WEBVTT 00:02.070 --> 00:05.470 Prof: All right, class, I thought I'd start as 00:05.473 --> 00:08.553 usual by telling you what happened last time. 00:08.550 --> 00:12.530 Not the whole thing, but just the highlights so you 00:12.526 --> 00:15.386 can follow what's happening today. 00:15.390 --> 00:20.520 The main things I did last time were the notion of an electric 00:20.521 --> 00:25.901 field, which is going to be with you from now till the end of the 00:25.904 --> 00:26.834 course. 00:26.830 --> 00:31.890 The idea of the electric field is that if you've got lots of 00:31.887 --> 00:34.347 charges, q_1, 00:34.347 --> 00:37.507 q_2, q_3, 00:37.510 --> 00:42.010 instead of worrying about the force they exert on each other, 00:42.010 --> 00:45.710 you ask yourself, at a generic point where 00:45.713 --> 00:49.403 there's nothing, if I put 1 coulomb here, 00:49.398 --> 00:51.448 what will it experience? 00:51.450 --> 00:52.850 What force will it experience? 00:52.850 --> 00:54.270 You compute that. 00:54.270 --> 00:56.100 So in your mind, imagine a coulomb, 00:56.100 --> 00:58.790 and you find the force due to q_1, 00:58.791 --> 01:00.031 it looks like that. 01:00.030 --> 01:02.730 q_2 may exert a force that way. 01:02.728 --> 01:04.648 q_3 could be of opposite sign, 01:04.653 --> 01:06.413 so maybe it will exert a force that way. 01:06.409 --> 01:09.519 You add all these vectors, they add up to something. 01:09.519 --> 01:15.859 That something is called the electric field at that point. 01:15.860 --> 01:18.880 There is nothing there except the electric field, 01:18.879 --> 01:21.899 but it's very real, because if you put something, 01:21.897 --> 01:23.657 something happens to it. 01:23.659 --> 01:26.079 So the electric field is everywhere. 01:26.080 --> 01:27.930 The charges are in a few places. 01:27.930 --> 01:31.360 Electric field is defined everywhere, except right on top 01:31.364 --> 01:34.064 of the charges, which is where it probably is 01:34.062 --> 01:34.862 infinite. 01:34.860 --> 01:36.830 Once you know the electric field anywhere, 01:36.830 --> 01:38.560 if you put another charge q, 01:38.560 --> 01:41.920 a real charge, it will experience a force 01:41.921 --> 01:45.331 equal to qE, because electric field was the 01:45.334 --> 01:47.634 force you would have had on the 1 coulomb, 01:47.629 --> 01:50.349 and if you put q coulombs, it will be qE. 01:50.349 --> 01:52.809 That's the electric field. 01:52.810 --> 01:56.400 So you can imagine computing it for any given distribution of 01:56.404 --> 01:59.704 charges, because you know what each one of them does. 01:59.700 --> 02:03.420 Then I said there's one nice way to visualize the electric 02:03.418 --> 02:06.158 field, which is to draw the field lines. 02:06.159 --> 02:09.619 You go to each point and you ask, if I put a charge here, 02:09.616 --> 02:12.636 a positive test charge, which way will it move? 02:12.639 --> 02:16.849 Then you follow that thing as it moves and you get that line 02:16.854 --> 02:20.144 and you get that line and you get that line. 02:20.139 --> 02:23.229 You can draw these lines. 02:23.229 --> 02:27.949 The lines give you one piece of information which is very 02:27.949 --> 02:30.649 obvious, namely, if you are here, 02:30.647 --> 02:33.427 the force is along that line. 02:33.430 --> 02:36.530 But I also pointed out to you, you get more than just the 02:36.526 --> 02:37.906 direction of the force. 02:37.910 --> 02:41.180 You also can understand the strength of the force. 02:41.180 --> 02:44.700 The strength of the force is contained in the density of 02:44.703 --> 02:45.603 these lines. 02:45.598 --> 02:47.158 Now density has to be defined carefully. 02:47.160 --> 02:49.080 It's not like mass per unit volume. 02:49.080 --> 02:50.570 It's an area density. 02:50.568 --> 02:53.998 That means if you get yourself 1 square meter, 02:54.000 --> 02:55.750 you know, meter by meter piece of wood, 02:55.750 --> 02:57.970 like a frame, and you hold it there, 02:57.970 --> 03:01.620 perpendicular to the lines and see how many lines go through, 03:01.620 --> 03:04.250 that's called the area density of lines. 03:04.250 --> 03:07.860 That you can see will fall like 1/r^(2), 03:07.860 --> 03:10.250 because if you draw a sphere of radius r, 03:10.250 --> 03:14.270 that area is 4Πr^(2), and the same number of lines 03:14.271 --> 03:16.271 are going through that as any other sphere, 03:16.270 --> 03:19.340 so it will be proportional to 1/r^(2). 03:19.340 --> 03:22.310 But so is the electrical field proportional to 1/r^(2), 03:22.310 --> 03:27.100 and I said, let us agree that we will draw 03:27.099 --> 03:33.059 1/ε_0 lines per coulomb. 03:33.060 --> 03:36.600 This is a necessity; it's just a convenience. 03:36.598 --> 03:40.568 It's like saying I want to measure distances in inches and 03:40.569 --> 03:42.589 centimeters for daily life. 03:42.590 --> 03:44.890 You can measure them in parsecs and angstroms, 03:44.890 --> 03:47.090 but you'll be dealing with nasty numbers. 03:47.090 --> 03:49.660 So it's a convenience, and the convenience here is, 03:49.656 --> 03:51.706 let's pick 1/ε_0 03:51.710 --> 03:52.840 lines per coulomb. 03:52.840 --> 03:56.060 You'll see the advantage of that, because if you ask, 03:56.060 --> 03:59.570 what's the density of lines here on a sphere of radius 03:59.574 --> 04:02.304 r, if there's charge q at 04:02.295 --> 04:04.675 the center, I got that many lines, 04:04.677 --> 04:08.127 and the area of the sphere is 4Πr^(2), 04:08.128 --> 04:10.518 so the density of lines per unit area, 04:10.520 --> 04:15.010 you see, is precisely equal in magnitude to the electric field 04:15.010 --> 04:16.190 at that point. 04:16.189 --> 04:18.809 If you had not drawn 1/ε_0, 04:18.810 --> 04:20.940 but maybe 5/ε_0 04:20.937 --> 04:22.827 lines, then the line density will be 5 04:22.827 --> 04:23.997 times the electric field. 04:24.000 --> 04:25.650 It will still represent the electric field, 04:25.649 --> 04:27.339 but we don't want to simply represent it. 04:27.339 --> 04:29.059 We want it to be the electric field. 04:29.060 --> 04:31.050 It makes it easier. 04:31.050 --> 04:34.940 Then I said, let's take a slightly more 04:34.935 --> 04:38.715 complicated situation, two charges. 04:38.720 --> 04:41.940 This is called a dipole, one pole and another pole, 04:41.935 --> 04:44.505 and you can draw the field lines here. 04:44.509 --> 04:48.029 You can see if you put a test charge, it'll run away from the 04:48.026 --> 04:49.196 plus to the minus. 04:49.199 --> 04:52.619 If you leave it somewhere here, it'll go like that and loop 04:52.620 --> 04:55.630 around and come back, and you can calculate them. 04:55.629 --> 04:57.009 This is no longer guesswork. 04:57.009 --> 04:59.809 If you had enough time, I hope you all agree, 04:59.810 --> 05:02.920 you can go to any point you like and find the force of 05:02.918 --> 05:06.168 attraction due to this one, the force of repulsion due to 05:06.173 --> 05:09.583 that one, add them up and you will get an 05:09.581 --> 05:11.601 arrow that direction. 05:11.600 --> 05:15.910 So one can draw these lines, and the lines tell you a story. 05:15.910 --> 05:19.940 Then I said, let us find the field in an 05:19.942 --> 05:24.082 analytic expression due to the dipole. 05:24.079 --> 05:25.029 Yes? 05:25.028 --> 05:26.118 Student: I have a question about the dipole. 05:26.120 --> 05:29.360 If you put a test particle going in the positive direction 05:29.358 --> 05:31.688 of the x axis, would it also _________? 05:31.689 --> 05:32.359 Prof: No. 05:32.360 --> 05:33.460 She's got a good point. 05:33.459 --> 05:37.859 If you put somebody here, it will never loop around. 05:37.860 --> 05:40.810 Can you see why? 05:40.810 --> 05:45.140 Because as it goes further away, this is trying to pull it 05:45.137 --> 05:45.667 back. 05:45.670 --> 05:48.450 It's always going to be closer to this guy, so it's never going 05:48.447 --> 05:49.117 to come back. 05:49.120 --> 05:55.000 This line will go like that and this line will go like this. 05:55.000 --> 06:01.630 But anything else at any other angle will loop around and come 06:01.629 --> 06:02.389 back. 06:02.389 --> 06:05.549 All right, now the field strength, if you want to 06:05.547 --> 06:07.527 calculate it, you can use the formula for 06:07.528 --> 06:09.588 E due to this one and E due to that one and add 06:09.589 --> 06:11.159 them, and I did that for you. 06:11.160 --> 06:12.560 I don't want to go into the details, 06:12.560 --> 06:18.990 but I remind you in all cases, the electric field fell like 06:18.992 --> 06:22.372 1/r^(3), because the field of each 06:22.367 --> 06:24.237 charges goes like 1/r^(2). 06:24.240 --> 06:27.620 And if these were on top of each other, they will completely 06:27.622 --> 06:28.772 cancel each other. 06:28.769 --> 06:32.799 So the reason you have a non-zero field is thanks to 06:32.800 --> 06:33.670 a. 06:33.670 --> 06:36.300 Therefore the answer has to contain an a in front of 06:36.295 --> 06:38.465 it, at least n the first approximation. 06:38.470 --> 06:40.850 But that a, from dimensional analysis, 06:40.848 --> 06:44.088 has to come with a 1/r so that the whole thing has the 06:44.091 --> 06:45.661 same dimension as before. 06:45.660 --> 06:48.210 That's where you get a/r^(3). 06:48.209 --> 06:53.329 That a times q times 2 and so on, 06:53.327 --> 06:56.887 that became the dipole moment. 06:56.889 --> 06:58.809 That was true here. 06:58.810 --> 07:00.730 We verified that's true here. 07:00.730 --> 07:04.020 Later on, we'll verify it everywhere, because there are 07:04.023 --> 07:06.893 easier ways to calculate than what I'm using. 