WEBVTT 00:00:00.060 --> 00:00:01.770 The following content is provided 00:00:01.770 --> 00:00:04.019 under a Creative Commons license. 00:00:04.019 --> 00:00:06.860 Your support will help MIT OpenCourseWare continue 00:00:06.860 --> 00:00:10.720 to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.226 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.226 --> 00:00:17.851 at ocw.mit.edu. 00:00:46.750 --> 00:00:48.310 PROFESSOR: So welcome back. 00:00:48.310 --> 00:00:54.280 And today, I'm going to do some examples of what 00:00:54.280 --> 00:00:58.790 happens when you have more than one charge radiating. 00:00:58.790 --> 00:01:04.129 This is a very important example, 00:01:04.129 --> 00:01:11.720 because whenever you have many charges radiating, if it's 00:01:11.720 --> 00:01:16.720 coherent, you get the phenomenon of interference, 00:01:16.720 --> 00:01:20.870 and it plays a very important role in many fields of physics, 00:01:20.870 --> 00:01:23.250 in particular, in optics. 00:01:23.250 --> 00:01:26.330 You may have heard terms like interference patterns, 00:01:26.330 --> 00:01:28.230 diffraction, et cetera. 00:01:28.230 --> 00:01:36.600 All that is to do with when you have more than one source 00:01:36.600 --> 00:01:38.130 radiating. 00:01:38.130 --> 00:01:42.990 So the example I'm going to consider is the following. 00:01:42.990 --> 00:01:49.090 Suppose you have two charges, both q. 00:01:49.090 --> 00:01:50.810 Each one is oscillating. 00:01:55.020 --> 00:01:59.190 It's oscillating with an angular frequency 2 pi over lambda. 00:01:59.190 --> 00:02:02.340 This tells you the position of the charge 00:02:02.340 --> 00:02:04.010 as a function of time. 00:02:04.010 --> 00:02:07.380 So it's centered to distance d from here 00:02:07.380 --> 00:02:11.380 and oscillating within amplitude a 00:02:11.380 --> 00:02:15.730 as a cosine 2 pi over lambda t. 00:02:15.730 --> 00:02:18.750 Well, lambda is, of course, the wavelength 00:02:18.750 --> 00:02:21.330 of the radiated electromagnetic wave. 00:02:29.630 --> 00:02:33.490 So this one is oscillating up and down like that, 00:02:33.490 --> 00:02:37.990 and there is another one over here oscillating just for fun 00:02:37.990 --> 00:02:39.940 in the other direction. 00:02:39.940 --> 00:02:44.350 This one is oscillating in along the x-axis, 00:02:44.350 --> 00:02:49.530 and this one is oscillating along the y-axis. 00:02:49.530 --> 00:02:52.600 And here is the formula, which tells you 00:02:52.600 --> 00:02:57.970 what is the position of this one at the given instant of time. 00:02:57.970 --> 00:03:00.940 They're both given in terms of the same time, 00:03:00.940 --> 00:03:02.830 with the same time. 00:03:02.830 --> 00:03:03.850 t equals 0. 00:03:03.850 --> 00:03:08.290 Therefore, these are two coherent sources. 00:03:08.290 --> 00:03:12.850 The question is, what will be the resultant field 00:03:12.850 --> 00:03:14.800 anywhere in space? 00:03:14.800 --> 00:03:20.640 At some point we want to know what is the resultant field. 00:03:20.640 --> 00:03:25.770 Now, each one of these-- you know what it does. 00:03:25.770 --> 00:03:32.230 It radiates a spherical wave with a certain angular 00:03:32.230 --> 00:03:34.060 distribution. 00:03:34.060 --> 00:03:34.560 Alright? 00:03:37.280 --> 00:03:41.400 And the waves overlap in space. 00:03:41.400 --> 00:03:45.210 Now because the electromagnetic waves 00:03:45.210 --> 00:03:48.920 are solutions of the electromagnetic wave 00:03:48.920 --> 00:03:55.890 equation, which is a linear equation, if we have 00:03:55.890 --> 00:04:00.010 a coherent source of radiation at any point, 00:04:00.010 --> 00:04:06.300 we simply add vectorially the electromagnetic waves. 00:04:06.300 --> 00:04:09.610 The electric fields you add vectorially 00:04:09.610 --> 00:04:11.145 all the magnetic field. 00:04:14.320 --> 00:04:25.910 Now in principle, we should go-- to first principles. 00:04:25.910 --> 00:04:27.920 Consider Maxwell's equations. 00:04:27.920 --> 00:04:32.190 Consider what happens when a charge is oscillating. 00:04:36.740 --> 00:04:41.110 And in fact, consider both charges, 00:04:41.110 --> 00:04:43.260 what they're doing at any instant of time, 00:04:43.260 --> 00:04:47.380 and solve in all of space Maxwell's equation 00:04:47.380 --> 00:04:50.790 to see what the electric field is everywhere. 00:04:50.790 --> 00:04:56.400 We build on our experience, and what we will do 00:04:56.400 --> 00:05:02.170 is we will assume that we've solved the problem of what 00:05:02.170 --> 00:05:06.710 happens when this single charge is accelerating. 00:05:06.710 --> 00:05:09.100 We've done this several times in the [? PAR sensor ?]. 00:05:09.100 --> 00:05:11.940 Professor Walter Lewin has done it in the course. 00:05:11.940 --> 00:05:18.640 Whenever a charge is accelerating, it radiates. 00:05:18.640 --> 00:05:21.210 And this is the formula, which tells 00:05:21.210 --> 00:05:26.370 you what is the electric field from the given charge, which 00:05:26.370 --> 00:05:31.640 is accelerating at some position, r and t. 00:05:31.640 --> 00:05:36.090 I'm just reminding you that what we find 00:05:36.090 --> 00:05:44.120 is that the electric field at any position at some time t 00:05:44.120 --> 00:05:50.870 is given by, or related to, the perpendicular 00:05:50.870 --> 00:05:56.050 component of the acceleration of the charge. 00:05:56.050 --> 00:06:01.650 It's perpendicular to the direction of propagation, 00:06:01.650 --> 00:06:09.010 but at a point t in space, the electric field there 00:06:09.010 --> 00:06:15.420 is related to the acceleration at an earlier time. 00:06:15.420 --> 00:06:20.320 At the time, t prime, which is earlier than the time when 00:06:20.320 --> 00:06:27.390 we are considering by the amount r/c. 00:06:27.390 --> 00:06:31.450 All that this does-- it reflects the fact 00:06:31.450 --> 00:06:34.710 that if I have some with an accelerated charge, 00:06:34.710 --> 00:06:37.610 at that instant, the electromagnetic wave starts 00:06:37.610 --> 00:06:46.780 to propagate, and it takes a time r/c to reach a distance r. 00:06:46.780 --> 00:06:50.250 That's all this is telling us. 00:06:50.250 --> 00:06:53.680 The other thing this is telling us-- the wave front 00:06:53.680 --> 00:06:56.260 is a sphere. 00:06:56.260 --> 00:07:01.470 The amplitude of the electric field drops off like 1/r. 00:07:01.470 --> 00:07:05.920 And it is not uniform in all direction, 00:07:05.920 --> 00:07:08.480 as we've seen in the past. 00:07:08.480 --> 00:07:09.450 Right? 00:07:09.450 --> 00:07:12.450 For example, if I take this charge 00:07:12.450 --> 00:07:18.350 as it's oscillating up and down, the maximum amplitude radiated 00:07:18.350 --> 00:07:22.400 is in this plane where we are looking, 00:07:22.400 --> 00:07:24.710 where the perpendicular component 00:07:24.710 --> 00:07:28.190 of the acceleration of this is the maximum. 00:07:28.190 --> 00:07:34.000 And its minimum upwards is 0, because this acceleration 00:07:34.000 --> 00:07:37.550 has no component of x perpendicular 00:07:37.550 --> 00:07:41.000 component to this direction. 