 So a little bit later, the radiation 594 00:41:07,970 --> 00:41:12,200 from this plus the radiation from this is getting here. 595 00:41:12,200 --> 00:41:15,140 And a little bit later, the radiation from this 596 00:41:15,140 --> 00:41:16,660 also gets there. 597 00:41:16,660 --> 00:41:19,900 So at the end, there is now radiation 598 00:41:19,900 --> 00:41:23,140 from all three charges getting to this point. 599 00:41:23,140 --> 00:41:26,010 And from then on, it'll continue forever 600 00:41:26,010 --> 00:41:27,550 as long as these are oscillating. 601 00:41:37,620 --> 00:41:42,100 So from then on, you have the radiation from the three. 602 00:41:42,100 --> 00:41:44,110 And the only difference between them 603 00:41:44,110 --> 00:41:47,675 is they all have a slightly different phase. 604 00:41:50,370 --> 00:41:54,020 So you might be satisfied with just knowing 605 00:41:54,020 --> 00:41:56,690 that these are the electric fields as sums 606 00:41:56,690 --> 00:42:01,180 of the algebraic sum between those 607 00:42:01,180 --> 00:42:03,300 all pointing the same direction. 608 00:42:03,300 --> 00:42:05,640 So I don't have to worry about the vector addition, 609 00:42:05,640 --> 00:42:10,760 but I have to worry about the addition of the amplitudes. 610 00:42:10,760 --> 00:42:14,630 If we want to find out what this sum is-- 611 00:42:14,630 --> 00:42:23,410 what happens when you add one or two or three oscillatory 612 00:42:23,410 --> 00:42:24,400 functions like this? 613 00:42:24,400 --> 00:42:28,890 What is the resultant oscillation? 614 00:42:28,890 --> 00:42:33,830 We have to algebraically, or trigonometrically, add 615 00:42:33,830 --> 00:42:36,780 these three or these two. 616 00:42:36,780 --> 00:42:42,850 And I do this example in order to introduce 617 00:42:42,850 --> 00:42:47,780 a mathematical technique, which in situations of this kind, 618 00:42:47,780 --> 00:42:51,716 makes life much, much easier than just going and adding 619 00:42:51,716 --> 00:42:52,215 cosines. 620 00:42:54,750 --> 00:43:02,190 And it is by using the so-called complex amplitudes. 621 00:43:02,190 --> 00:43:10,560 So the issue is, how do we add these three cosine functions 622 00:43:10,560 --> 00:43:16,000 and where they each have a slightly different phase? 623 00:43:16,000 --> 00:43:20,110 One way is brute force. 624 00:43:20,110 --> 00:43:23,610 Do it by using trigonometrical [? formulae ?]. 625 00:43:23,610 --> 00:43:26,980 For example, you could add the first and second 626 00:43:26,980 --> 00:43:30,530 by using the formula cosine a plus cosine b equals twice 627 00:43:30,530 --> 00:43:32,880 cosine half the sum of the angles 628 00:43:32,880 --> 00:43:36,830 cosine half the difference of the angle. 629 00:43:36,830 --> 00:43:40,171 And you could do that for adding the first to the second, 630 00:43:40,171 --> 00:43:41,920 and then later, once you've got an answer, 631 00:43:41,920 --> 00:43:45,240 you could add to that the third, et cetera. 632 00:43:45,240 --> 00:43:48,530 But you can see that if you have five, six, seven, 633 00:43:48,530 --> 00:43:54,045 eight sources, or many, many more, this becomes cumbersome. 634 00:43:56,930 --> 00:44:00,150 There's a nice mathematical trick. 635 00:44:00,150 --> 00:44:04,210 And that is by using complex numbers. 636 00:44:04,210 --> 00:44:07,580 You know that this is De Moivre's theorem 637 00:44:07,580 --> 00:44:12,480 that E to the j theta can be written 638 00:44:12,480 --> 00:44:18,640 as the cosine of an angle plus j sine of an angle. 