07:06.889 --> 07:10.949 Then I said, forget about the field due to 07:10.949 --> 07:11.939 charges. 07:11.939 --> 07:15.849 Let's look at what charges do when you put them in a field. 07:15.850 --> 07:17.220 So I took two examples. 07:17.220 --> 07:19.680 One was a very simple example. 07:19.680 --> 07:22.210 These are two parallel plates. 07:22.209 --> 07:24.629 They are not two lines; they are plates coming out of 07:24.630 --> 07:25.410 the blackboard. 07:25.410 --> 07:29.290 They're filled with charge and this has got charge on it. 07:29.290 --> 07:31.010 This has got - charge on it. 07:31.009 --> 07:34.819 And therefore, the electric field will look 07:34.824 --> 07:36.554 like this, right? 07:36.550 --> 07:39.850 Because if you leave a test charge, it will go away from the 07:39.851 --> 07:41.701 positive, towards the negative. 07:41.699 --> 07:43.919 Then I said, suppose there really is a 07:43.916 --> 07:47.146 particle here with some velocity, v_0, 07:47.151 --> 07:48.351 what will it do? 07:48.350 --> 07:52.820 You can see that the force on this guy is going to be q 07:52.815 --> 07:54.055 times E. 07:54.060 --> 07:56.060 E is pointing down. 07:56.060 --> 07:59.130 If you divide by the mass, that's the acceleration, 07:59.125 --> 08:01.575 also pointing down, and it's constant. 08:01.579 --> 08:04.239 So that's like a particle in a gravitational field, 08:04.237 --> 08:06.627 except g is replaced by this number. 08:06.629 --> 08:11.499 So it will just curve like that, and you can calculate the 08:11.499 --> 08:12.609 trajectory. 08:12.610 --> 08:17.270 The final thing I did was, what happens when you put a 08:17.274 --> 08:19.744 dipole in a uniform field. 08:19.740 --> 08:21.890 Here as well, I think I was rushing near the 08:21.894 --> 08:24.654 end, and even I couldn't read my stuff in the corner. 08:24.649 --> 08:26.169 So I will go over that. 08:26.170 --> 08:28.150 If there's something that you didn't follow, 08:28.154 --> 08:30.514 then I will be happy to repeat that part for you. 08:30.509 --> 08:33.059 But you should understand what the question is. 08:33.058 --> 08:37.618 There is an electric field which is pointing like this, 08:37.619 --> 08:41.499 as if you have two plates here, charge is here, 08:41.501 --> 08:43.361 - charge is there. 08:43.360 --> 08:48.410 They're producing a constant electric field in the horizontal 08:48.408 --> 08:49.418 direction. 08:49.418 --> 08:54.828 In that environment, I take an electric dipole whose 08:54.828 --> 08:59.388 - charge and charge, q and -q, 08:59.390 --> 09:03.210 happen to be oriented like that. 09:03.210 --> 09:05.080 Question is, what will happen to this guy? 09:05.080 --> 09:07.820 If you want, you can imagine that it's a 09:07.816 --> 09:11.056 little massless stick, and one end you glue q 09:11.062 --> 09:13.642 coulombs, other end you glue -q 09:13.639 --> 09:15.949 coulombs, and your let it sit there. 09:15.950 --> 09:17.200 What will it do? 09:17.200 --> 09:20.070 First of all, it won't feel any net force, 09:20.070 --> 09:22.700 because the force in this direction is q times 09:22.702 --> 09:25.692 E and the force that direction is also q times 09:25.691 --> 09:27.181 E, if you want, 09:27.182 --> 09:30.252 it's -q times E and they cancel. 09:30.250 --> 09:32.240 But that doesn't mean it won't react. 09:32.240 --> 09:34.830 It will react, because you can all see 09:34.832 --> 09:38.622 intuitively, it's trying to straighten this guy out and 09:38.616 --> 09:40.856 applying a torque like that. 09:40.860 --> 09:42.550 You follow that? 09:42.549 --> 09:44.289 That's what it will do. 09:44.288 --> 09:49.078 And the way to find the torque, the torque is the product of 09:49.076 --> 09:53.536 the force and the distance between the point of rotation 09:53.538 --> 09:56.518 and the force, and the sine of the angle 09:56.523 --> 09:58.503 between them, that is to say, 09:58.495 --> 09:59.965 sine of this angle. 09:59.970 --> 10:03.790 What the sine of the angle does is to take the component of the 10:03.791 --> 10:06.011 force perpendicular to this axis, 10:06.009 --> 10:08.819 because if you resolve the force into that part and that 10:08.821 --> 10:10.781 part, this part is no good for 10:10.782 --> 10:11.432 rotation. 10:11.428 --> 10:13.868 That's trying to stretch the dipole along its own length. 10:13.870 --> 10:17.440 It's the perpendicular part that's going to rotate 10:17.443 --> 10:20.803 something, so you get that times sine theta. 10:20.798 --> 10:23.558 That you can write now as the vector equation 10:23.559 --> 10:26.929 p x E. 10:26.928 --> 10:31.408 Because p is equal to--I'm sorry. 10:31.409 --> 10:33.509 I need a 2 here. 10:33.509 --> 10:35.749 I forgot the 2, because this charge will have a 10:35.753 --> 10:38.883 torque and that charge will have a torque and the two torques are 10:38.875 --> 10:39.505 additive. 10:39.509 --> 10:41.239 They are both going the same way. 10:41.240 --> 10:44.650 Then 2q times a is p, and this is 10:44.645 --> 10:47.595 E and the sinθ comes in the 10:47.602 --> 10:48.762 cross product. 10:48.759 --> 10:51.799 I'm assuming all of you know about the cross product. 10:51.798 --> 10:56.708 Okay, final thing I did, which is, if you have a force, 10:56.714 --> 11:01.724 you can associate with that force a potential energy. 11:01.720 --> 11:05.970 Again, this is something you must have seen last time, 11:05.972 --> 11:07.902 but I will remind you. 11:07.899 --> 11:11.309 As long as it's not a frictional force, 11:11.312 --> 11:16.342 you can say the force is connected to potential energy in 11:16.344 --> 11:18.774 this following fashion. 11:18.769 --> 11:22.529 Or the potential energy at x minus potential energy 11:22.532 --> 11:25.702 at some starting point x_0 -- 11:25.700 --> 11:30.320 I'm sorry, x_0 - x is the integral of 11:30.320 --> 11:33.170 the force from x_0 to 11:33.168 --> 11:34.168 x. 11:34.168 --> 11:37.618 This is the relation between the force, as delivered to the 11:37.615 --> 11:41.175 potential, and the potential is the integral of the force. 11:41.178 --> 11:44.348 For example, for a spring, 11:48.534 --> 11:52.344 and F = -kx. 11:52.340 --> 11:55.210 If you go to this one, we tell you 11:55.207 --> 11:59.727 U(x_0) - U(x) is equal to the 11:59.729 --> 12:04.949 integral of -kx dx from x_0 to 12:04.945 --> 12:06.245 x. 12:06.250 --> 12:11.180 So that gives you kx_0^(2)/2 12:11.177 --> 12:14.087 − kx^(2)/2. 12:14.090 --> 12:20.150 And by comparison, you can see U(x) = 12:23.538 --> 12:25.468 Actually, this is not the unique answer. 12:25.470 --> 12:27.560 Do you know why? 12:27.558 --> 12:30.568 Given this formula, can I immediately say this guy 12:30.570 --> 12:33.520 corresponds to that, this one corresponds to that 12:33.518 --> 12:34.008 one? 12:34.009 --> 12:39.139 Is there some latitude here? 12:39.139 --> 12:40.809 Yes? 12:40.808 --> 12:41.928 Student: You can always add in a constant. 12:41.928 --> 12:44.198 Prof: You can add a constant to both, 12:44.198 --> 12:46.678 because if I said that, that certainly works. 12:46.678 --> 12:48.938 If I add 92 to both, it still works, 12:48.943 --> 12:52.313 because the 92 extra doesn't matter when you take the 12:52.308 --> 12:53.278 difference. 12:53.279 --> 12:57.859 So it's conventional to simply pick the constant so that the 12:57.863 --> 12:59.653 formula looks simple. 12:59.649 --> 13:03.879 Coming to other expression, if you had a torque, 13:03.879 --> 13:06.979 which is pEsinθ, 13:06.980 --> 13:09.910 you can ask--that's the torque--it's minus, 13:09.908 --> 13:12.338 because it's trying to reduce the angle θ-- 13:12.340 --> 13:14.760 you can ask what U leads to that. 13:14.759 --> 13:18.959 And you can see it's -pEcosθ. 13:18.960 --> 13:21.480 See that, take the -U and take the derivative, 13:21.476 --> 13:22.636 you'll get the torque. 13:22.639 --> 13:25.709 But if you got two vectors, p and E and you 13:25.714 --> 13:29.124 see the cosθ, I hope you guys know that you 13:29.116 --> 13:31.156 can write it as a dot product. 13:31.158 --> 13:37.048 So that's the end of what I did last time, okay? 13:37.048 --> 13:39.538 The potential energy is proportional to the dot product 13:39.538 --> 13:40.828 of p with E. 13:40.830 --> 13:45.170 The torque is equal to the cross product of p with 13:45.167 --> 13:46.017 E. 13:46.019 --> 13:49.369 And what does pEcosθ mean 13:49.366 --> 13:52.406 if you plot it as a function of angle? 13:52.409 --> 13:56.369 It will look like this. 13:56.370 --> 14:00.710 This is Π and this is 0. 14:00.710 --> 14:04.890 That means it likes to sit here and if you deviate a little bit 14:04.890 --> 14:08.130 and let it go there, it'll rattle back and forth, 14:08.129 --> 14:10.489 just like a mass spring system. 14:10.490 --> 14:14.720 In fact, you can very easily show, near the bottom of the 14:14.715 --> 14:18.025 well, the potential energy is proportional to 14:18.033 --> 14:19.773 θ^(2). 14:19.769 --> 14:23.239 That's because cosθ can be written 14:23.239 --> 14:25.649 as 1 − θ^(2)/2 14:25.653 --> 14:28.433 θ^(4)/4!, etc. 14:28.429 --> 14:30.189 for small angles. 14:30.190 --> 14:32.780 If you just keep that term, you will find it looks like 14:32.783 --> 14:33.123 this. 14:33.120 --> 14:37.590 Not very different from U = kx^(2). 14:37.590 --> 14:42.450 So what x does with forces, θ does with 14:42.445 --> 14:43.445 rotations. 14:43.450 --> 14:46.000 All right, so this is what we did last time. 14:46.000 --> 14:52.270 Now I'm going to do the new stuff. 14:52.269 --> 14:56.759 So new stuff is going to give you--I think it's useful, 14:56.763 --> 15:01.753 because it tells you the level at which you should be able to 15:01.754 --> 15:03.424 do calculations. 