00:07:41.000 --> 00:07:44.470 And similarly, for this. 00:07:44.470 --> 00:07:46.900 The other thing which I wish to point out 00:07:46.900 --> 00:07:53.730 is that the polarization of the electromagnetic wave, 00:07:53.730 --> 00:07:57.640 the electric vector, is parallel to the perpendicular 00:07:57.640 --> 00:07:59.560 component of a. 00:07:59.560 --> 00:08:03.630 So from this, the electric vector, 00:08:03.630 --> 00:08:06.490 when it gets to here or here, will 00:08:06.490 --> 00:08:10.630 be in the direction of x, while from this one, 00:08:10.630 --> 00:08:14.540 it'll polarized in the direction y. 00:08:14.540 --> 00:08:16.540 The other thing which I want to stress, 00:08:16.540 --> 00:08:19.830 that we will in the other examples-- 00:08:19.830 --> 00:08:22.520 always consider a situation where 00:08:22.520 --> 00:08:27.670 the point where we're looking at the electromagnetic field 00:08:27.670 --> 00:08:32.500 is far away from where the charges are. 00:08:32.500 --> 00:08:36.720 The formula, which I just showed you here, 00:08:36.720 --> 00:08:39.809 applies only in the far field. 00:08:39.809 --> 00:08:42.840 Nearby, the electric field and the magnetic field 00:08:42.840 --> 00:08:44.580 are very complicated. 00:08:44.580 --> 00:08:47.900 They can be calculated, but they cannot be summarized in that 00:08:47.900 --> 00:08:48.980 simple form. 00:08:48.980 --> 00:08:54.030 But if I'm far away from the accelerating charge then, 00:08:54.030 --> 00:08:58.380 it is possible to write it in this simple form. 00:08:58.380 --> 00:09:05.330 So our discussion is only valid if the wave length radiated 00:09:05.330 --> 00:09:09.060 is much smaller than the distance 00:09:09.060 --> 00:09:10.365 from where we're considering. 00:09:13.590 --> 00:09:18.700 This distance is also much smaller than that. 00:09:18.700 --> 00:09:21.950 And this is to allow us to approximate sine theta 00:09:21.950 --> 00:09:23.710 by theta, et cetera. 00:09:23.710 --> 00:09:25.730 And that's the case, alright? 00:09:28.380 --> 00:09:29.760 OK. 00:09:29.760 --> 00:09:34.680 With that, we can now immediately solve the problem. 00:09:34.680 --> 00:09:40.040 Simply consider the acceleration of each charge 00:09:40.040 --> 00:09:43.630 from our knowledge of where the charge is. 00:09:43.630 --> 00:09:46.140 We can differentiate it twice, and we 00:09:46.140 --> 00:09:47.570 can get the acceleration. 00:09:47.570 --> 00:09:52.785 So we know that the acceleration of the first charge, 00:09:52.785 --> 00:09:55.360 which is oscillating along the x-axis, 00:09:55.360 --> 00:09:58.290 is given by minus a omega squared 00:09:58.290 --> 00:10:01.400 cosine omega in the x-direction. 00:10:01.400 --> 00:10:03.480 I want to emphasize, because this 00:10:03.480 --> 00:10:09.760 is in the same direction as the perpendicular component of a1. 00:10:14.240 --> 00:10:15.820 And, I'm sorry. 00:10:15.820 --> 00:10:17.110 I misspoke. 00:10:17.110 --> 00:10:18.950 The answer-- what I said is correct 00:10:18.950 --> 00:10:20.120 but for the wrong reason. 00:10:20.120 --> 00:10:25.760 In this case, by definition, the acceleration of the first one 00:10:25.760 --> 00:10:31.450 is this because the displacement of the first one 00:10:31.450 --> 00:10:33.260 is in the x-direction. 00:10:33.260 --> 00:10:36.310 Similarly, the acceleration on the second one 00:10:36.310 --> 00:10:39.390 is in the y-direction, because the displacement 00:10:39.390 --> 00:10:41.800 is in the y-direction. 00:10:41.800 --> 00:10:42.370 Alright? 00:10:42.370 --> 00:10:44.970 So these are the two accelerations. 00:10:44.970 --> 00:10:49.170 This is just some-- occasionally I'll use omega, 00:10:49.170 --> 00:10:52.580 occasionally lambda just to remind you of that. 00:10:52.580 --> 00:10:53.080 OK. 00:10:53.080 --> 00:10:59.736 So all I now have to do for each of the charges 00:10:59.736 --> 00:11:05.170 is calculate this quantity at the position 00:11:05.170 --> 00:11:08.720 where I want to calculate the electric and the magnetic 00:11:08.720 --> 00:11:10.040 field. 00:11:10.040 --> 00:11:13.040 So, first of all in this problem, 00:11:13.040 --> 00:11:15.170 we're asked to do it in two places. 00:11:15.170 --> 00:11:18.120 We're asked to do it at position p1 00:11:18.120 --> 00:11:21.290 and at p2. p1 is along the z-axis, 00:11:21.290 --> 00:11:24.450 and p2 is off the z-axis. 00:11:24.450 --> 00:11:27.570 First, let's do it for p1. 00:11:27.570 --> 00:11:33.740 So, the electric field, due to the first charge 00:11:33.740 --> 00:11:40.780 at position p1 at time t, is given by this formula 00:11:40.780 --> 00:11:49.610 if I insert in this for the distance r, which is L. OK? 00:11:49.610 --> 00:11:54.050 And I put in the acceleration of this one 00:11:54.050 --> 00:11:59.400 and the perpendicular component of the acceleration 00:11:59.400 --> 00:12:04.870 perpendicular to the line drawing my two charges. 00:12:04.870 --> 00:12:08.300 And the position p1 is-- in fact, 00:12:08.300 --> 00:12:10.250 that line is along the z-axis. 00:12:10.250 --> 00:12:13.790 So the perpendicular direction is, in fact, 00:12:13.790 --> 00:12:15.650 in the x-direction. 00:12:15.650 --> 00:12:16.150 OK. 00:12:16.150 --> 00:12:21.670 So I don't even have to take the-- the actual acceleration 00:12:21.670 --> 00:12:23.870 is the perpendicular component. 00:12:23.870 --> 00:12:24.580 OK. 00:12:24.580 --> 00:12:30.570 And I have to calculate it at a time which 00:12:30.570 --> 00:12:40.230 is t minus the distance p1 is from in my charges, which is L. 00:12:40.230 --> 00:12:43.730 So it's minus omega L over c. 00:12:43.730 --> 00:12:47.660 I calculate t prime in here. 00:12:47.660 --> 00:12:51.620 Notice, I have not tried to calculate 00:12:51.620 --> 00:12:56.110 the distance between the charge and p1 exactly, 00:12:56.110 --> 00:13:00.060 because I told you that the distance d is very much 00:13:00.060 --> 00:13:05.900 smaller than L. And so the overall distance is negligently 00:13:05.900 --> 00:13:11.250 different from being just L. So that's 00:13:11.250 --> 00:13:13.540 the electric field due to the first one. 00:13:13.540 --> 00:13:17.250 Similarly, I can do the same for the second one. 00:13:17.250 --> 00:13:19.150 Everything is the same. 00:13:19.150 --> 00:13:22.650 The only difference in that case is 00:13:22.650 --> 00:13:27.720 that now the electric vector-- the acceleration 00:13:27.720 --> 00:13:32.470 is along the y-axis and therefore, the electric field 00:13:32.470 --> 00:13:35.710 is polarized along the y-direction. 00:13:35.710 --> 00:13:42.290 So at the point p1, the two electric fields are the same. 00:13:42.290 --> 00:13:44.830 They're both oscillating in phase 00:13:44.830 --> 00:13:47.260 with the same frequency, et cetera. 00:13:47.260 --> 00:13:50.270 But this one is pointing in the x-direction. 00:13:50.270 --> 00:13:52.940 This in the y-direction. 00:13:52.940 --> 00:13:56.870 Now, as I mentioned at the beginning, if at any point 00:13:56.870 --> 00:14:01.450 you have electric field due to two sources, 00:14:01.450 --> 00:14:05.780 they simply add because the system is a linear system. 00:14:05.780 --> 00:14:08.800 So we simply have to add the electric fields. 00:14:08.800 --> 00:14:11.060 Electric fields are vectors, therefore 00:14:11.060 --> 00:14:16.390 we don't add them algebraic as scalars, 00:14:16.390 --> 00:14:19.750 but we have to add them vectorially. 