639 00:44:18,640 --> 00:44:22,130 I can use this mathematical trick 640 00:44:22,130 --> 00:44:25,960 to solve the problem of adding these. 641 00:44:25,960 --> 00:44:31,140 Remember that at this stage, this is pure mathematics. 642 00:44:31,140 --> 00:44:35,140 As always, we've converted an experimental situation 643 00:44:35,140 --> 00:44:38,140 into a mathematical problem. 644 00:44:38,140 --> 00:44:41,630 And we've got to solve this using mathematics. 645 00:44:41,630 --> 00:44:43,320 You don't have to ask yourself what's 646 00:44:43,320 --> 00:44:46,500 the meaning of j sine theta something. 647 00:44:46,500 --> 00:44:48,915 It is a mathematical expression. 648 00:44:48,915 --> 00:44:51,120 And we're going to use mathematics 649 00:44:51,120 --> 00:44:54,200 to solve this problem. 650 00:44:54,200 --> 00:44:58,200 Using this, I could always write-- 651 00:44:58,200 --> 00:45:01,590 that suppose I have the cosine of some function, 652 00:45:01,590 --> 00:45:10,030 I can always write it as the real part of E 653 00:45:10,030 --> 00:45:11,890 to the j that angle. 654 00:45:11,890 --> 00:45:14,760 So if cosine, for example, of omega 655 00:45:14,760 --> 00:45:20,470 t minus kL, and you'll see it somewhere up there for example. 656 00:45:20,470 --> 00:45:30,690 It can be written as the real part of E to the j omega t 657 00:45:30,690 --> 00:45:35,100 minus j to the kL, but that is the same as multiplying these 658 00:45:35,100 --> 00:45:41,170 as E to the j omega t times E to the minus jkL, where here I'm 659 00:45:41,170 --> 00:45:43,875 just reminding you of k omega c, et cetera, is. 660 00:45:46,630 --> 00:45:50,940 I can do this for every-- if I want 661 00:45:50,940 --> 00:45:54,920 to d-- let's say we're doing this one first. 662 00:45:54,920 --> 00:45:56,850 I want this to that. 663 00:45:56,850 --> 00:46:02,900 I can write this as the real part of a complex number. 664 00:46:02,900 --> 00:46:09,090 And I can write this as a real part of the complex number. 665 00:46:09,090 --> 00:46:15,500 What I will then do-- I will first solve this. 666 00:46:15,500 --> 00:46:21,030 I will do the addition by adding the two complex numbers, 667 00:46:21,030 --> 00:46:27,610 knowing full well that if I added the complex numbers, 668 00:46:27,610 --> 00:46:31,910 I will have in the process added the real parts and also 669 00:46:31,910 --> 00:46:34,160 the imaginary parts. 670 00:46:34,160 --> 00:46:40,510 And since you cannot have a real number equal to an imaginary 671 00:46:40,510 --> 00:46:48,890 number in any way, if I take during the process of addition, 672 00:46:48,890 --> 00:46:53,740 I would have continuously kept separate the real and imaginary 673 00:46:53,740 --> 00:46:55,210 part. 674 00:46:55,210 --> 00:47:02,000 So for example, for this third part, this addition here, 675 00:47:02,000 --> 00:47:11,082 I can write the first term as the real part of each of the E 676 00:47:11,082 --> 00:47:16,600 to the j omega t, E to the minus jkL in the y-direction times 677 00:47:16,600 --> 00:47:17,510 E0. 678 00:47:17,510 --> 00:47:19,840 This is, of course, nothing other 679 00:47:19,840 --> 00:47:24,680 than E cosine omega t minus Lc. 680 00:47:24,680 --> 00:47:27,720 The second one I can do the same. 681 00:47:27,720 --> 00:47:29,300 This is the same. 682 00:47:29,300 --> 00:47:31,730 And the only difference between those two 683 00:47:31,730 --> 00:47:41,850 is that phase minus d/c, which is kd [? using ?] here. 684 00:47:41,850 --> 00:47:47,880 And so, the real part of this will 685 00:47:47,880 --> 00:47:50,815 be the answer to the sum of those. 