15:03.419 --> 15:06.599 So here's a typical problem. 15:06.600 --> 15:13.440 You have an infinite line of charge, of which I will show 15:13.436 --> 15:19.656 that part, and somebody has sprinkled on it λ 15:19.662 --> 15:22.472 coulombs per meter. 15:22.470 --> 15:25.280 So it is not a discrete set of charges; 15:25.278 --> 15:29.068 it's assumed to be continuous and it's everywhere. 15:29.070 --> 15:32.450 I'm just showing a few of them, and if you cut out one meter, 15:32.448 --> 15:35.038 you'll find there's λ coulombs there. 15:35.038 --> 15:38.948 And you want to compute the electric field you will get due 15:38.950 --> 15:41.310 to this distribution, everywhere. 15:41.308 --> 15:45.248 So you want to go somewhere here and ask what's the electric 15:45.245 --> 15:45.775 field. 15:45.779 --> 15:51.849 That's what we're going to do. 15:51.850 --> 15:52.960 Let's go here. 15:52.960 --> 15:54.030 You will see why. 15:54.029 --> 15:58.379 Now first of all, you've got to have an intuition 15:58.379 --> 16:02.459 on which way the electric field will point. 16:02.460 --> 16:05.320 You have a feeling? 16:05.320 --> 16:06.200 Yes. 16:06.200 --> 16:09.270 It will point here, this way. 16:09.269 --> 16:13.399 Why not like that? 16:13.399 --> 16:14.209 Yeah? 16:14.210 --> 16:20.000 Student: > 16:20.000 --> 16:21.040 Prof: Okay. 16:21.038 --> 16:23.168 She said the horizontal parts will cancel. 16:23.169 --> 16:24.589 That's correct. 16:24.590 --> 16:28.640 Another argument from symmetry is that if anybody can give you 16:28.636 --> 16:31.486 a reason why it should tilt to the left, 16:31.490 --> 16:33.910 I can say, "Why don't you use the same argument to say it 16:33.909 --> 16:35.139 will tilt to the right?" 16:35.139 --> 16:38.249 Because this is an infinitely long wire and things look the 16:38.245 --> 16:41.455 same if you look to the left and if you look to the right. 16:41.460 --> 16:43.790 And the field you get should have the same property. 16:43.788 --> 16:46.708 If this was a finite wire, I wouldn't say that, 16:46.714 --> 16:49.514 because in a finite wire, that can be a tilt, 16:49.511 --> 16:50.721 somewhere here. 16:50.720 --> 16:53.360 But infinite wire, it cannot tilt to the left or 16:53.356 --> 16:55.826 to the right, because each point has the same 16:55.826 --> 16:58.796 symmetric situation to its right and to its left. 16:58.799 --> 17:00.849 In a finite wire, it's not true. 17:00.850 --> 17:03.510 Life to the left is different from the life to the right, 17:03.513 --> 17:05.893 but for infinite wire, you know it cannot be biased 17:05.891 --> 17:07.081 one way or the other. 17:07.079 --> 17:08.429 It's got to go straight up. 17:08.430 --> 17:11.570 Secondly, you can find the field here, here, 17:11.567 --> 17:16.087 here or at anywhere at the same distance, you've got to get the 17:16.092 --> 17:17.262 same answer. 17:17.259 --> 17:19.449 Again, because if you move two inches to the right, 17:19.450 --> 17:21.860 it doesn't make any difference with an infinite wire. 17:21.858 --> 17:25.568 You've still got infinite wire on either side. 17:25.568 --> 17:29.008 So we'll pick a typical point and calculate the field and we 17:29.005 --> 17:32.495 know that answer is going to be good throughout that line. 17:32.500 --> 17:35.140 So now I take this point here. 17:35.140 --> 17:36.870 I want the field here. 17:36.868 --> 17:38.738 I'm going to make that my origin. 17:38.740 --> 17:45.200 Then I take a piece of wire of length dx. 17:45.200 --> 17:51.410 dx is so small that I can treat it as a point. 17:51.410 --> 17:54.180 Now the dx I've drawn is not a point, but in the end, 17:54.182 --> 17:56.962 we're going to make dx arbitrarily small so that it's 17:56.957 --> 17:57.707 good enough. 17:57.710 --> 18:03.080 It's like a point and it is at a distance x from the 18:03.079 --> 18:07.709 origin, and let's say that distance is a. 18:07.710 --> 18:10.320 So let's find the field due to just this guy, 18:10.321 --> 18:11.511 the shaded region. 18:11.509 --> 18:12.529 Think of a charge. 18:12.529 --> 18:14.699 How much charge is sitting here? 18:14.700 --> 18:17.110 I hope you all agree, the charge sitting in this 18:17.113 --> 18:19.273 region is just λ times dx. 18:19.269 --> 18:21.399 That's just the definition. 18:21.400 --> 18:25.080 If you've got that many coulombs, imagine a test charge 18:25.080 --> 18:25.900 of 1 here. 18:25.900 --> 18:29.360 Well, it will push it this way, and the field due to that, 18:29.355 --> 18:31.415 I'm going to draw as infinitesimal, 18:31.415 --> 18:33.715 so I'm going to call it dE. 18:33.720 --> 18:37.510 You can call it E, but dE is to remind you, 18:37.510 --> 18:40.150 it's a tiny field, due to a tiny section 18:40.152 --> 18:41.102 dx. 18:41.098 --> 18:44.778 Now that electric field is biased to the left, 18:44.779 --> 18:47.129 but for every such section you find here, 18:47.130 --> 18:50.380 I'll find a section on the other side that's precisely 18:50.376 --> 18:52.886 biased to the right by the same amount. 18:52.890 --> 18:56.060 Therefore the only part that's going to survive due to this 18:56.057 --> 18:57.717 guy, combined with this, 18:57.720 --> 19:05.090 will be the portion here, which is the vertical part of 19:05.085 --> 19:07.375 that force. 19:07.380 --> 19:11.000 So let me find the contribution first, only from this one, 19:10.998 --> 19:14.108 then we will add the contribution for this one. 19:14.108 --> 19:16.718 For that one, you just find the vertical 19:16.720 --> 19:17.590 projection. 19:17.589 --> 19:19.469 So how much is that? 19:19.470 --> 19:23.000 Remember, the Coulomb's law for the electric field is 19:23.002 --> 19:26.472 q/4Πε _0r^(2). 19:26.470 --> 19:28.360 So this is the q. 19:28.358 --> 19:31.338 That's the 4Πε _0. 19:31.338 --> 19:38.318 r^(2) is that distance squared, which is x^(2) 19:38.324 --> 19:40.104 a^(2). 19:40.098 --> 19:44.368 That's really like the field of a point charge at that distance. 19:44.368 --> 19:48.328 But now this is the magnitude of the electric field vector at 19:48.326 --> 19:52.476 this angle, but I want the part along the y direction. 19:52.480 --> 19:56.520 So I've got to take cosine of that θ. 19:56.519 --> 19:57.379 You guys follow that? 19:57.380 --> 19:59.600 If you took this vector, this part is 19:59.599 --> 20:01.389 dEcosθ. 20:01.390 --> 20:03.290 That part is dEsinθ. 20:03.288 --> 20:05.588 But that angle and that angle are equal, 20:05.588 --> 20:08.018 and cosθ for this triangle, 20:08.019 --> 20:13.969 you can see, is a divided by 20:21.140 --> 20:25.310 So this is the electric field in the y direction. 20:25.308 --> 20:27.948 I'm going to call it dE, 20:27.945 --> 20:32.075 in the y direction, due to the segment dx. 20:32.078 --> 20:35.368 The total electric field is obtained by adding all the 20:35.368 --> 20:39.158 dEs or adding all the contributions from all the 20:39.155 --> 20:40.765 segments on this line. 20:40.769 --> 20:51.379 And that goes from - infinity to infinity. 20:51.380 --> 20:57.480 All right, so now it's a matter of just doing this integral. 20:57.480 --> 21:02.520 So this gives me (λa/4 21:02.521 --> 21:07.281 Πε _0) 21:07.277 --> 21:22.257 dx/(x^(2) a^(2))^(3/2), integrated from - to infinity. 21:22.259 --> 21:26.479 Now I can make life a little easier by saying that this 21:26.484 --> 21:29.854 function is an even function of x. 21:29.848 --> 21:32.768 That means when you change x to -x, 21:32.772 --> 21:33.872 it doesn't care. 21:33.868 --> 21:37.098 Therefore the contribution from a positive x region is 21:37.098 --> 21:40.488 the same as the contribution from a negative x region. 21:40.490 --> 21:42.320 That also makes sense in this picture here, 21:42.318 --> 21:44.958 because if you look at the field I'm computing, 21:44.960 --> 21:47.980 this section and this section give equal contributions in the 21:47.981 --> 21:49.041 y direction. 21:49.038 --> 21:51.168 But even if you did not know any of that background, 21:51.170 --> 21:53.320 as a mathematician, if you see this integral, 21:53.318 --> 21:55.668 you would say, "Hey, put a 0 there and 21:55.671 --> 21:58.361 put a 2 here" namely,integrate over half the 21:58.357 --> 22:00.377 region, because the second half is 22:00.377 --> 22:01.827 giving you the same answer. 22:01.828 --> 22:05.548 So you double the integral, but cut the region of 22:05.553 --> 22:07.263 integration in half. 22:07.259 --> 22:11.369 So at this point, you are free to look up a book, 22:11.366 --> 22:15.556 if it was an exam, but maybe not even if it was an 22:15.560 --> 22:17.530 exam at your level. 22:17.528 --> 22:22.228 You should be able to do this integral. 22:22.230 --> 22:27.770 So integrals have been around from the time of Newton and the 22:27.769 --> 22:32.979 question of an integral is, find the area of some graph 22:32.978 --> 22:36.408 with this particular functional form. 22:36.410 --> 22:39.440 And the answer to any integral is that function whose 22:39.441 --> 22:41.191 derivative is the integrand. 22:41.190 --> 22:45.030 So what you have to do is guess many answers until you get the 22:45.032 --> 22:45.792 right one. 22:45.788 --> 22:48.288 But people have been guessing for hundreds of years, 22:48.288 --> 22:51.078 and there's big tables of integrals with all the integrals 22:51.082 --> 22:51.722 you want. 22:51.720 --> 22:55.110 But you should still be able to do some integrals from scratch 22:55.107 --> 22:57.717 and I'm going to tell you how to do this one. 22:57.720 --> 23:01.150 But before you do the integral, you've got to have some idea 23:01.145 --> 23:03.465 what the answer is going to look like. 23:03.470 --> 23:07.610 I want you to get some feeling about this. 23:07.608 --> 23:10.638 Answer depends on what, is the first question. 