00:14:19.750 --> 00:14:24.220 And so, we take this vector and that vector and add it. 00:14:24.220 --> 00:14:28.910 And that gives us the total electric field at point p1. 00:14:28.910 --> 00:14:32.390 And simply from here, I get d0 like that, 00:14:32.390 --> 00:14:38.840 which I can rewrite in terms of a unit vector. 00:14:38.840 --> 00:14:39.750 Alright? 00:14:39.750 --> 00:14:45.980 This is a unit vector, which is at 45 degrees. 00:14:45.980 --> 00:14:51.680 In other words, if that's the z-direction x and y, 00:14:51.680 --> 00:14:56.180 it's at 45 degrees to both the x and y-axes. 00:14:56.180 --> 00:14:57.830 That's the unit vector. 00:14:57.830 --> 00:14:59.780 This is the amplitude of it. 00:14:59.780 --> 00:15:05.280 And this is d-- tell us what is the oscillating frequency 00:15:05.280 --> 00:15:08.730 and what is the phase. 00:15:08.730 --> 00:15:13.670 Notice, although each one of these sources 00:15:13.670 --> 00:15:19.430 radiated there an electric field of magnitude E0, 00:15:19.430 --> 00:15:25.080 when I add them, I did not get to 2E0, because as I mentioned, 00:15:25.080 --> 00:15:26.980 electric fields are vectors, and we 00:15:26.980 --> 00:15:28.550 have to add them vectorially. 00:15:28.550 --> 00:15:31.770 And that's why you get here the root 2. 00:15:31.770 --> 00:15:36.940 So this is the first example, which I've done. 00:15:36.940 --> 00:15:40.870 Now I want to take the same geometry 00:15:40.870 --> 00:15:46.050 but calculate what is the electric field. 00:15:46.050 --> 00:15:49.170 What we've just done, we've calculated here. 00:15:49.170 --> 00:15:51.750 Now I want to do the same situation-- 00:15:51.750 --> 00:15:57.530 two sources oscillating coherently. 00:15:57.530 --> 00:16:01.240 But I want to look at what is the electric field at the point 00:16:01.240 --> 00:16:02.560 p2. 00:16:02.560 --> 00:16:09.620 And I've taken p2 to be in that the xz-plane, 00:16:09.620 --> 00:16:14.986 but in a direction that's an angle lambda over a to d. 00:16:14.986 --> 00:16:16.900 d is this distance. 00:16:16.900 --> 00:16:22.490 Lambda is the wavelength of the radiated electromagnetic wave. 00:16:22.490 --> 00:16:25.760 And I've taken this crazy number simply 00:16:25.760 --> 00:16:29.690 to make the arithmetic come out easier at the end. 00:16:29.690 --> 00:16:34.020 And so we now want to calculate the electric field here. 00:16:34.020 --> 00:16:37.320 And we do exactly the same except we go a little faster. 00:16:37.320 --> 00:16:41.330 What I will do is I will calculate the electric field 00:16:41.330 --> 00:16:42.310 here. 00:16:42.310 --> 00:16:45.610 Due to this charge oscillating, I'll 00:16:45.610 --> 00:16:47.990 then calculate the electric field here. 00:16:47.990 --> 00:16:51.120 Due to this charge, this one is oscillating like that. 00:16:51.120 --> 00:16:54.210 I'm reminding you this one is oscillating like that. 00:16:54.210 --> 00:16:57.740 And we will vectorially add the two. 00:16:57.740 --> 00:17:02.790 Now, from the point of view of the amplitude, 00:17:02.790 --> 00:17:10.630 the difference in distance between this and here or this 00:17:10.630 --> 00:17:15.940 and here or this and here is insignificant. 00:17:15.940 --> 00:17:19.270 And so I have ignored that in the previous calculations, 00:17:19.270 --> 00:17:21.599 and I'll do the same here. 00:17:21.599 --> 00:17:28.470 But when you add to the two, we have 00:17:28.470 --> 00:17:30.990 to worry about the relative phase 00:17:30.990 --> 00:17:34.050 of the alternating the electric fields. 00:17:34.050 --> 00:17:38.370 And the difference of this distance compared 00:17:38.370 --> 00:17:44.360 to that distance is no longer insignificant 00:17:44.360 --> 00:17:50.700 when it comes to the relative phase of the two radiations. 00:17:50.700 --> 00:17:58.490 So, in this formula when I'm calculating the electric field, 00:17:58.490 --> 00:18:03.630 this r-- it doesn't matter whether I 00:18:03.630 --> 00:18:07.390 call this L or L plus a little bit or less. 00:18:07.390 --> 00:18:09.730 This doesn't change very much. 00:18:09.730 --> 00:18:19.330 But in calculating the phase of the radiation, 00:18:19.330 --> 00:18:21.990 I cannot ignore that. 00:18:21.990 --> 00:18:25.350 In the previous case, the phase was the same 00:18:25.350 --> 00:18:28.890 because this distance and that distance 00:18:28.890 --> 00:18:32.050 are exactly the same by symmetry. 00:18:32.050 --> 00:18:37.660 For this position, that distance and that distance 00:18:37.660 --> 00:18:39.380 are not the same. 00:18:39.380 --> 00:18:45.670 And so I what I will do is I'll use as reference that distance, 00:18:45.670 --> 00:18:48.420 and when I'm calculating this distance, 00:18:48.420 --> 00:18:52.870 I will calculate by how much I have to subtract from here 00:18:52.870 --> 00:18:57.690 to end up with this length and for this one, by how much 00:18:57.690 --> 00:19:01.750 I've add to it to get to here. 00:19:01.750 --> 00:19:05.590 Pay attention to that when I'm doing that in a second. 00:19:05.590 --> 00:19:11.410 And the rest is very straightforward and similar 00:19:11.410 --> 00:19:12.740 to what I've just done. 00:19:17.960 --> 00:19:21.650 So now, as I say, I want to calculate the electric field 00:19:21.650 --> 00:19:25.510 at the point p2, which is this. 00:19:25.510 --> 00:19:28.350 This is the x-coordinate, y, and z-coordinate. 00:19:28.350 --> 00:19:31.950 And I pointed it out to you a second ago. 00:19:31.950 --> 00:19:32.450 Alright. 00:19:36.050 --> 00:19:41.210 So I don't have to rewrite all those q's and omegas, 00:19:41.210 --> 00:19:44.290 et cetera, by analogy, what we've 00:19:44.290 --> 00:19:52.290 done-- the electric field due to the first charge at position p2 00:19:52.290 --> 00:19:55.845 will be that the amplitude that we've got 00:19:55.845 --> 00:19:59.530 is the same for the previous case of P1. 00:19:59.530 --> 00:20:02.340 So it's-- E0 is the amplitude. 00:20:02.340 --> 00:20:07.800 It's pointing in the x-direction as before. 00:20:07.800 --> 00:20:18.780 And we know that the description of the electric field 00:20:18.780 --> 00:20:28.745 will be given by cosine omega t1 prime where t is-- so far 00:20:28.745 --> 00:20:30.660 I've always talked about t prime, 00:20:30.660 --> 00:20:34.740 because I only had one when I was considering just one 00:20:34.740 --> 00:20:35.490 charge. 00:20:35.490 --> 00:20:42.680 But here the t1 is different to t2. 00:20:42.680 --> 00:20:51.920 So quickly, this term comes from the perpendicular component 00:20:51.920 --> 00:20:55.630 of a right as before. 00:20:55.630 --> 00:21:00.770 And t1 prime is equal to the time 00:21:00.770 --> 00:21:11.120 when I'm looking at the electric field minus the time it takes 00:21:11.120 --> 00:21:17.030 for this signal from the charge to get to the point 00:21:17.030 --> 00:21:20.850 where I'm looking at the electric field to point p2. 00:21:20.850 --> 00:21:26.320 So, it will be-- as I told you a second ago, 00:21:26.320 --> 00:21:29.440 I'll take L as the reference. 00:21:29.440 --> 00:21:35.940 And I'm subtracting from it that distance d sine lambda over 8d. 00:21:35.940 --> 00:21:38.380 For a second, let me go back to that picture 00:21:38.380 --> 00:21:42.080 because you can easily get confused. 