686 00:47:55,450 --> 00:47:58,580 So I'll do this in a second, but then 687 00:47:58,580 --> 00:48:01,530 let me immediately go to the third one so I do both of them 688 00:48:01,530 --> 00:48:02,790 at the same time. 689 00:48:02,790 --> 00:48:07,330 And in this third case, the answer I'll want 690 00:48:07,330 --> 00:48:10,410 is the real part. 691 00:48:10,410 --> 00:48:12,040 This is the same as before. 692 00:48:12,040 --> 00:48:14,850 And here the three terms. 693 00:48:14,850 --> 00:48:19,260 The first term is just E0, because all the phases are out 694 00:48:19,260 --> 00:48:20,370 here. 695 00:48:20,370 --> 00:48:25,380 The second one differs from that by minus d/c, which 696 00:48:25,380 --> 00:48:29,410 is minus kd, so it's E to the minus jkd. 697 00:48:29,410 --> 00:48:34,150 And the last one is E to the minus j times 2 kd, 698 00:48:34,150 --> 00:48:35,510 because here we have a 2d. 699 00:48:42,600 --> 00:48:45,530 Now, why did I bother to do this? 700 00:48:45,530 --> 00:48:51,480 Because this addition is trivial while the other one 701 00:48:51,480 --> 00:48:52,660 was not trivial. 702 00:48:52,660 --> 00:48:54,960 It needed hard labor. 703 00:48:54,960 --> 00:48:57,180 Why do I say this is trivial? 704 00:48:57,180 --> 00:49:00,830 Because I've converted this algebraic problem 705 00:49:00,830 --> 00:49:02,580 into a geometry one. 706 00:49:02,580 --> 00:49:07,990 I can represent each one of these terms 707 00:49:07,990 --> 00:49:10,830 on an Argand diagram. 708 00:49:10,830 --> 00:49:16,920 So for example, here to here. 709 00:49:16,920 --> 00:49:22,450 Let's take the third case. 710 00:49:22,450 --> 00:49:25,940 We're adding these two vectors. 711 00:49:25,940 --> 00:49:30,310 This is only a real part, so it's E0. 712 00:49:30,310 --> 00:49:34,180 That's a vector of length E0 along the real axis. 713 00:49:34,180 --> 00:49:40,510 I'm reminding you on an Argand diagram, 714 00:49:40,510 --> 00:49:46,880 this direction is the real axis, and this is the imaginary axis. 715 00:49:46,880 --> 00:49:53,460 With that, so the E0 is only real, 716 00:49:53,460 --> 00:50:00,670 and I'm adding to it E to the minus jkd, which is what? 717 00:50:00,670 --> 00:50:03,420 Has a magnitude of E0. 718 00:50:03,420 --> 00:50:09,820 And this is the angle, theta, with respect 719 00:50:09,820 --> 00:50:16,170 to the real axis, which in this case is minus kd. 720 00:50:16,170 --> 00:50:18,005 So this is the angle, so it's minus. 721 00:50:18,005 --> 00:50:20,250 So I'm going down. 722 00:50:20,250 --> 00:50:25,910 So it's a vector, which is length E0, 723 00:50:25,910 --> 00:50:31,944 and it's pointing in an angle of minus kd. 724 00:50:31,944 --> 00:50:32,610 It's this angle. 725 00:50:32,610 --> 00:50:38,220 I suppose, not to confuse you, I'll call it minus kd. 726 00:50:38,220 --> 00:50:41,160 This is minus and minus, just so not to confuse you. 727 00:50:51,110 --> 00:50:56,374 With the distance between the charges of d, 728 00:50:56,374 --> 00:51:01,800 kd is 2 pi over 3, which is a 120 degrees. 729 00:51:01,800 --> 00:51:06,400 So this term is a vector like that. 730 00:51:06,400 --> 00:51:09,240 And we have to add those two vectors. 