23:10.640 --> 23:14.230 What's the answer going to depend on? 23:14.230 --> 23:15.490 Student: a. 23:15.490 --> 23:17.610 Prof: a, you understand? 23:17.608 --> 23:20.988 Whatever this is depends only on a because 0 and 23:20.994 --> 23:24.824 infinity are not going to be present as part of the answer. 23:24.818 --> 23:27.148 If the lower limit was 5, it can depend on 5, 23:27.145 --> 23:29.835 but it doesn't depend on any other thing, other than 23:29.842 --> 23:30.532 a. 23:30.528 --> 23:34.668 Then from dimension analysis, I got a length squared to the 23:34.670 --> 23:36.140 3/2, that's length cubed, 23:36.138 --> 23:39.288 and a length on the top, so whole thing should look like 23:39.288 --> 23:41.178 something over length squared. 23:41.180 --> 23:45.030 The only length I have is a, so it's going to look 23:45.025 --> 23:47.905 like 1 over a^(2) times a number. 23:47.910 --> 23:51.830 Once you got the number, you're done. 23:51.828 --> 23:55.008 So I'm going to do all the work now to show you that the number 23:55.011 --> 23:56.091 is actually just 1. 23:56.088 --> 24:00.248 This number will turn out to be 1, in which case, 24:00.253 --> 24:04.073 you will find it's λ/2Πε 24:04.070 --> 24:05.980 _0a. 24:05.980 --> 24:10.870 Well, let's see how we get the number to be 1. 24:10.868 --> 24:15.918 So does anybody know what trick you use to do this integral? 24:15.920 --> 24:19.460 This is whatever, math 120 or-- yes? 24:19.460 --> 24:20.670 Student: Use substitution? 24:20.670 --> 24:21.340 Prof: Yes. 24:21.339 --> 24:22.629 What substitution? 24:22.630 --> 24:24.730 Student: x^(2) a^(2) = U. 24:24.730 --> 24:27.610 Prof: That won't help you. 24:27.609 --> 24:28.869 Yes? 24:28.868 --> 24:30.378 Student: Can you use trigonometric substitution? 24:30.380 --> 24:31.050 Prof: Yes. 24:31.049 --> 24:32.259 Trigonometric substitution. 24:32.259 --> 24:37.859 Which one? 24:37.858 --> 24:41.628 Okay, look--no, no, I don't blame you. 24:41.630 --> 24:44.190 I know the answer because I've seen it, but if I have to work 24:44.190 --> 24:46.410 on it, I'll try for a while before I got it right. 24:46.410 --> 24:48.800 The whole idea is, we don't like all these 3 24:48.804 --> 24:49.644 _______ here. 24:49.640 --> 24:51.190 We want to turn that into something nice. 24:51.190 --> 24:52.700 So I'll tell you what the answer is. 24:52.700 --> 24:55.740 You can all marvel at how wonderfully it works. 24:55.740 --> 25:00.210 So what we are going to do is to introduce an angle 25:00.205 --> 25:05.565 theta--nothing to do with the angle in the problem--so that x 25:05.566 --> 25:07.706 = tanθ. 25:07.710 --> 25:10.270 That means instead of going over all values of x, 25:10.269 --> 25:13.319 I'll go with the suitable values of θ-- 25:13.318 --> 25:16.118 I'm sorry, this would not even be correct dimensionally. 25:16.118 --> 25:20.638 x = atanθ. 25:20.640 --> 25:22.420 You can see that every x that I want, 25:22.420 --> 25:25.440 I can get by some choice of theta, because tanθ 25:25.439 --> 25:28.359 goes from 0 to infinity when θ goes from 0 to 25:28.356 --> 25:28.916 Π/2. 25:28.920 --> 25:31.110 You cannot say x = acosθ, 25:31.108 --> 25:31.708 for example. 25:31.710 --> 25:33.030 You are doomed. 25:33.029 --> 25:34.379 If x is acosθ, 25:34.380 --> 25:36.400 the biggest x you can get is a, 25:36.400 --> 25:38.780 whereas I want this x to go from 0 to infinity. 25:38.779 --> 25:40.129 So when you make the change of variables, 25:40.130 --> 25:42.310 you've got to make sure that for every x you want, 25:42.308 --> 25:44.858 there is some θ that will do it. 25:44.858 --> 25:47.378 Then the next thing you do, you say 25:47.375 --> 25:51.445 dx/dθ = a times derivative of 25:51.446 --> 25:55.366 tanθ which is sec^(2)θ. 25:55.368 --> 26:01.618 Then you write that as dx = that. 26:01.618 --> 26:04.948 What that means is an integral dx is related to an 26:04.951 --> 26:07.451 integral dθ by this factor. 26:07.450 --> 26:09.720 Therefore going to the integration here, 26:09.720 --> 26:13.360 I'm just doing that part, which is going to be a 26:13.355 --> 26:16.445 sec^(2)θ dθ. 26:16.450 --> 26:19.740 θ goes from 0 to Π/2. 26:19.740 --> 26:21.840 Now let's look downstairs. 26:21.838 --> 26:25.178 Downstairs I've got x^(2) a^(2). 26:25.180 --> 26:29.020 x itself is atanθ, 26:29.015 --> 26:33.195 therefore a^(2) times 1 tan^(2)θ. 26:33.200 --> 26:38.210 1 tan^(2)θ happens to be sec^(2)θ. 26:38.210 --> 26:44.180 That to the power 3/2, which is what I want, 26:44.180 --> 26:48.480 will give me an a^(3)sec^ 26:48.484 --> 26:51.544 (3)θ. 26:51.539 --> 26:52.539 So what do we get? 26:52.538 --> 26:56.598 You can see as promised I get a 1/a^(2) and I get integral of 26:56.602 --> 26:59.352 1/secθ, which is 26:59.348 --> 27:05.888 cosθdθ from 0 to Π/2. 27:05.890 --> 27:08.900 Yes. 27:08.900 --> 27:13.400 And integral of cosθ is 27:13.395 --> 27:18.145 sinθ from Π/2 to 0. 27:18.150 --> 27:21.490 That just happens to be 1. 27:21.490 --> 27:26.760 So the final answer is what I gave you here, 27:26.759 --> 27:28.589 E. 27:28.589 --> 27:29.529 Well, E is the vector. 27:29.528 --> 27:32.688 I've just shown you the magnitude, but we've all agreed 27:32.692 --> 27:34.102 what the direction is. 27:34.098 --> 27:38.658 The direction is away from the wire. 27:38.660 --> 27:45.770 So if you like, if you look at this wire from 27:45.771 --> 27:52.401 the end, the lines will look like this. 27:52.400 --> 27:55.470 If the infinite wire is coming out of the blackboard towards 27:55.474 --> 27:58.194 you and you look at it this way, if you go too close, 27:58.185 --> 27:59.745 you'll poke your eyes out. 27:59.750 --> 28:02.660 Look from here, you'll see the lines are going 28:02.657 --> 28:04.207 out radial everywhere. 28:04.210 --> 28:06.410 The question is, how do the fields get weak? 28:06.410 --> 28:07.940 How does it weaken with distance? 28:07.940 --> 28:12.440 It weakens like 1/a. 28:12.440 --> 28:14.110 That's a big of a surprise, right? 28:14.108 --> 28:17.478 The field away from the wire doesn't fall like 1 over 28:17.481 --> 28:20.791 distance squared, but falls like 1 over distance. 28:20.788 --> 28:23.858 The reason is that every individual portion of the wire 28:23.864 --> 28:26.884 has a contribution that does fall like 1 over distance 28:26.882 --> 28:29.162 squared, but it is an infinite wire. 28:29.160 --> 28:32.340 When you add it all up, the net answer goes like 1 over 28:32.336 --> 28:33.216 the distance. 28:33.220 --> 28:35.780 The field away from a wire falls like the distance from the 28:35.778 --> 28:38.418 wire, on the perpendicular from the wire and there's pointing 28:38.424 --> 28:39.444 away from the wire. 28:39.440 --> 28:43.150 That's it. 28:43.150 --> 28:49.730 Okay, so that's one calculation. 28:49.730 --> 28:53.650 Then I'm going to do one more and that's going to be the end 28:53.648 --> 28:55.508 of the tough calculations. 28:55.509 --> 29:04.149 Second calculation is going to be an infinite sheet. 29:04.150 --> 29:08.320 On the infinite sheet, the appropriate quantity is 29:08.315 --> 29:12.815 called the charge density, which is coulombs per meter 29:12.818 --> 29:13.838 squared. 29:13.838 --> 29:16.538 That means if you cut out a tiny piece, the charge on it 29:16.535 --> 29:18.735 will be sigma times the area of that piece. 29:18.740 --> 29:22.150 So there is positive charge everywhere here, 29:22.150 --> 29:26.830 and the number of coulombs per unit area is called sigma. 29:26.829 --> 29:28.009 These are standard. 29:28.009 --> 29:32.339 λ is coulombs per meter, σ is used for coulombs 29:32.336 --> 29:33.506 per unit area. 29:33.509 --> 29:37.949 The question is, what's going to be the electric 29:37.948 --> 29:42.008 field at some point away from that plane? 29:42.009 --> 29:46.349 Once again, I think we can all agree that the electric field at 29:46.348 --> 29:50.618 some point from the plane will not depend on where in front of 29:50.618 --> 29:52.788 the plane you are standing. 29:52.788 --> 29:55.128 Are you standing here or are you standing there? 29:55.130 --> 29:57.780 It doesn't matter, because it's an infinite plane. 29:57.779 --> 30:01.539 If I moved 1 inch--I'll tell you why it won't matter. 30:01.538 --> 30:04.458 If I moved 1 inch and the answer changed, 30:04.458 --> 30:09.058 I should get the same change if I didn't move and somebody moved 30:09.056 --> 30:11.606 the sheet 1 inch the other way. 30:11.608 --> 30:14.548 But when I move an infinite sheet the other way by 1 inch, 30:14.553 --> 30:16.003 it looks exactly the same. 30:16.000 --> 30:19.060 It's got to produce exactly the same field. 30:19.058 --> 30:22.028 So you can always ask, what will happen if I move to 30:22.025 --> 30:25.685 the left, the same as what will happen if the sheet moves to the 30:25.688 --> 30:26.268 right? 30:26.269 --> 30:28.429 The sheet moving to the right looks exactly like the sheet 30:28.429 --> 30:28.769 before. 30:28.769 --> 30:31.229 The answer won't change, therefore the answer won't 30:31.230 --> 30:33.250 change for you if you move to the left. 30:33.250 --> 30:35.290 I've got infinite plane below you. 30:35.288 --> 30:38.228 As long as you don't change the distance from the plane, 30:38.232 --> 30:41.282 you navigate perpendicular to it, no matter where you are, 30:41.282 --> 30:43.052 you will get the same answer. 30:43.048 --> 30:46.778 Same answer, meaning same direction of the 30:46.777 --> 30:48.957 field, same magnitude. 