00:21:42.080 --> 00:21:44.790 So I'm coming back here. 00:21:44.790 --> 00:21:53.160 What we're trying to calculate-- the distance from here to here. 00:21:53.160 --> 00:21:58.400 So I'm taking this distance, which is L, 00:21:58.400 --> 00:22:04.680 and I'm subtracting from it this distance, which 00:22:04.680 --> 00:22:11.500 is d sine this angle here, which is the same as this angle. 00:22:11.500 --> 00:22:15.150 So that's what I'm calculating there. 00:22:15.150 --> 00:22:21.840 So this is L minus d sine lambda over 8d divided by c. 00:22:21.840 --> 00:22:26.570 Then the electric field is then immediately followed 00:22:26.570 --> 00:22:33.540 from this d0 in the x-direction, cosine omega t minus this. 00:22:33.540 --> 00:22:37.990 And if you assume that this angle is small, 00:22:37.990 --> 00:22:43.470 and I told you that that's given in the problem, 00:22:43.470 --> 00:22:46.150 the sine of an angle is equal to an angle. 00:22:46.150 --> 00:22:48.030 Approximately, they're small. 00:22:48.030 --> 00:22:52.500 And if I multiply this out, I simply get an angle here. 00:22:52.500 --> 00:22:54.320 That's pi/4. 00:22:54.320 --> 00:22:59.680 So I find that the electric field at point p2 00:22:59.680 --> 00:23:03.620 is pointing, due to the first charge, is pointing 00:23:03.620 --> 00:23:05.630 in the x-direction, as you'd expect, 00:23:05.630 --> 00:23:08.350 because of the direction which the charge is moving 00:23:08.350 --> 00:23:14.085 times E0 times cosine omega t minus omega L over c 00:23:14.085 --> 00:23:20.120 plus a phase like that. 00:23:20.120 --> 00:23:22.320 How about from the other charge? 00:23:22.320 --> 00:23:25.800 If I take the other charge, everything 00:23:25.800 --> 00:23:30.720 will be the same except now this charge is oscillating 00:23:30.720 --> 00:23:34.670 in the y-direction, so the electric field will 00:23:34.670 --> 00:23:36.960 be in the y-direction. 00:23:36.960 --> 00:23:42.070 And this time, it's omega t2 prime 00:23:42.070 --> 00:23:46.890 where t2 will be very similar to t1 00:23:46.890 --> 00:23:52.180 but now the distance is greater by this amount. 00:23:52.180 --> 00:23:55.350 For a second, let's go back to this picture 00:23:55.350 --> 00:23:57.810 so you see what I'm talking about. 00:23:57.810 --> 00:24:02.360 If I calculate this distance, it's 00:24:02.360 --> 00:24:12.400 the same as that distance plus d sine this angle, which 00:24:12.400 --> 00:24:17.340 is this lambda over 8d, and I'm subtracting. 00:24:17.340 --> 00:24:20.390 And I want to emphasize that what 00:24:20.390 --> 00:24:23.740 I'm focusing on here, what is important, 00:24:23.740 --> 00:24:28.640 is the difference between this and that 00:24:28.640 --> 00:24:31.040 and not the absolute value. 00:24:31.040 --> 00:24:35.640 So when I'm approximating-- if you 00:24:35.640 --> 00:24:37.160 do this calculation for yourself, 00:24:37.160 --> 00:24:39.540 you'll see I'm doing a slight approximation 00:24:39.540 --> 00:24:40.750 to calculate that. 00:24:40.750 --> 00:24:44.530 But this and this is the same quantity. 00:24:44.530 --> 00:24:47.610 But the important thing is the difference of this phase 00:24:47.610 --> 00:24:49.320 here is plus pi/4. 00:24:49.320 --> 00:24:52.330 Here is minus pi/4, because here it's 00:24:52.330 --> 00:24:54.460 minus that little bit of a distance, 00:24:54.460 --> 00:24:58.960 and here it's plus a little bit of the distance. 00:24:58.960 --> 00:25:00.470 So we finished. 00:25:00.470 --> 00:25:05.820 The total electric field at p2 and time t 00:25:05.820 --> 00:25:10.700 is the sum of the electric field due to the first charge plus 00:25:10.700 --> 00:25:12.000 due to the second challenge. 00:25:15.170 --> 00:25:19.810 And as I mentioned a second ago, electric fields are vectors. 00:25:19.810 --> 00:25:23.230 Thus, we have to add these two vectorially. 00:25:23.230 --> 00:25:26.140 This is in the x-direction. 00:25:26.140 --> 00:25:28.350 This is in the y-direction. 00:25:28.350 --> 00:25:30.620 And so what we have at that point 00:25:30.620 --> 00:25:34.850 is two electric oscillating electric fields. 00:25:34.850 --> 00:25:39.430 They're oscillating coherently but out of phase. 00:25:39.430 --> 00:25:41.700 This is a plus pi/4 phase. 00:25:41.700 --> 00:25:44.600 This is a minus pi/4 phase. 00:25:44.600 --> 00:25:48.900 So the difference of phase between those two is pi/2. 00:25:48.900 --> 00:25:50.680 It's 90 degrees. 00:25:50.680 --> 00:25:55.860 So this is out of phase with, one with respect to the other, 00:25:55.860 --> 00:25:57.280 by 90 degrees. 00:25:57.280 --> 00:26:01.740 They're coherent, but 90 degrees out of phase. 00:26:01.740 --> 00:26:04.510 The magnitudes are the same, but this one 00:26:04.510 --> 00:26:07.830 is pointing in the x-direction and this in the y-direction, 00:26:07.830 --> 00:26:10.900 and you know what that corresponds to. 00:26:10.900 --> 00:26:15.480 If I have at the location two electric fields, one 00:26:15.480 --> 00:26:20.410 doing that and the other one doing this, 00:26:20.410 --> 00:26:27.960 if they're out of phase by 90 degrees, and I add the two, 00:26:27.960 --> 00:26:29.010 what do I get? 00:26:29.010 --> 00:26:34.010 I get a constant size of a radius, 00:26:34.010 --> 00:26:35.900 and the thing is rotating. 00:26:35.900 --> 00:26:42.100 That is the description of a rotating 00:26:42.100 --> 00:26:45.340 vector of magnitude E0. 00:26:45.340 --> 00:26:53.310 In the other case, at P1, there was not this phase difference, 00:26:53.310 --> 00:26:59.280 so the two were-- one was oscillating like this. 00:26:59.280 --> 00:27:02.140 The other one was oscillating like that. 00:27:02.140 --> 00:27:05.330 But they were oscillating at the same phase. 00:27:05.330 --> 00:27:07.230 So this is what they were doing. 00:27:07.230 --> 00:27:12.200 And if you add those up, clearly you get a line diagonally. 00:27:12.200 --> 00:27:17.700 And that's why before, the result was a linearly polarized 00:27:17.700 --> 00:27:21.990 electric field there at 45 degrees to the x and y-axis. 00:27:21.990 --> 00:27:28.740 In this case, these two are out of phase by 90 degrees. 00:27:28.740 --> 00:27:32.440 So when one is doing this, the other one 00:27:32.440 --> 00:27:33.790 is also but out of phase. 00:27:33.790 --> 00:27:35.420 And so when this one's at the maximum, 00:27:35.420 --> 00:27:38.670 this one's at the minimum, and so that's what's happening. 00:27:38.670 --> 00:27:39.810 So they're out of phase. 00:27:39.810 --> 00:27:42.030 And then if you add the two up, you 00:27:42.030 --> 00:27:43.710 get something doing this, which we 00:27:43.710 --> 00:27:46.000 called circularly polarized light. 00:27:46.000 --> 00:27:48.600 And that's why I chose that crazy angle, 00:27:48.600 --> 00:27:51.920 because it came out like this. 00:27:51.920 --> 00:27:53.910 So that's the end of that problem. 00:27:53.910 --> 00:27:57.840 I will now do another one, which to emphasize 00:27:57.840 --> 00:28:04.110 some technique-- a different technique of doing it. 00:28:04.110 --> 00:28:04.850 OK. 00:28:04.850 --> 00:28:06.170 Let's move. 00:28:06.170 --> 00:28:12.300 So the next problem is the following. 00:28:12.300 --> 00:28:20.670 We're again dealing with several charges oscillating coherently. 00:28:20.670 --> 00:28:22.790 Their cohering sources. 00:28:22.790 --> 00:28:26.440 And we're asking, what are the electric fields? 