731 00:51:09,240 --> 00:51:12,180 Well, this is easy to do. 732 00:51:12,180 --> 00:51:14,880 This angle is 60 degrees. 733 00:51:14,880 --> 00:51:17,030 This is 120. 734 00:51:17,030 --> 00:51:17,780 OK? 735 00:51:17,780 --> 00:51:20,680 And this length is equal to that, 736 00:51:20,680 --> 00:51:25,050 so the result of this plus of that 737 00:51:25,050 --> 00:51:31,780 will be this red vector here, which has magnitude E0. 738 00:51:31,780 --> 00:51:33,550 And at what angle is it? 739 00:51:33,550 --> 00:51:37,685 This angle here 240 degrees. 740 00:51:47,300 --> 00:51:47,800 I'm sorry. 741 00:51:47,800 --> 00:51:50,970 I need that later in the second for the next part. 742 00:51:50,970 --> 00:51:52,590 I don't need it at this stage. 743 00:51:52,590 --> 00:51:53,910 I'm sorry. 744 00:51:53,910 --> 00:51:57,020 I'm adding this to that. 745 00:51:57,020 --> 00:52:00,110 The result is this vector. 746 00:52:00,110 --> 00:52:03,610 And this vector has a magnitude E0. 747 00:52:03,610 --> 00:52:09,000 And this angle here is, of course, 748 00:52:09,000 --> 00:52:13,070 60 degrees, which is 2 pi over 6. 749 00:52:13,070 --> 00:52:22,330 And so, so adding those two will give me just the real part 750 00:52:22,330 --> 00:52:25,740 on each of E to the j omega t to the minus 751 00:52:25,740 --> 00:52:28,960 jkL in the y-direction. 752 00:52:28,960 --> 00:52:37,300 And adding those two is E0 in the direction 753 00:52:37,300 --> 00:52:43,330 of 60 degrees, which is minus 2 pi over 6. 754 00:52:43,330 --> 00:52:45,450 Bracket right there. 755 00:52:45,450 --> 00:52:48,920 So this describes this addition. 756 00:52:48,920 --> 00:52:53,860 And this, of course, I can now take the real part 757 00:52:53,860 --> 00:52:58,140 of this whole thing, and there is the answer. 758 00:52:58,140 --> 00:53:00,680 E0 in the y-direction cosine omega 759 00:53:00,680 --> 00:53:05,670 t minus omega over c and with a phase of minus 2 pi 760 00:53:05,670 --> 00:53:12,240 over 6, which is of course, 60 degrees. 761 00:53:12,240 --> 00:53:15,570 For the first one, so this is the answer. 762 00:53:15,570 --> 00:53:18,810 And we didn't have to add to any cosine. 763 00:53:18,810 --> 00:53:25,090 We just do a vector addition on the Argand diagram. 764 00:53:25,090 --> 00:53:27,040 Let's do the last case. 765 00:53:27,040 --> 00:53:31,115 In the last case we have three terms-- one, two, three. 766 00:53:34,150 --> 00:53:44,810 If I write these as the real part of a complex amplitudes, 767 00:53:44,810 --> 00:53:49,020 by analogy with this, the first term is the same. 768 00:53:49,020 --> 00:53:51,290 The second term is the same. 769 00:53:51,290 --> 00:53:59,750 And the third one is almost the same as the second time, 770 00:53:59,750 --> 00:54:02,610 except the d has now become 2kd. 771 00:54:02,610 --> 00:54:04,210 This was d/c. 772 00:54:04,210 --> 00:54:05,940 Here it's 2d over c. 773 00:54:05,940 --> 00:54:09,150 So here we have minus j 2kd. 774 00:54:09,150 --> 00:54:14,110 So now I have to add these three complex numbers. 775 00:54:14,110 --> 00:54:19,220 And again, this is much easier to do it geometrically then 776 00:54:19,220 --> 00:54:21,370 trigonometrically. 777 00:54:21,370 --> 00:54:25,970 The first one I can represent by this on the Argand diagram. 778 00:54:25,970 --> 00:54:28,270 The second one by this. 779 00:54:28,270 --> 00:54:34,000 And the third one, of course, is at 240 degrees to here. 780 00:54:34,000 --> 00:54:35,670 This was 120. 