30:48.960 --> 30:53.170 And that direction has to be perpendicular to the plane, 30:53.170 --> 30:55.390 again for symmetry reasons. 30:55.390 --> 30:59.080 If you tilt it in any one direction, you have no reason to 30:59.080 --> 30:59.600 do it. 30:59.598 --> 31:02.668 For example, if you tilt it this way, 31:02.670 --> 31:05.010 I can take the infinite plane and rotate it, 31:05.009 --> 31:06.909 then the tilt will be in some other direction, 31:06.910 --> 31:09.790 maybe like that, but the rotated infinite plane 31:09.786 --> 31:10.846 looks the same. 31:10.848 --> 31:13.138 In other words, if the cause does not change, 31:13.136 --> 31:14.746 the effect should not change. 31:14.750 --> 31:18.600 If I can do certain things to the infinite plane that leave it 31:18.598 --> 31:20.578 invariant, then I can do the same 31:20.575 --> 31:22.985 transformation to the location of the point, 31:22.990 --> 31:26.040 and that shouldn't have a different answer. 31:26.038 --> 31:28.778 So the plane has the property that when you slide it up and 31:28.776 --> 31:31.526 down parallel to itself, or twist it and turn it, 31:31.532 --> 31:34.372 it looks the same, therefore the field pattern 31:34.365 --> 31:35.955 should have that property. 31:35.960 --> 31:39.130 Therefore the field has to be the same at all distances from 31:39.134 --> 31:42.474 the plane anywhere on top of the plane, and it's going to point 31:42.470 --> 31:43.170 this way. 31:43.170 --> 31:45.900 But you can also find out in a minute--by the way, 31:45.900 --> 31:48.520 you don't need any of the symmetry arguments. 31:48.519 --> 31:51.749 You just do the calculation by brute force, it will have these 31:51.750 --> 31:52.440 properties. 31:52.440 --> 31:55.290 But it's good to know what to anticipate, because maybe you 31:55.285 --> 31:56.605 made a mistake somewhere. 31:56.608 --> 31:58.378 It's good to know some broad features. 31:58.380 --> 32:01.930 So none of this is needed to calculate, even in that problem. 32:01.930 --> 32:04.800 Go ahead and find the electric field not where I found it, 32:04.799 --> 32:06.209 but 2 inches to the right. 32:06.210 --> 32:08.690 You'll find the answer looks the same. 32:08.690 --> 32:11.830 So those symmetry properties will come out of the wash, 32:11.828 --> 32:14.588 but it's good for you to anticipate that, 32:14.588 --> 32:16.758 and that's where you should look at the symmetry of the 32:16.758 --> 32:17.118 source. 32:17.118 --> 32:19.338 For example, the source was a ball of 32:19.336 --> 32:19.886 charge. 32:19.890 --> 32:22.830 You know if you rotate the ball, when I'm not looking to 32:22.834 --> 32:25.354 rotate the ball, it's going to look the same. 32:25.348 --> 32:27.718 That means the field pattern should have the property, 32:27.723 --> 32:29.383 when you rotate it, it looks the same, 32:29.381 --> 32:31.981 because the same cause should produce the same effect. 32:31.980 --> 32:36.020 Anyway, going to this problem now, let's find the electric 32:36.018 --> 32:36.938 field here. 32:36.940 --> 32:40.370 Okay, now this is going to be a stretch for me to draw, 32:40.368 --> 32:42.788 so I'm going to try, but you'll have to go look at 32:42.791 --> 32:45.711 some textbook if you want a really nice looking picture, 32:45.710 --> 32:48.180 but this is the best I can do. 32:48.180 --> 32:54.800 I take a ring of radius r and thickness 32:54.795 --> 32:56.555 dr. 32:56.558 --> 33:08.418 I take an annulus, and I ask, what will that ring 33:08.423 --> 33:13.123 do to this point? 33:13.118 --> 33:19.908 So let's take a tiny part of that ring, this guy. 33:19.910 --> 33:22.050 Well, for that, you just did what you did, 33:22.053 --> 33:23.363 you draw the line here. 33:23.358 --> 33:29.558 You'll produce a dE that looks like this. 33:29.559 --> 33:32.829 What is its magnitude? 33:32.828 --> 33:36.328 Magnitude is just given by Coulomb's law. 33:36.328 --> 33:39.818 The q there is sigma times the tiny area, 33:39.818 --> 33:40.708 dA. 33:40.710 --> 33:46.230 Let's call this dA. 33:46.230 --> 33:48.910 dA is the name for a small area. 33:48.910 --> 33:51.460 σ times dA is the name for a small charge. 33:51.460 --> 33:54.380 That charge will produce a force, 33:54.377 --> 33:59.117 1/4Πε_0, square of the distance, 33:59.119 --> 34:00.579 r^(2) a^(2). 34:00.578 --> 34:03.168 Finally, here is where the symmetry comes in, 34:03.170 --> 34:05.410 can you see that for every section here, 34:05.410 --> 34:09.710 I can find an opposite section that will cancel everything but 34:09.710 --> 34:12.390 the part perpendicular to the plane? 34:12.389 --> 34:15.179 So I should only keep this portion of it. 34:15.179 --> 34:18.409 Namely, I should take the cosθ. 34:18.409 --> 34:21.669 The cosine of that θ is the distance 34:21.672 --> 34:27.992 a, just like in the other problem, 34:35.449 --> 34:37.039 This is now dE. 34:37.039 --> 34:38.729 If you want, you can put this following 34:38.733 --> 34:40.163 symbol, dE_perp, 34:40.161 --> 34:41.901 meaning perpendicular to the plane. 34:41.900 --> 34:42.840 Yes? 34:42.840 --> 34:46.990 Student: Do you need to multiply by 2 again, 34:46.990 --> 34:49.270 because you're __________? 34:49.269 --> 34:51.499 Prof: Let's be careful. 34:51.500 --> 34:53.810 Her question was, should I multiply by 2, 34:53.813 --> 34:55.783 because of this guy here, right? 34:55.780 --> 34:58.870 In fact, I should multiply by all kinds of other numbers, 34:58.865 --> 35:01.285 because so far, I've found the field only due 35:01.291 --> 35:02.671 to this segment here. 35:02.670 --> 35:07.480 I've got to add the field due to that and that and that and so 35:07.483 --> 35:08.433 on, right? 35:08.429 --> 35:10.429 What will that contribution be? 35:10.429 --> 35:13.189 For every one of them, this factor, 35:13.186 --> 35:17.076 (r^(2) a^(2))^(3/2) is the same. 35:17.079 --> 35:20.919 They all contribute to the same factor, so when I added the 35:20.920 --> 35:23.770 shaded region, I'll just get the area of the 35:23.768 --> 35:24.958 shaded region. 35:24.960 --> 35:28.800 All these dAs, if you add them up, 35:28.800 --> 35:30.530 what will I get? 35:30.530 --> 35:34.170 It will be sigma over 4Πε 35:34.170 --> 35:35.990 _0. 35:35.989 --> 35:40.329 Now you've got to ask yourself, what's the area of an annulus 35:40.331 --> 35:43.591 of radius r and thickness dr? 35:43.590 --> 35:46.440 So take that annulus, take a pair of scissors and you 35:46.438 --> 35:49.668 cut it, and you stretch it out like that, it's going to look 35:49.672 --> 35:50.442 like this. 35:50.440 --> 35:55.680 This is dr and this is 2Πr. 35:55.679 --> 35:58.559 So the area of an annulus is just 2Πr dr. 35:58.559 --> 36:05.189 So the sum of all these areas is 2Πr dr and then 36:05.192 --> 36:11.012 I've got here (r^(2) a^(2))^(3/2). 36:11.010 --> 36:15.080 But now this is the dE, due to annulus of thickness of 36:15.076 --> 36:15.886 dr. 36:15.889 --> 36:20.069 Then I've got to integrate over all values of r, 36:20.070 --> 36:23.090 but r goes from 0 to infinity. 36:23.090 --> 36:31.570 So I have here σ/2ε_0 36:31.572 --> 36:37.772 times rdr, divided by (r^(2) 36:37.771 --> 36:43.391 a^(2))^(3/2), 0 to infinity. 36:43.389 --> 36:45.639 Student: > 36:45.639 --> 36:47.629 Prof: Did I miss a pi? 36:47.630 --> 36:51.300 Student: Shouldn't it be sigma over ____ pi? 36:51.300 --> 36:52.630 Prof: There is a 2 pi here. 36:52.630 --> 36:53.050 Student: Oh, okay, yeah. 36:53.050 --> 36:59.050 Prof: 2ε_0. 36:59.050 --> 37:00.930 So do you understand what I did? 37:00.929 --> 37:05.899 I broke the plane into concentric rings and I took one 37:05.902 --> 37:09.752 ring and looking head on at that ring, 37:09.750 --> 37:13.740 I took a portion of that ring and see what field is produces. 37:13.739 --> 37:16.359 And I know that even though the field due to that is at an 37:16.356 --> 37:18.116 angle, the only part that's going to 37:18.121 --> 37:20.361 remain is the part perpendicular to the plane, 37:20.360 --> 37:23.540 because the counterpart to this one on the other side will 37:23.543 --> 37:26.953 produce a similar field with the opposite angle here that will 37:26.949 --> 37:29.759 cancel, so only the part perpendicular 37:29.760 --> 37:30.760 will survive. 37:30.760 --> 37:34.040 Then I found out that the contribution from every 37:34.041 --> 37:36.641 dA had exactly these factors. 37:36.639 --> 37:38.649 They all had the same r and they all had the same 37:38.650 --> 37:40.910 a, so some of all the dAs, 37:40.905 --> 37:43.685 all I have to add is 2Πr dr. 37:43.690 --> 37:46.390 And that's the contribution from this annulus, 37:46.385 --> 37:49.495 then I still have to look at annulus of every radius, 37:49.501 --> 37:51.661 so that's the integral over dr. 37:51.659 --> 37:52.849 Yes? 37:52.849 --> 37:55.029 Student: What happened to the a _______? 37:55.030 --> 37:56.690 I thought it was a over-- 37:56.690 --> 37:57.690 Prof: Oh, I'm sorry. 37:57.690 --> 37:58.540 It's there. 37:58.539 --> 37:59.879 Thank you. 37:59.880 --> 38:08.450 There is an a still here. 38:08.449 --> 38:10.479 Yeah, I would have caught that guy in a while, 38:10.481 --> 38:12.291 but I'm always happy when you do that. 38:12.289 --> 38:13.819 That's correct. 38:13.820 --> 38:17.740 Okay, so now about how this integral. 38:17.739 --> 38:24.169 Do you have any idea what you might do now? 38:24.170 --> 38:24.810 Yes? 38:24.809 --> 38:26.089 Student: Use substitution. 38:26.090 --> 38:26.590 Prof: Right. 38:26.590 --> 38:27.090 What substitution? 38:27.090 --> 38:30.390 "Use substitution" is a pretty safe answer, 38:30.387 --> 38:33.177 but you've got to go a little beyond that. 38:33.179 --> 38:37.359 Student: Substitute r^(2) a^(2) for 38:37.360 --> 38:38.480 the ________. 