00:28:26.440 --> 00:28:27.920 Somewhere in space? 00:28:27.920 --> 00:28:29.880 And if we found the electric field, of course, 00:28:29.880 --> 00:28:33.990 we could always calculate the magnetic field 00:28:33.990 --> 00:28:37.420 at that location using Maxwell's equations or our knowledge 00:28:37.420 --> 00:28:39.960 of the relation between electromagnetic field 00:28:39.960 --> 00:28:43.790 in a progressive electromagnetic way. 00:28:43.790 --> 00:28:47.540 Now I'll take three charges. 00:28:47.540 --> 00:28:50.630 But that's not the hope. 00:28:50.630 --> 00:28:55.780 It's more different, but what this problem adds 00:28:55.780 --> 00:29:03.060 is I'll use it to show a technique that is often used, 00:29:03.060 --> 00:29:05.910 which helps in the solution of such problems. 00:29:05.910 --> 00:29:09.190 So the problem now is the following. 00:29:09.190 --> 00:29:11.010 I have three charges. 00:29:11.010 --> 00:29:13.540 Each of the same magnitude. 00:29:13.540 --> 00:29:17.482 They're located along the x-axis at position 0 00:29:17.482 --> 00:29:20.300 minus d and minus 2d. 00:29:20.300 --> 00:29:25.100 So these distances are the same-- each distance d. 00:29:25.100 --> 00:29:31.260 And what the problem is-- up to t equals 0, 00:29:31.260 --> 00:29:37.080 these three charges are displaced from equilibrium. 00:29:37.080 --> 00:29:39.000 I put y is 0. 00:29:39.000 --> 00:29:40.715 So, at the height, y0 here. 00:29:44.060 --> 00:29:49.740 Then, at t0-- don't ask me how. 00:29:49.740 --> 00:29:56.190 Magically, I get these three charges to start oscillating. 00:29:56.190 --> 00:30:00.720 Such that at time yt, the displacement 00:30:00.720 --> 00:30:04.600 is y0 cosine omega t. 00:30:04.600 --> 00:30:09.590 And they continue-- this continues like that forever. 00:30:09.590 --> 00:30:16.910 The question is, as a result of this oscillation of charges, 00:30:16.910 --> 00:30:23.180 what will be the electric field a long way away from here 00:30:23.180 --> 00:30:28.530 at position L along the x-axis, so we call that position p. 00:30:28.530 --> 00:30:35.720 So the problem is, calculate E at position p for all times. 00:30:35.720 --> 00:30:38.730 Once we've calculated, we could calculate the magnetic field, 00:30:38.730 --> 00:30:43.060 But to save you time--I mean, you could do it for yourself. 00:30:43.060 --> 00:30:45.220 You know if you calculate the electric field, 00:30:45.220 --> 00:30:47.390 there is a progressive wave over here. 00:30:47.390 --> 00:30:49.790 if E is like this, then b will be 00:30:49.790 --> 00:30:52.250 perpendicular to it of the amplitude which 00:30:52.250 --> 00:30:54.300 is just a dE over c. 00:30:54.300 --> 00:30:57.350 So that will be straightforward, so I'm not asking it. 00:31:00.030 --> 00:31:02.280 Now, let's just think for a second 00:31:02.280 --> 00:31:04.070 what goes on in this problem. 00:31:04.070 --> 00:31:07.820 Initially, the charges are stationary, 00:31:07.820 --> 00:31:10.290 so there will be a Coulomb field around. 00:31:10.290 --> 00:31:14.210 But we are specifically asked to ignore static fields. 00:31:14.210 --> 00:31:18.440 So in the last problem too, I ignored static fields. 00:31:18.440 --> 00:31:22.590 We were only considering the time-dependent fields. 00:31:25.410 --> 00:31:29.200 You can always have superimposed on the time-dependent field 00:31:29.200 --> 00:31:33.690 some static field that doesn't add anything. 00:31:33.690 --> 00:31:39.620 So what we have here is at time up to time t equals 0, 00:31:39.620 --> 00:31:41.030 the charges are here. 00:31:41.030 --> 00:31:43.450 So let's take one of the charges here. 00:31:43.450 --> 00:31:47.920 Then it starts moving. 00:31:47.920 --> 00:31:51.880 It will have an acceleration, which 00:31:51.880 --> 00:31:54.750 is perpendicular to the direction in which I'm 00:31:54.750 --> 00:31:58.990 interested the propagation of the electromagnetic wave. 00:31:58.990 --> 00:32:02.840 So it will certainly radiate in this direction. 00:32:02.840 --> 00:32:08.860 So this charge, which was oscillating, will radiate. 00:32:08.860 --> 00:32:12.470 Over here at the point p, the electric field 00:32:12.470 --> 00:32:14.690 will be initially 0. 00:32:14.690 --> 00:32:23.380 And it will continue being 0 until the electric field, which 00:32:23.380 --> 00:32:28.530 is generated here, propagates that distance. 00:32:28.530 --> 00:32:42.970 So, only after a time, L/c, will the electric vector get here? 00:32:42.970 --> 00:32:49.440 So up to the time L/c, there will be no electric field here. 00:32:53.670 --> 00:32:55.230 How about this charge? 00:32:55.230 --> 00:32:59.520 This charge is initially is displaced at y0. 00:32:59.520 --> 00:33:02.990 Then it starts oscillating the same. 00:33:02.990 --> 00:33:05.670 Initially, it's stationary. 00:33:05.670 --> 00:33:08.450 It'll produce a Coulomb field, which is a static field. 00:33:08.450 --> 00:33:10.330 We're not interested in it. 00:33:10.330 --> 00:33:16.010 But once it starts accelerating, it starts radiating. 00:33:16.010 --> 00:33:18.800 And that radiating progresses. 00:33:18.800 --> 00:33:23.410 So that will also produce a field over here. 00:33:23.410 --> 00:33:29.660 But since these two distances are different, 00:33:29.660 --> 00:33:32.700 the radiation from here, first of all, will get there a little 00:33:32.700 --> 00:33:33.970 later. 00:33:33.970 --> 00:33:39.240 But also, once it gets here, it will have a different phase 00:33:39.240 --> 00:33:43.780 because the radiation traveled a different distance. 00:33:43.780 --> 00:33:45.420 And same for the last one. 00:33:48.190 --> 00:33:50.520 So now, how do we do this? 00:33:54.030 --> 00:33:58.350 I almost sound like a broken record. 00:33:58.350 --> 00:34:02.240 As always, I can calculate the electric field 00:34:02.240 --> 00:34:06.940 from each charge, and it'll be given by this formula, which 00:34:06.940 --> 00:34:11.620 we've seen over and over again where t prime is 00:34:11.620 --> 00:34:17.469 the distance from the charge to the point p. 00:34:17.469 --> 00:34:21.880 Furthermore, I know the perpendicular acceleration 00:34:21.880 --> 00:34:24.719 and also the perpendicular component of it, 00:34:24.719 --> 00:34:27.560 because in this problem, the perpendicular component 00:34:27.560 --> 00:34:29.404 is the same as the actual acceleration. 00:34:32.429 --> 00:34:37.380 After time t equals 0, the acceleration 00:34:37.380 --> 00:34:42.630 of every one of these charges a1, a2, a3, is the same. 00:34:42.630 --> 00:34:44.179 It's the same direction. 00:34:44.179 --> 00:34:46.980 And it's given by that simply because we 00:34:46.980 --> 00:34:50.620 know what is the displacement y of t. 00:34:50.620 --> 00:34:54.389 If I differentiate it twice, I get the acceleration. 00:34:54.389 --> 00:34:58.460 If I plug this into here for each charge, 00:34:58.460 --> 00:35:02.400 I will know what is the electric fields from each charge. 00:35:02.400 --> 00:35:06.550 I can then add the electric fields from each charge, 00:35:06.550 --> 00:35:09.730 and I'll get the total electric field. 00:35:09.730 --> 00:35:12.380 You have to add them vectorially. 00:35:12.380 --> 00:35:18.900 So, once again, for t less than L/c, 00:35:18.900 --> 00:35:23.880 the electric field at position p will be 0. 