781 00:54:35,670 --> 00:54:41,110 The next one is 240, which means that the last one, all 782 00:54:41,110 --> 00:54:44,230 that it is, I'd remove this vector, 783 00:54:44,230 --> 00:54:46,210 and it's in this direction. 784 00:54:46,210 --> 00:54:49,440 So now we're adding this to that to this, 785 00:54:49,440 --> 00:54:52,280 and even I can solve that in my head. 786 00:54:52,280 --> 00:54:55,430 If I add this vector to that to that, I get 0. 787 00:54:55,430 --> 00:54:57,170 I'm back where I started. 788 00:54:57,170 --> 00:55:00,140 So this quantity is 0. 789 00:55:00,140 --> 00:55:05,050 And so the electric field will be 0. 790 00:55:05,050 --> 00:55:07,980 And see how relatively easy it is 791 00:55:07,980 --> 00:55:13,700 to do if you use this complex amplitude method? 792 00:55:13,700 --> 00:55:19,980 One can do many problems to do with waves and vibrations 793 00:55:19,980 --> 00:55:22,390 using complex amplitudes. 794 00:55:22,390 --> 00:55:25,810 In general, it makes the algebra easier. 795 00:55:25,810 --> 00:55:32,103 But for most cases, I have not done that in order 796 00:55:32,103 --> 00:55:38,200 to-- that you don't have the double difficulty of trying 797 00:55:38,200 --> 00:55:41,630 to understand the physics and struggling with the mathematics 798 00:55:41,630 --> 00:55:44,200 that you may not be so familiar. 799 00:55:44,200 --> 00:55:48,500 By the time we come to many sources of the radiation, 800 00:55:48,500 --> 00:55:52,980 it is so much easier to do using complex amplitude 801 00:55:52,980 --> 00:55:57,780 that I would urge you to learn it on simple cases like this, 802 00:55:57,780 --> 00:56:02,390 and then use it in more complicated situation. 803 00:56:02,390 --> 00:56:06,260 Finally, I just want to save one word, 804 00:56:06,260 --> 00:56:09,670 and that is the following. 805 00:56:09,670 --> 00:56:11,210 Some of you may be surprised. 806 00:56:11,210 --> 00:56:16,110 How is it that I got-- I have three charges. 807 00:56:16,110 --> 00:56:22,980 Three charges oscillating, radiating. 808 00:56:22,980 --> 00:56:26,130 How is it that when you get to here, 809 00:56:26,130 --> 00:56:31,070 you get after a certain time, you get to [INAUDIBLE]. 810 00:56:31,070 --> 00:56:33,530 And the answer is-- pictorially you 811 00:56:33,530 --> 00:56:37,390 can see what happens-- at this point, 812 00:56:37,390 --> 00:56:41,960 the radiation from one of the charges looks like that. 813 00:56:41,960 --> 00:56:45,770 From the second one, it's out the phase 814 00:56:45,770 --> 00:56:50,790 by 1/3 of the wavelength, and the next one by 2/3. 815 00:56:50,790 --> 00:56:56,550 And so you're adding three waves, three oscillating 816 00:56:56,550 --> 00:56:59,270 motions on top of each other. 817 00:56:59,270 --> 00:57:02,670 And if you add these, the result is 0. 818 00:57:02,670 --> 00:57:07,110 That is, pictorially, the same as what 819 00:57:07,110 --> 00:57:08,940 I did here in this diagram. 820 00:57:08,940 --> 00:57:14,470 You simply-- you do have three waves arriving from the three 821 00:57:14,470 --> 00:57:18,510 charges, but they add to 0 because each 822 00:57:18,510 --> 00:57:22,620 has a different phase from the previous one. 823 00:57:22,620 --> 00:57:24,720 And you can see it here what happens. 824 00:57:24,720 --> 00:57:28,010 Here I'm plotting as a function of time. 825 00:57:28,010 --> 00:57:30,410 The amplitude at the point p.