38:38.480 --> 38:39.170 Prof: Yes. 38:39.170 --> 38:43.490 You can do that in this problem, because there's an r 38:43.485 --> 38:44.815 on the top. 38:44.820 --> 38:46.640 If you didn't have the r, you couldn't do that, 38:46.641 --> 38:47.261 but now you can. 38:47.260 --> 38:48.400 I'll tell you how it works. 38:48.400 --> 38:50.930 First of all, you can always do that 38:50.925 --> 38:54.025 tanθ substitution even here. 38:54.030 --> 38:55.350 It will always work. 38:55.349 --> 38:57.279 The tanθ substitution, 38:57.284 --> 38:59.624 if you put it here, it will still work. 38:59.619 --> 39:02.169 You can go home and verify that, but I will do it a 39:02.170 --> 39:03.190 different way now. 39:03.190 --> 39:08.850 I will say, let w = r^(2), 39:08.851 --> 39:14.951 then dw is equal to 2r dr. 39:14.949 --> 39:17.999 So if I come here, I can write it as 39:18.000 --> 39:21.660 aσ/ 4ε_0. 39:21.659 --> 39:25.429 I borrow a 2 top and bottom to make it dw. 39:25.429 --> 39:31.969 w also goes from 0 to infinity, but now I get 39:31.965 --> 39:35.165 (w a^(2))^(3/2). 39:35.170 --> 39:39.760 Now this is simple integral, dx/x some number 39:39.757 --> 39:44.337 to some power is x^(n 1)/(n 1), but n is now -3 over 39:44.344 --> 39:44.824 2. 39:44.820 --> 39:48.790 So you get aσ/ 4ε_0, 39:58.510 --> 40:05.130 and that goes from infinity to 0. 40:05.130 --> 40:07.220 So I'm not going to do this much slower than this. 40:07.219 --> 40:10.009 This is the kind of integral that you can see right away, 40:10.007 --> 40:12.047 or you can go and work out the details. 40:12.050 --> 40:14.510 This is something you should do. 40:14.510 --> 40:16.130 If you have trouble with such integrals, 40:16.130 --> 40:18.410 then you should work harder than people who don't have 40:18.407 --> 40:22.337 trouble with such integrals, because you should be able to 40:26.940 --> 40:30.850 and when it comes upstairs, it becomes -2. 40:30.849 --> 40:34.069 Now if you look at this integral, in the upper limit 40:34.067 --> 40:36.647 omega's infinity, you get 1 over infinity, 40:36.652 --> 40:37.602 which is 0. 40:37.599 --> 40:41.139 The lower limit when omega is 0, you get 1/a, 40:41.139 --> 40:43.109 and that will cancel the a here, 40:43.110 --> 40:49.120 and you will get σ/2ε_0. 40:49.119 --> 40:52.029 So that's the final answer. 40:52.030 --> 40:57.270 So the electric field of this infinite plane, 40:57.271 --> 41:03.351 if you look at it from the side, looks like this. 41:03.349 --> 41:08.299 The σ/2ε_0. 41:08.300 --> 41:13.180 So what do you notice about this one that's interesting? 41:13.179 --> 41:14.719 Student: It doesn't depend on the distance. 41:14.719 --> 41:16.849 Prof: It doesn't fall with distance. 41:16.849 --> 41:19.999 No matter how far you go from this infinite plane, 41:19.998 --> 41:21.538 the field is the same. 41:21.539 --> 41:24.749 Again, each part of it makes a contribution that falls like 1 41:24.753 --> 41:26.043 over distance squared. 41:26.039 --> 41:29.499 As you go further and further out, you might think the field 41:29.496 --> 41:31.076 should get weaker, right? 41:31.079 --> 41:32.379 How could it not get weaker? 41:32.380 --> 41:34.710 They're moving away from everything. 41:34.710 --> 41:37.530 At least with the line charge, it didn't go weaker like 41:37.530 --> 41:39.620 1/r^(2), but it did get weaker. 41:39.619 --> 41:43.039 How can you go further and further from a plane? 41:43.039 --> 41:45.929 You are going further away from everybody? 41:45.929 --> 41:49.569 How could it not matter? 41:49.570 --> 41:54.740 Yes, any ideas? 41:54.739 --> 41:56.599 For example, if you go very close to the 41:56.599 --> 41:57.649 plane, what happens? 41:57.650 --> 42:02.910 If you go really close to the plane what happens is, 42:02.913 --> 42:07.973 the field in each section here looks like this. 42:07.969 --> 42:12.439 Therefore the part that's useful is very small. 42:12.440 --> 42:15.630 If you go further away, you get things like that. 42:15.630 --> 42:18.790 Maybe it's a little weaker, but the part that's useful, 42:18.791 --> 42:20.551 this one, is getting bigger. 42:20.550 --> 42:23.950 So by magic, these tendencies cancel in the 42:23.952 --> 42:24.442 end. 42:24.440 --> 42:26.000 It doesn't depend on distance. 42:26.000 --> 42:28.050 Now unless you do the integral, you will not know it doesn't 42:28.054 --> 42:30.104 depend on the distance, because you can give arguments 42:30.101 --> 42:32.731 for why it'll get weaker, arguments for why it'll get 42:32.728 --> 42:33.318 stronger. 42:33.320 --> 42:36.190 The fact that it'll precisely be independent of distance, 42:36.188 --> 42:37.878 you have to do the calculation. 42:37.880 --> 42:38.830 Yes. 42:38.829 --> 42:41.339 Student: What's the negative sign? 42:41.340 --> 42:42.570 Prof: Negative sign where, here? 42:42.570 --> 42:43.910 Student: Yes. 42:43.909 --> 42:47.549 Prof: -2 is there, but the upper limit is 42:47.547 --> 42:48.397 infinity. 42:48.400 --> 42:55.960 Student: Oh, okay. 42:55.960 --> 42:56.500 Prof: All right. 42:56.500 --> 42:58.930 Now here's the third problem, and the good news is, 42:58.925 --> 43:01.785 I'm not going to solve it for you, but I'll tell you what it 43:01.789 --> 43:02.129 is. 43:02.130 --> 43:06.770 Here is a solid ball of charge. 43:06.768 --> 43:12.678 It's got some charge density ρ coulombs per meter cubed. 43:12.679 --> 43:14.059 So ρ is the standard name. 43:14.059 --> 43:17.949 You use density for mass over volume and you use the same 43:17.945 --> 43:20.715 symbol rho for charge per unit volume. 43:20.719 --> 43:24.369 So somebody's assembled a blob of electrical charge, 43:24.373 --> 43:27.743 and 1 cubic meter of that has ρ coulombs. 43:27.739 --> 43:31.409 You want to find the field due to this one. 43:31.409 --> 43:34.039 Now when you do a similar problem in gravitation, 43:34.039 --> 43:37.919 it is generally assumed that when you're outside the sphere, 43:37.920 --> 43:41.230 the whole sphere acts like a point charge with the entire 43:41.233 --> 43:43.073 charge sitting at the center. 43:43.070 --> 43:45.370 But you actually have to prove that. 43:45.369 --> 43:48.159 That's what took Newton a long time to prove. 43:48.159 --> 43:50.679 He knew it was true but he couldn't prove it, 43:50.684 --> 43:53.214 because for that, you've got to be able to do 43:53.208 --> 43:54.468 integral calculus. 43:54.469 --> 43:56.519 And even today, to find the field due to a 43:56.518 --> 43:58.818 sphere using integration is quite difficult. 43:58.820 --> 44:00.520 Think about what you have to do. 44:00.519 --> 44:02.059 You want to sit somewhere here. 44:02.059 --> 44:03.519 First of all, for a sphere, 44:03.516 --> 44:05.866 we know the field is going to be radial. 44:05.869 --> 44:08.369 It doesn't matter where you pick, everything looks the same. 44:08.369 --> 44:13.059 You can decide to be horizontally here at that point. 44:13.059 --> 44:16.919 Then you've got to divide the sphere into tiny pieces, 44:16.920 --> 44:19.980 tiny little cubes, each with some charge rho times 44:19.983 --> 44:22.903 the volume of the cube, and that will exert a force 44:22.896 --> 44:23.416 like this. 44:23.420 --> 44:26.530 And you've got to integrate over the volume of the sphere, 44:26.525 --> 44:29.135 but each portion is at a different distance and a 44:29.139 --> 44:30.229 different angle. 44:30.230 --> 44:31.810 You've got to add it all up. 44:31.809 --> 44:35.529 That's why it's a tough problem. 44:35.530 --> 44:39.810 So to solve that tough problem, we're now going to use a very 44:39.813 --> 44:43.673 powerful trick and that trick is called Gauss's law. 44:43.670 --> 44:55.590 So we're going to learn today about the Gauss's law. 44:55.590 --> 44:59.230 Now a prelude to that, you need a little more 44:59.228 --> 45:03.198 mathematical definitions, but they're not bad. 45:03.199 --> 45:05.909 I just have to tell you what the definition is. 45:05.909 --> 45:12.179 Suppose I have in three dimensions a tiny little area, 45:12.184 --> 45:16.804 like a snowflake, but it's flat and it's 45:16.802 --> 45:20.002 rectangular, let's say. 45:20.000 --> 45:24.930 I want to tell you everything about it. 45:24.929 --> 45:29.029 I want you to be able to visualize the area. 45:29.030 --> 45:33.790 What can I do to specify this little thing? 45:33.789 --> 45:36.719 First I have to tell you how big it is. 45:36.719 --> 45:40.679 If it's a tiny area, let this area be dA 45:40.679 --> 45:43.419 meters squared, but that doesn't tell me the 45:43.416 --> 45:45.926 orientation of this area, because that area could be like 45:45.929 --> 45:47.459 this, it could be like this, 45:47.463 --> 45:49.303 it could be tilted in many ways. 45:49.300 --> 45:54.470 So I want to tell you it's an area in a certain plane, 45:54.467 --> 45:56.317 what should I do? 45:56.320 --> 46:03.260 How do I nail down the plane in which the area is located? 46:03.260 --> 46:04.180 Yeah? 46:04.179 --> 46:06.209 Student: Define the vector that's perpendicular to 46:06.213 --> 46:06.753 that surface. 46:06.750 --> 46:10.170 Prof: Define a vector normal to that surface, 46:10.170 --> 46:16.200 because if you draw that vector, then there's only one 46:16.195 --> 46:19.375 plane perpendicular to it. 46:19.380 --> 46:22.020 Then we can follow, we can then form a vector, 46:22.019 --> 46:23.039 dA. 46:23.039 --> 46:28.729 It's a tiny vector whose direction is perpendicular to 46:28.731 --> 46:35.501 its area and whose magnitude is the value of the area itself. 46:35.500 --> 46:39.800 So areas can be associated with vectors. 