00:35:23.880 --> 00:35:27.820 There was no earlier accelerated charge 00:35:27.820 --> 00:35:30.830 which radiated an electric field which 00:35:30.830 --> 00:35:35.110 got here before this time. 00:35:35.110 --> 00:35:37.330 So that's nice and easy. 00:35:37.330 --> 00:35:45.110 How about for a later period, a period between L/c or just 00:35:45.110 --> 00:35:47.010 after L/c. 00:35:47.010 --> 00:35:53.320 For t just after L/c, the acceleration 00:35:53.320 --> 00:36:01.870 of the charge at x equals 0 would produce an electric field 00:36:01.870 --> 00:36:07.720 which propagated and would have got to my point p. 00:36:07.720 --> 00:36:14.220 So between that time L/c and the time when 00:36:14.220 --> 00:36:20.420 the radiation from the second charge got to the point p, 00:36:20.420 --> 00:36:24.050 I will have an electric field but only 00:36:24.050 --> 00:36:28.470 from the first radiation due to the first charge. 00:36:28.470 --> 00:36:32.630 I can forget the second and third charge at minus d 00:36:32.630 --> 00:36:38.530 and a minus 2d but not the charge at x equals 0. 00:36:38.530 --> 00:36:40.980 So that will be I use this formula. 00:36:40.980 --> 00:36:41.830 I plug it in. 00:36:41.830 --> 00:36:43.880 I use the acceleration of a1. 00:36:43.880 --> 00:36:50.771 I put it in here, and I know that t prime is t minus r/c. 00:36:50.771 --> 00:36:55.300 r in this case is L. So this formula 00:36:55.300 --> 00:37:00.180 immediately tells me that's what it is and lets me save-- 00:37:00.180 --> 00:37:02.640 I don't want to write this over and over again. 00:37:02.640 --> 00:37:04.400 This-- I'll call that E0. 00:37:04.400 --> 00:37:08.310 So from the first charge, this is 00:37:08.310 --> 00:37:12.600 what the electric field-- it's an oscillating electric field-- 00:37:12.600 --> 00:37:15.440 what it looks like at position p. 00:37:15.440 --> 00:37:24.350 But I'm reminding you, this is only true for this time window 00:37:24.350 --> 00:37:27.410 when the radiation from the charge at q 00:37:27.410 --> 00:37:33.700 equals 0 has got to point p, but from the other one's not. 00:37:33.700 --> 00:37:36.880 How about from a little later? 00:37:36.880 --> 00:37:42.040 From the time L/c plus d over c-- in other words, 00:37:42.040 --> 00:37:49.390 now radiation over a distance L plus d 00:37:49.390 --> 00:37:52.570 has had time to reach my point p. 00:37:52.570 --> 00:38:00.190 So now, the second charge, the one that minused d, 00:38:00.190 --> 00:38:05.190 has had enough time to reach my point p. 00:38:05.190 --> 00:38:11.370 But if I limit to the window, this and that, in other words, 00:38:11.370 --> 00:38:16.950 there's still a difference of d/c in time between those. 00:38:16.950 --> 00:38:21.630 I will get at point p the radiation as 00:38:21.630 --> 00:38:23.970 from the first charge. 00:38:23.970 --> 00:38:27.010 That's exactly the same as this. 00:38:27.010 --> 00:38:28.830 That's obviously got there. 00:38:28.830 --> 00:38:32.850 But now from the second charge, it has got there. 00:38:32.850 --> 00:38:35.110 And they're very similar. 00:38:35.110 --> 00:38:41.800 They are both-- the charges are accelerating 00:38:41.800 --> 00:38:50.060 along y-direction in phase. 00:38:50.060 --> 00:38:56.150 So they have the same frequency, and the amplitude 00:38:56.150 --> 00:39:00.790 is going to be the same, because in this formula, 00:39:00.790 --> 00:39:05.790 this r is the total distance between the charge 00:39:05.790 --> 00:39:07.570 and the point p. 00:39:07.570 --> 00:39:11.210 Now, there is a tiny difference in that distance 00:39:11.210 --> 00:39:14.130 for the first and second charge. 00:39:14.130 --> 00:39:17.420 But if I take them here a tiny difference, 00:39:17.420 --> 00:39:19.465 this will not change very much. 00:39:19.465 --> 00:39:21.680 I'm ignoring that difference. 00:39:21.680 --> 00:39:24.790 So I'm calling both of them E0 and ignoring 00:39:24.790 --> 00:39:26.210 that tiny difference. 00:39:26.210 --> 00:39:31.140 But I cannot ignore the difference on the phase. 00:39:31.140 --> 00:39:36.920 So the t prime into two cases is different. 00:39:36.920 --> 00:39:40.580 In one case, it's t minus L/c. 00:39:40.580 --> 00:39:44.640 And the other case, it's minus L/c 00:39:44.640 --> 00:39:49.510 minus this little extra time where the radiation 00:39:49.510 --> 00:39:54.070 takes from the second to the first charge. 00:39:54.070 --> 00:39:57.140 And the third one, the radiation could 00:39:57.140 --> 00:39:59.360 not to reach the point p yet. 00:39:59.360 --> 00:40:04.805 So that is now the total electric field from the two. 00:40:08.400 --> 00:40:11.380 And finally-- and I have to do this. 00:40:11.380 --> 00:40:12.870 I haven't done the addition. 00:40:12.870 --> 00:40:20.030 And finally, if we take a time which is greater than this, 00:40:20.030 --> 00:40:26.140 then we are in a situation where the oscillation 00:40:26.140 --> 00:40:29.900 on the last charge has had enough time 00:40:29.900 --> 00:40:31.880 to get to the point p. 00:40:31.880 --> 00:40:37.650 Let me go back to this picture and just repeat this 00:40:37.650 --> 00:40:39.030 so that you don't get lost. 00:40:39.030 --> 00:40:45.480 So these charges have started to move all at the same time. 00:40:45.480 --> 00:40:51.460 After a time L/c, the radiation from this one 00:40:51.460 --> 00:40:52.580 will have got to here. 00:40:55.220 --> 00:41:00.740 If I add to the time d/c, that's how long 00:41:00.740 --> 00:41:04.380 the electromagnetic radiation takes to get to from here 00:41:04.380 --> 00:41:05.060 to here. 00:41:05.060 --> 00:41:07.970 So a little bit later, the radiation 00:41:07.970 --> 00:41:12.200 from this plus the radiation from this is getting here. 00:41:12.200 --> 00:41:15.140 And a little bit later, the radiation from this 00:41:15.140 --> 00:41:16.660 also gets there. 00:41:16.660 --> 00:41:19.900 So at the end, there is now radiation 00:41:19.900 --> 00:41:23.140 from all three charges getting to this point. 00:41:23.140 --> 00:41:26.010 And from then on, it'll continue forever 00:41:26.010 --> 00:41:27.550 as long as these are oscillating. 00:41:37.620 --> 00:41:42.100 So from then on, you have the radiation from the three. 00:41:42.100 --> 00:41:44.110 And the only difference between them 00:41:44.110 --> 00:41:47.675 is they all have a slightly different phase. 00:41:50.370 --> 00:41:54.020 So you might be satisfied with just knowing 00:41:54.020 --> 00:41:56.690 that these are the electric fields as sums 00:41:56.690 --> 00:42:01.180 of the algebraic sum between those 00:42:01.180 --> 00:42:03.300 all pointing the same direction. 00:42:03.300 --> 00:42:05.640 So I don't have to worry about the vector addition, 00:42:05.640 --> 00:42:10.760 but I have to worry about the addition of the amplitudes. 00:42:10.760 --> 00:42:14.630 If we want to find out what this sum is-- 00:42:14.630 --> 00:42:23.410 what happens when you add one or two or three oscillatory 00:42:23.410 --> 00:42:24.400 functions like this? 00:42:24.400 --> 00:42:28.890 What is the resultant oscillation? 00:42:28.890 --> 00:42:33.830 We have to algebraically, or trigonometrically, add 00:42:33.830 --> 00:42:36.780 these three or these two. 00:42:36.780 --> 00:42:42.850 And I do this example in order to introduce 00:42:42.850 --> 00:42:47.780 a mathematical technique, which in situations of this kind, 00:42:47.780 --> 00:42:51.716 makes life much, much easier than just going and adding 00:42:51.716 --> 00:42:52.215 cosines. 00:42:54.750 --> 00:43:02.190 And it is by using the so-called complex amplitudes. 00:43:02.190 --> 00:43:10.560 So the issue is, how do we add these three cosine functions 00:43:10.560 --> 00:43:16.000 and where they each have a slightly different phase? 00:43:16.000 --> 00:43:20.110 One way is brute force. 00:43:20.110 --> 00:43:23.610 Do it by using trigonometrical [? formulae ?]. 00:43:23.610 --> 00:43:26.980 For example, you could add the first and second 00:43:26.980 --> 00:43:30.530 by using the formula cosine a plus cosine b equals twice 00:43:30.530 --> 00:43:32.880 cosine half the sum of the angles 00:43:32.880 --> 00:43:36.830 cosine half the difference of the angle. 00:43:36.830 --> 00:43:40.171 And you could do that for adding the first to the second, 00:43:40.171 --> 00:43:41.920 and then later, once you've got an answer, 00:43:41.920 --> 00:43:45.240 you could add to that the third, et cetera. 00:43:45.240 --> 00:43:48.530 But you can see that if you have five, six, seven, 00:43:48.530 --> 00:43:54.045 eight sources, or many, many more, this becomes cumbersome. 00:43:56.930 --> 00:44:00.150 There's a nice mathematical trick. 00:44:00.150 --> 00:44:04.210 And that is by using complex numbers. 00:44:04.210 --> 00:44:07.580 You know that this is De Moivre's theorem 00:44:07.580 --> 00:44:12.480 that E to the j theta can be written 00:44:12.480 --> 00:44:18.640 as the cosine of an angle plus j sine of an angle. 00:44:18.640 --> 00:44:22.130 I can use this mathematical trick 00:44:22.130 --> 00:44:25.960 to solve the problem of adding these. 00:44:25.960 --> 00:44:31.140 Remember that at this stage, this is pure mathematics. 00:44:31.140 --> 00:44:35.140 As always, we've converted an experimental situation 00:44:35.140 --> 00:44:38.140 into a mathematical problem. 00:44:38.140 --> 00:44:41.630 And we've got to solve this using mathematics. 00:44:41.630 --> 00:44:43.320 You don't have to ask yourself what's 00:44:43.320 --> 00:44:46.500 the meaning of j sine theta something. 00:44:46.500 --> 00:44:48.915 It is a mathematical expression. 00:44:48.915 --> 00:44:51.120 And we're going to use mathematics 00:44:51.120 --> 00:44:54.200 to solve this problem. 00:44:54.200 --> 00:44:58.200 Using this, I could always write-- 00:44:58.200 --> 00:45:01.590 that suppose I have the cosine of some function, 00:45:01.590 --> 00:45:10.030 I can always write it as the real part of E 00:45:10.030 --> 00:45:11.890 to the j that angle. 00:45:11.890 --> 00:45:14.760 So if cosine, for example, of omega 00:45:14.760 --> 00:45:20.470 t minus kL, and you'll see it somewhere up there for example. 00:45:20.470 --> 00:45:30.690 It can be written as the real part of E to the j omega t 00:45:30.690 --> 00:45:35.100 minus j to the kL, but that is the same as multiplying these 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 00:45:41.170 --> 00:45:43.875 just reminding you of k omega c, et cetera, is. 00:45:46.630 --> 00:45:50.940 I can do this for every-- if I want 00:45:50.940 --> 00:45:54.920 to d-- let's say we're doing this one first. 00:45:54.920 --> 00:45:56.850 I want this to that. 00:45:56.850 --> 00:46:02.900 I can write this as the real part of a complex number. 00:46:02.900 --> 00:46:09.090 And I can write this as a real part of the complex number. 00:46:09.090 --> 00:46:15.500 What I will then do-- I will first solve this. 00:46:15.500 --> 00:46:21.030 I will do the addition by adding the two complex numbers, 00:46:21.030 --> 00:46:27.610 knowing full well that if I added the complex numbers, 00:46:27.610 --> 00:46:31.910 I will have in the process added the real parts and also 00:46:31.910 --> 00:46:34.160 the imaginary parts. 00:46:34.160 --> 00:46:40.510 And since you cannot have a real number equal to an imaginary 00:46:40.510 --> 00:46:48.890 number in any way, if I take during the process of addition, 00:46:48.890 --> 00:46:53.740 I would have continuously kept separate the real and imaginary 00:46:53.740 --> 00:46:55.210 part. 00:46:55.210 --> 00:47:02.000 So for example, for this third part, this addition here, 00:47:02.000 --> 00:47:11.082 I can write the first term as the real part of each of the E 00:47:11.082 --> 00:47:16.600 to the j omega t, E to the minus jkL in the y-direction times 00:47:16.600 --> 00:47:17.510 E0. 00:47:17.510 --> 00:47:19.840 This is, of course, nothing other 00:47:19.840 --> 00:47:24.680 than E cosine omega t minus Lc. 00:47:24.680 --> 00:47:27.720 The second one I can do the same. 00:47:27.720 --> 00:47:29.300 This is the same. 00:47:29.300 --> 00:47:31.730 And the only difference between those two 00:47:31.730 --> 00:47:41.850 is that phase minus d/c, which is kd [? using ?] here. 00:47:41.850 --> 00:47:47.880 And so, the real part of this will 00:47:47.880 --> 00:47:50.815 be the answer to the sum of those. 00:47:55.450 --> 00:47:58.580 So I'll do this in a second, but then 00:47:58.580 --> 00:48:01.530 let me immediately go to the third one so I do both of them 00:48:01.530 --> 00:48:02.790 at the same time. 00:48:02.790 --> 00:48:07.330 And in this third case, the answer I'll want 00:48:07.330 --> 00:48:10.410 is the real part. 00:48:10.410 --> 00:48:12.040 This is the same as before. 00:48:12.040 --> 00:48:14.850 And here the three terms. 00:48:14.850 --> 00:48:19.260 The first term is just E0, because all the phases are out 00:48:19.260 --> 00:48:20.370 here. 00:48:20.370 --> 00:48:25.380 The second one differs from that by minus d/c, which 00:48:25.380 --> 00:48:29.410 is minus kd, so it's E to the minus jkd. 00:48:29.410 --> 00:48:34.150 And the last one is E to the minus j times 2 kd, 00:48:34.150 --> 00:48:35.510 because here we have a 2d. 00:48:42.600 --> 00:48:45.530 Now, why did I bother to do this? 00:48:45.530 --> 00:48:51.480 Because this addition is trivial while the other one 00:48:51.480 --> 00:48:52.660 was not trivial. 00:48:52.660 --> 00:48:54.960 It needed hard labor. 00:48:54.960 --> 00:48:57.180 Why do I say this is trivial? 00:48:57.180 --> 00:49:00.830 Because I've converted this algebraic problem 00:49:00.830 --> 00:49:02.580 into a geometry one. 00:49:02.580 --> 00:49:07.990 I can represent each one of these terms 00:49:07.990 --> 00:49:10.830 on an Argand diagram. 00:49:10.830 --> 00:49:16.920 So for example, here to here. 00:49:16.920 --> 00:49:22.450 Let's take the third case. 00:49:22.450 --> 00:49:25.940 We're adding these two vectors. 00:49:25.940 --> 00:49:30.310 This is only a real part, so it's E0. 00:49:30.310 --> 00:49:34.180 That's a vector of length E0 along the real axis. 00:49:34.180 --> 00:49:40.510 I'm reminding you on an Argand diagram, 00:49:40.510 --> 00:49:46.880 this direction is the real axis, and this is the imaginary axis. 00:49:46.880 --> 00:49:53.460 With that, so the E0 is only real, 00:49:53.460 --> 00:50:00.670 and I'm adding to it E to the minus jkd, which is what? 00:50:00.670 --> 00:50:03.420 Has a magnitude of E0. 00:50:03.420 --> 00:50:09.820 And this is the angle, theta, with respect 00:50:09.820 --> 00:50:16.170 to the real axis, which in this case is minus kd. 00:50:16.170 --> 00:50:18.005 So this is the angle, so it's minus. 