46:39.800 --> 46:42.500 You may not have thought about it that way, but you can by this 46:42.503 --> 46:42.943 process. 46:42.940 --> 46:47.620 I've told you, there's only one ambiguity even 46:47.615 --> 46:48.235 now. 46:48.239 --> 46:50.089 Do you know what that one-- yes? 46:50.090 --> 46:51.150 Student: Which direction. 46:51.150 --> 46:53.140 Prof: There are two normal's you can draw to an 46:53.143 --> 46:53.673 area, right? 46:53.670 --> 46:57.650 We've got an area like this, it can come out towards you or 46:57.654 --> 46:58.964 go away from you. 46:58.960 --> 47:02.520 Therefore simply drawing that rectangular patch is not enough 47:02.518 --> 47:03.408 to nail that. 47:03.409 --> 47:06.829 That's like saying, "Here is a vector." 47:06.829 --> 47:07.789 That's not enough. 47:07.789 --> 47:09.749 Where is the head and where is the tail? 47:09.750 --> 47:11.760 That's not a vector. 47:11.760 --> 47:13.240 That is a vector. 47:13.239 --> 47:16.569 Similarly, this area has to be specified some more and here is 47:16.574 --> 47:18.164 what you're supposed to do. 47:18.159 --> 47:20.959 You take that area here, I'm just drawing it another 47:20.958 --> 47:23.978 place, draw some arrows, then circulate around it in one 47:23.976 --> 47:25.126 sense or another. 47:25.130 --> 47:28.220 I picked a particular sense in which they're going around. 47:28.219 --> 47:30.569 Then use the famous right hand rule, 47:30.570 --> 47:34.170 where your fingers curl along the arrow and your thumb points 47:34.170 --> 47:37.840 in some direction, that is the direction, 47:37.840 --> 47:40.470 the area of the vector. 47:40.469 --> 47:43.319 If the arrows are running round the opposite way, 47:43.315 --> 47:46.215 then your thumb will point into the blackboard. 47:46.219 --> 47:50.109 So an area like this is like a vector without a head. 47:50.110 --> 47:52.070 Area like this is a signed area. 47:52.070 --> 47:56.620 It's an area that's got a magnitude and unique direction. 47:56.619 --> 48:01.199 So get used to the notion that a tiny planar area can be 48:01.195 --> 48:03.355 represented as a vector. 48:03.360 --> 48:06.370 Another way to see that is, if you took any area, 48:06.369 --> 48:11.969 a rectangular area like this--square is a special case-- 48:11.969 --> 48:15.139 if you took two vectors A and B that form 48:15.141 --> 48:18.481 the two edges, then A x B is 48:18.481 --> 48:20.391 just double the area. 48:20.389 --> 48:22.249 It's the fact that given two vectors, 48:22.250 --> 48:24.410 I can find the third vector perpendicular to them, 48:24.409 --> 48:27.769 up to a sign, is what makes a cross product 48:27.768 --> 48:30.028 possible, only in three dimensions. 48:30.030 --> 48:32.510 You cannot have a cross product of two vectors and four 48:32.507 --> 48:34.387 dimensions because in four dimensions, 48:34.389 --> 48:37.569 if I pick two vectors, they'll be two other directions 48:37.570 --> 48:39.370 perpendicular to that plane. 48:39.369 --> 48:42.009 Only in 3D, there's only one direction left. 48:42.010 --> 48:43.700 The question is, is it in or out? 48:43.699 --> 48:45.749 That you pick a sign in the cross product, 48:45.753 --> 48:48.613 A x B is something that goes from A to B. 48:48.610 --> 48:51.370 Or for an area, you draw arrows around the edge 48:51.369 --> 48:52.869 in a certain direction. 48:52.869 --> 48:54.409 So area is a vector. 48:54.409 --> 48:57.989 You have to get used to that notion, along with all the other 48:57.987 --> 48:59.117 vectors you know. 48:59.119 --> 49:02.259 Now I'll tell you why that becomes useful. 49:02.260 --> 49:14.040 So we're going to take--let's see-- 49:14.039 --> 49:18.669 there is a rectangular tube which has got a height h 49:18.672 --> 49:22.652 and a width w, and some fluid is flowing along 49:22.650 --> 49:26.510 the tube with the velocity v along the length of the 49:26.507 --> 49:27.037 tube. 49:27.039 --> 49:27.789 You got that? 49:27.789 --> 49:29.479 It's like an air duct. 49:29.480 --> 49:31.040 Stuff is going through that tube. 49:31.039 --> 49:33.429 It's got a rectangular cross section. 49:33.429 --> 49:40.499 The cross section area is hw. 49:40.500 --> 49:44.260 If the fluid is going velocity V along the length of the 49:44.255 --> 49:46.795 tube, what is the flow rate, 49:46.800 --> 49:52.330 which is equal to meter cubed of stuff flowing per second. 49:52.329 --> 49:56.009 I'm going to denote it by the symbol Φ. 49:56.010 --> 49:59.860 If I wait 1 second and I watch all the fluid go by me, 49:59.855 --> 50:03.405 past any cross section, how much stuff goes by? 50:03.409 --> 50:06.379 I think you can all see, if I wait 1 second, 50:06.380 --> 50:11.240 the fluid whose front was here would have advanced to here, 50:11.239 --> 50:13.259 and the volume here will be v times 1, 50:13.260 --> 50:16.080 because in 1 second, it goes a distance v. 50:16.079 --> 50:24.799 So the flow rate will be hwv or area times 50:24.800 --> 50:26.800 v. 50:26.800 --> 50:27.970 That makes sense? 50:27.969 --> 50:30.019 The faster it's going, the more stuff you get. 50:30.018 --> 50:32.048 The bigger the cross section, the more stuff you get. 50:32.050 --> 50:34.210 This doesn't depend on a rectangular cross section. 50:34.210 --> 50:36.530 It can be cylindrical pipe carrying oil. 50:36.530 --> 50:40.590 Again, the flow rate is area times velocity. 50:40.590 --> 50:44.820 But I'm going to write this in terms of vectors, 50:44.820 --> 50:48.600 because I know the velocity is a vector. 50:48.599 --> 50:50.589 But now I have also learned area is a vector, 50:50.590 --> 50:54.360 because area vector here, you can draw a vector 50:54.360 --> 50:58.100 perpendicular to this area, and I'll draw up this 50:58.103 --> 51:01.223 convention that it's area vectors along this way, 51:01.219 --> 51:02.859 rather than the other way. 51:02.860 --> 51:08.300 Then I can write this v⋅A. 51:08.300 --> 51:10.630 Let's check that v⋅A is 51:10.632 --> 51:11.122 correct. 51:11.119 --> 51:15.029 v⋅A is the length of v, 51:15.025 --> 51:18.715 the size of v, the size of A and cosine 51:18.719 --> 51:20.849 of the angle between them. 51:20.849 --> 51:22.929 Here we've got to be very careful. 51:22.929 --> 51:25.079 The velocity vector is like this. 51:25.079 --> 51:28.229 It's perpendicular to the plane but don't say it's cosine of 90 51:28.231 --> 51:28.741 degrees. 51:28.739 --> 51:31.839 The area vector is perpendicular to the area 51:31.840 --> 51:32.490 itself. 51:32.489 --> 51:33.649 Do you understand that? 51:33.650 --> 51:38.050 When I take the dot product and I will ask for the angle between 51:38.048 --> 51:42.168 the area vector and the velocity vector, that angle is 0. 51:42.170 --> 51:44.360 For the area, there's a little confusion. 51:44.360 --> 51:47.470 The vector representing it happens to be perpendicular to 51:47.467 --> 51:49.187 the plane of the area itself. 51:49.190 --> 51:51.160 So if you remember that, then 51:51.159 --> 51:54.889 v⋅Ais vA times cos 0, 51:54.891 --> 51:58.271 so this indeed is one way to write the flow. 51:58.269 --> 52:02.049 This flow is also called a flux. 52:02.050 --> 52:06.470 But now let us do the following. 52:06.469 --> 52:23.389 Let's take the same problem, and I have this area here. 52:23.389 --> 52:29.029 Let me now take a tilted area like this. 52:29.030 --> 52:32.500 It also goes from the ceiling to the floor but still turned at 52:32.500 --> 52:33.980 an angle θ. 52:33.980 --> 52:38.400 So it's a bigger area than the original one. 52:38.400 --> 52:40.190 How much bigger? 52:40.190 --> 52:44.400 That area prime, I claim, is equal to the base 52:44.400 --> 52:47.020 w, times this side. 52:47.018 --> 52:52.708 This side is h/cosθ. 52:52.710 --> 52:56.720 θ is the angle between these two planes. 52:56.719 --> 52:59.269 It's a bigger area, but you all know that just 52:59.266 --> 53:02.546 because it's a bigger area, it doesn't intercept more fluid 53:02.550 --> 53:03.400 per second. 53:03.400 --> 53:08.030 Any stuff crossing this guy also crosses this at the same 53:08.030 --> 53:08.610 rate. 53:08.610 --> 53:10.610 So how am I going to get the same rate? 53:10.610 --> 53:13.330 The flux is not going to be v times A'. 53:13.329 --> 53:17.689 It's going to be vA'cosθ. 53:17.690 --> 53:22.840 But θ is the angle between the area vector and the 53:22.840 --> 53:26.830 velocity vector, which is the same angle here. 53:26.829 --> 53:31.499 So the moral of the story is, v⋅A', 53:31.500 --> 53:36.590 or v⋅A in general is the flux or the flow, 53:36.590 --> 53:40.570 off of any vector across an area. 53:40.570 --> 53:43.770 If it's a fluid that's flowing, then v⋅A is 53:43.769 --> 53:45.449 the fluid flow past that area. 53:45.449 --> 53:48.509 If you need the dot product, you need the cosine of the 53:48.510 --> 53:51.460 angle, because the area, if it's not perpendicular to 53:51.458 --> 53:53.158 the flow, it's not useful. 53:53.159 --> 53:55.859 In fact, you can take a huge area parallel to the flow and 53:55.858 --> 53:57.088 nothing goes through it. 53:57.090 --> 54:00.440 So area is most effective if the plane of the area is normal 54:00.442 --> 54:02.892 to the fluid, or the area vector is parallel 54:02.885 --> 54:04.755 to the velocity of the fluid. 54:04.760 --> 54:07.570 So that's that lesson. 