00:50:18.005 --> 00:50:20.250 So I'm going down. 00:50:20.250 --> 00:50:25.910 So it's a vector, which is length E0, 00:50:25.910 --> 00:50:31.944 and it's pointing in an angle of minus kd. 00:50:31.944 --> 00:50:32.610 It's this angle. 00:50:32.610 --> 00:50:38.220 I suppose, not to confuse you, I'll call it minus kd. 00:50:38.220 --> 00:50:41.160 This is minus and minus, just so not to confuse you. 00:50:51.110 --> 00:50:56.374 With the distance between the charges of d, 00:50:56.374 --> 00:51:01.800 kd is 2 pi over 3, which is a 120 degrees. 00:51:01.800 --> 00:51:06.400 So this term is a vector like that. 00:51:06.400 --> 00:51:09.240 And we have to add those two vectors. 00:51:09.240 --> 00:51:12.180 Well, this is easy to do. 00:51:12.180 --> 00:51:14.880 This angle is 60 degrees. 00:51:14.880 --> 00:51:17.030 This is 120. 00:51:17.030 --> 00:51:17.780 OK? 00:51:17.780 --> 00:51:20.680 And this length is equal to that, 00:51:20.680 --> 00:51:25.050 so the result of this plus of that 00:51:25.050 --> 00:51:31.780 will be this red vector here, which has magnitude E0. 00:51:31.780 --> 00:51:33.550 And at what angle is it? 00:51:33.550 --> 00:51:37.685 This angle here 240 degrees. 00:51:47.300 --> 00:51:47.800 I'm sorry. 00:51:47.800 --> 00:51:50.970 I need that later in the second for the next part. 00:51:50.970 --> 00:51:52.590 I don't need it at this stage. 00:51:52.590 --> 00:51:53.910 I'm sorry. 00:51:53.910 --> 00:51:57.020 I'm adding this to that. 00:51:57.020 --> 00:52:00.110 The result is this vector. 00:52:00.110 --> 00:52:03.610 And this vector has a magnitude E0. 00:52:03.610 --> 00:52:09.000 And this angle here is, of course, 00:52:09.000 --> 00:52:13.070 60 degrees, which is 2 pi over 6. 00:52:13.070 --> 00:52:22.330 And so, so adding those two will give me just the real part 00:52:22.330 --> 00:52:25.740 on each of E to the j omega t to the minus 00:52:25.740 --> 00:52:28.960 jkL in the y-direction. 00:52:28.960 --> 00:52:37.300 And adding those two is E0 in the direction 00:52:37.300 --> 00:52:43.330 of 60 degrees, which is minus 2 pi over 6. 00:52:43.330 --> 00:52:45.450 Bracket right there. 00:52:45.450 --> 00:52:48.920 So this describes this addition. 00:52:48.920 --> 00:52:53.860 And this, of course, I can now take the real part 00:52:53.860 --> 00:52:58.140 of this whole thing, and there is the answer. 00:52:58.140 --> 00:53:00.680 E0 in the y-direction cosine omega 00:53:00.680 --> 00:53:05.670 t minus omega over c and with a phase of minus 2 pi 00:53:05.670 --> 00:53:12.240 over 6, which is of course, 60 degrees. 00:53:12.240 --> 00:53:15.570 For the first one, so this is the answer. 00:53:15.570 --> 00:53:18.810 And we didn't have to add to any cosine. 00:53:18.810 --> 00:53:25.090 We just do a vector addition on the Argand diagram. 00:53:25.090 --> 00:53:27.040 Let's do the last case. 00:53:27.040 --> 00:53:31.115 In the last case we have three terms-- one, two, three. 00:53:34.150 --> 00:53:44.810 If I write these as the real part of a complex amplitudes, 00:53:44.810 --> 00:53:49.020 by analogy with this, the first term is the same. 00:53:49.020 --> 00:53:51.290 The second term is the same. 00:53:51.290 --> 00:53:59.750 And the third one is almost the same as the second time, 00:53:59.750 --> 00:54:02.610 except the d has now become 2kd. 00:54:02.610 --> 00:54:04.210 This was d/c. 00:54:04.210 --> 00:54:05.940 Here it's 2d over c. 00:54:05.940 --> 00:54:09.150 So here we have minus j 2kd. 00:54:09.150 --> 00:54:14.110 So now I have to add these three complex numbers. 00:54:14.110 --> 00:54:19.220 And again, this is much easier to do it geometrically then 00:54:19.220 --> 00:54:21.370 trigonometrically. 00:54:21.370 --> 00:54:25.970 The first one I can represent by this on the Argand diagram. 00:54:25.970 --> 00:54:28.270 The second one by this. 00:54:28.270 --> 00:54:34.000 And the third one, of course, is at 240 degrees to here. 00:54:34.000 --> 00:54:35.670 This was 120. 00:54:35.670 --> 00:54:41.110 The next one is 240, which means that the last one, all 00:54:41.110 --> 00:54:44.230 that it is, I'd remove this vector, 00:54:44.230 --> 00:54:46.210 and it's in this direction. 00:54:46.210 --> 00:54:49.440 So now we're adding this to that to this, 00:54:49.440 --> 00:54:52.280 and even I can solve that in my head. 00:54:52.280 --> 00:54:55.430 If I add this vector to that to that, I get 0. 00:54:55.430 --> 00:54:57.170 I'm back where I started. 00:54:57.170 --> 00:55:00.140 So this quantity is 0. 00:55:00.140 --> 00:55:05.050 And so the electric field will be 0. 00:55:05.050 --> 00:55:07.980 And see how relatively easy it is 00:55:07.980 --> 00:55:13.700 to do if you use this complex amplitude method? 00:55:13.700 --> 00:55:19.980 One can do many problems to do with waves and vibrations 00:55:19.980 --> 00:55:22.390 using complex amplitudes. 00:55:22.390 --> 00:55:25.810 In general, it makes the algebra easier. 00:55:25.810 --> 00:55:32.103 But for most cases, I have not done that in order 00:55:32.103 --> 00:55:38.200 to-- that you don't have the double difficulty of trying 00:55:38.200 --> 00:55:41.630 to understand the physics and struggling with the mathematics 00:55:41.630 --> 00:55:44.200 that you may not be so familiar. 00:55:44.200 --> 00:55:48.500 By the time we come to many sources of the radiation, 00:55:48.500 --> 00:55:52.980 it is so much easier to do using complex amplitude 00:55:52.980 --> 00:55:57.780 that I would urge you to learn it on simple cases like this, 00:55:57.780 --> 00:56:02.390 and then use it in more complicated situation. 00:56:02.390 --> 00:56:06.260 Finally, I just want to save one word, 00:56:06.260 --> 00:56:09.670 and that is the following. 00:56:09.670 --> 00:56:11.210 Some of you may be surprised. 00:56:11.210 --> 00:56:16.110 How is it that I got-- I have three charges. 00:56:16.110 --> 00:56:22.980 Three charges oscillating, radiating. 00:56:22.980 --> 00:56:26.130 How is it that when you get to here, 00:56:26.130 --> 00:56:31.070 you get after a certain time, you get to [INAUDIBLE]. 00:56:31.070 --> 00:56:33.530 And the answer is-- pictorially you 00:56:33.530 --> 00:56:37.390 can see what happens-- at this point, 00:56:37.390 --> 00:56:41.960 the radiation from one of the charges looks like that. 00:56:41.960 --> 00:56:45.770 From the second one, it's out the phase 00:56:45.770 --> 00:56:50.790 by 1/3 of the wavelength, and the next one by 2/3. 00:56:50.790 --> 00:56:56.550 And so you're adding three waves, three oscillating 00:56:56.550 --> 00:56:59.270 motions on top of each other. 00:56:59.270 --> 00:57:02.670 And if you add these, the result is 0. 00:57:02.670 --> 00:57:07.110 That is, pictorially, the same as what 00:57:07.110 --> 00:57:08.940 I did here in this diagram. 00:57:08.940 --> 00:57:14.470 You simply-- you do have three waves arriving from the three 00:57:14.470 --> 00:57:18.510 charges, but they add to 0 because each 00:57:18.510 --> 00:57:22.620 has a different phase from the previous one. 00:57:22.620 --> 00:57:24.720 And you can see it here what happens. 00:57:24.720 --> 00:57:28.010 Here I'm plotting as a function of time. 00:57:28.010 --> 00:57:30.410 The amplitude at the point p.