54:07.570 --> 54:09.940 Okay, this had nothing to do with the electric field, 54:09.940 --> 54:12.310 but we're going to come to the electric field now. 54:12.309 --> 54:14.999 This is just a warm up. 54:15.000 --> 54:19.000 Let's come to the electric field and see what's going on. 54:19.000 --> 54:32.670 So I take a charge q and I draw the lines coming out of 54:32.668 --> 54:33.788 it. 54:33.789 --> 54:43.029 How many lines do I get crossing a sphere S? 54:43.030 --> 54:45.440 Well, we know that we have agreed to draw 54:45.443 --> 54:48.163 1/ε_0 lines per coulomb, 54:48.159 --> 54:54.199 so this q here, that many lines cross the 54:54.202 --> 54:55.362 sphere. 54:55.360 --> 55:00.130 I'm now going to relate it to something I can do with the 55:00.126 --> 55:02.506 electric field as follows. 55:02.510 --> 55:09.400 I'm going to say that if I go to that sphere, 55:09.396 --> 55:14.246 I look at the electric field. 55:14.250 --> 55:19.640 Electric field is in that direction, E. 55:19.639 --> 55:22.549 And any portion of the sphere--and I want this is where 55:22.554 --> 55:25.424 you got to __________ imagine--take a tiny part of the 55:25.416 --> 55:26.006 sphere. 55:26.010 --> 55:31.970 There's a tiny area that's got a size dA. 55:31.969 --> 55:35.949 What is its direction? 55:35.949 --> 55:39.769 Direction of the area vector is radial. 55:39.769 --> 55:40.459 You understand? 55:40.460 --> 55:44.130 The area is on the surface and normal to that is the radial 55:44.132 --> 55:47.682 vector, which I always denote by e_r. 55:47.679 --> 55:54.729 The electric field is equal to q/4Πε 55:54.726 --> 55:59.256 _0(1/ r^(2))e 55:59.257 --> 56:01.897 _r. 56:01.900 --> 56:06.080 Therefore E⋅dA = 56:06.077 --> 56:10.657 q/4Πε _0(1/ 56:10.661 --> 56:14.231 r^(2))dAe _r 56:14.226 --> 56:17.586 ⋅e_r. 56:17.590 --> 56:20.230 This is a dot product of the area vector with the electric 56:20.226 --> 56:20.916 field vector. 56:20.920 --> 56:22.850 The two unit vectors are parallel. 56:22.849 --> 56:29.259 The dot product of that guy with itself is just 1. 56:29.260 --> 56:32.840 So this is the number of lines crossing the tiny area, 56:32.840 --> 56:36.490 because electric field numerically is equal to the line 56:36.489 --> 56:37.299 density. 56:37.300 --> 56:40.190 So those lines crossing this area, this is the number of 56:40.186 --> 56:42.436 lines crossing that patch, which is given by 56:42.443 --> 56:44.283 E⋅dA. 56:44.280 --> 56:49.830 If you add up all the lines, you must add up all the 56:49.826 --> 56:51.236 dAs. 56:51.239 --> 56:55.739 Sum of all the dAs on a sphere of radius r is 56:55.740 --> 57:00.550 just q/4Πε _0r^(2) times sum 57:00.552 --> 57:03.582 of all the dAs on a sphere, 57:03.579 --> 57:06.669 that is just 4Πr^(2) . 57:06.670 --> 57:10.770 So you get q/ε 57:10.773 --> 57:13.293 _0. 57:13.289 --> 57:16.069 In other words, either you can draw the picture 57:16.067 --> 57:19.447 and it's immediately obvious to you the lines crossing is 57:19.447 --> 57:21.857 q/ε _0 by 57:21.862 --> 57:24.402 construction, or you can remember, "Hey, 57:24.400 --> 57:27.420 the electric field is a direct measure of the number of lines 57:27.416 --> 57:29.826 per unit area," and the electric field times 57:29.831 --> 57:32.751 area times the cosine of the angle between the area and the 57:32.748 --> 57:36.168 electric field will count the lines going through a tiny area. 57:36.170 --> 57:38.530 If I add them all up, I'd better get 57:38.534 --> 57:41.104 q/ε _0, 57:41.103 --> 57:42.593 and indeed you do. 57:42.590 --> 57:45.370 So the moral of this little exercise is that 57:45.369 --> 57:48.419 the surface integral, let's call it the surface 57:48.422 --> 57:54.432 integral of the electric field, on a surface is equal to 57:54.429 --> 58:02.549 q/ε _0 where this was 58:02.554 --> 58:04.434 a sphere. 58:04.429 --> 58:06.479 In other words, even if you've never heard of 58:06.483 --> 58:08.863 field lines, just take the electric field and do the 58:08.864 --> 58:10.504 surface integral, you get this. 58:10.500 --> 58:13.170 So surface integral is a new concept. 58:13.170 --> 58:16.100 You probably have not done that before and I've got to remind 58:16.099 --> 58:18.249 you how it is done at least operationally. 58:18.250 --> 58:21.250 If you've got a computer, you can find the surface 58:21.253 --> 58:23.343 integral of anything as follows. 58:23.340 --> 58:27.070 Take the surface over which you're doing an integration. 58:27.070 --> 58:31.580 Divide it into tiny pieces, each is a little area 58:31.581 --> 58:32.711 dA. 58:32.710 --> 58:35.560 Take the dot product of the dA with the electric 58:35.561 --> 58:38.941 field there, and sum over all the patches covering the sphere. 58:38.940 --> 58:41.740 Then take the limit in which every patch gets vanishingly 58:41.739 --> 58:42.139 small. 58:42.139 --> 58:44.829 That is called the surface integral of the electric field 58:44.827 --> 58:48.437 and we see that if you do that, you get the charge inside 58:48.436 --> 58:51.676 divided by ε_0. 58:51.679 --> 58:54.399 But that was on a sphere, and the interesting thing was, 58:54.400 --> 58:57.220 the answer was independent of the radius of the sphere, 58:57.219 --> 59:01.069 because the 1/r^(2) in the field canceled the 59:01.070 --> 59:03.110 r^(2) in the area. 59:03.110 --> 59:05.500 But it is even better than that, because it's clear to you 59:05.503 --> 59:07.563 know, without any calculation, 59:07.561 --> 59:10.861 that if I took some crazy surface like this, 59:10.860 --> 59:14.870 the lines crossing that is also the same. 59:14.869 --> 59:17.579 Because the lines leave the origin, they go radially 59:17.576 --> 59:18.156 outwards. 59:18.159 --> 59:19.769 They don't terminate on anything. 59:19.768 --> 59:22.128 They're only supposed to terminate on other charges, 59:22.130 --> 59:24.400 therefore you can count them anywhere you like. 59:24.400 --> 59:27.490 You can take a census here, you can take a census there or 59:27.494 --> 59:27.934 there. 59:27.929 --> 59:30.949 You're always going to get the same number of lines. 59:30.949 --> 59:33.909 Therefore q/ε_0 59:33.911 --> 59:38.931 is also going to be equal to the line count on some 59:38.931 --> 59:42.141 weird surface enclosing the charge. 59:42.139 --> 59:45.479 So how am I going to count the lines on a weird surface? 59:45.480 --> 59:48.190 I take the surface, divide it into little pieces, 59:48.190 --> 59:53.100 but now the area vector and the electric field vector may not be 59:53.103 --> 59:55.713 parallel, because it's not a sphere. 59:55.710 --> 59:58.780 But it is still true that if you're trying to count the lines 59:58.777 --> 1:00:00.817 going through, just like in the velocity, 1:00:00.820 --> 1:00:03.020 you must take the dot product of these. 1:00:03.018 --> 1:00:07.488 Therefore you will again find E⋅dA 1:00:07.489 --> 1:00:11.629 for any surface, any closed surface, 1:00:11.632 --> 1:00:17.042 is equal to q/ε_0 1:00:17.038 --> 1:00:19.028 . 1:00:19.030 --> 1:00:21.520 If you got lost in the mathematics, the physical 1:00:21.518 --> 1:00:22.788 picture is very clear. 1:00:22.789 --> 1:00:25.369 If you want to count how many lines leave the charge, 1:00:25.369 --> 1:00:27.749 you can surround it with any surface you like. 1:00:27.750 --> 1:00:30.160 If it's a sphere, it's very easy for you to do 1:00:30.157 --> 1:00:30.797 the check. 1:00:30.800 --> 1:00:33.460 If it's a crazy surface, it's harder but they all come 1:00:33.460 --> 1:00:36.120 from the fact that E⋅dA, 1:00:36.119 --> 1:00:41.289 count the number of lines and that's independent of the 1:00:41.289 --> 1:00:43.109 surface at stake. 1:00:43.110 --> 1:00:49.120 So now that it what is called Gauss's law, so I'm going to 1:00:49.123 --> 1:00:51.343 write it down here. 1:00:51.340 --> 1:00:54.030 What I've shown you is the following. 1:00:54.030 --> 1:00:58.630 Here is some strange closed surface that's a charge q 1:00:58.628 --> 1:00:59.328 inside. 1:00:59.329 --> 1:01:07.219 Then E⋅dA on that surface is equal to 1:01:07.224 --> 1:01:12.954 q_inside /ε 1:01:12.954 --> 1:01:15.634 _0. 1:01:15.630 --> 1:01:21.770 This is not yet the theorem, but this is the case for 1 1:01:21.766 --> 1:01:22.786 charge. 1:01:22.789 --> 1:01:29.589 If the charge were outside, suppose the charge were here, 1:01:29.590 --> 1:01:33.880 then the lines would go like this, and the total number 1:01:33.876 --> 1:01:37.206 leaving the surface would actually be 0. 1:01:37.210 --> 1:01:38.090 You might say, "How is that? 1:01:38.090 --> 1:01:40.100 I see all these lines penetrating the surface," 1:01:40.097 --> 1:01:42.697 but you've got to remember, if you take an area vector 1:01:42.697 --> 1:01:45.367 here, with the definition for a closed surface, 1:01:45.369 --> 1:01:49.739 the area vector is the outward pointing normal. 1:01:49.739 --> 1:01:52.459 For a closed surface, every area is a vector pointing 1:01:52.460 --> 1:01:54.030 out from the closed surface. 1:01:54.030 --> 1:01:57.390 Then you can see that in this surface, lines are coming out; 1:01:57.389 --> 1:02:00.489 on this surface, lines are going in. 1:02:00.489 --> 1:02:02.749 It's very clear what's going in is coming out, 1:02:02.751 --> 1:02:04.411 because nothing is terminating. 1:02:04.409 --> 1:02:07.989 Therefore if you took a surface that did not enclose the charge, 1:02:07.989 --> 1:02:10.629 this answer would be 0, but if it enclosed the charge, 1:02:10.630 --> 1:02:14.800 it will be the charge inside, but it won't matter where it 1:02:14.802 --> 1:02:15.172 is. 1:02:15.170 --> 1:02:18.830 You can also see that you can move this charge around anywhere 1:02:18.829 --> 1:02:22.429 you like, you don't change the number of lines coming out. 1:02:22.429 --> 1:02:26.159 Then the most important generalization is, 1:02:26.164 --> 1:02:29.814 if instead of 1 charge, I have 2 charges, 1:02:29.809 --> 1:02:31.539 what do you do? 1:02:31.539 --> 1:02:36.029 If you had 2 charges, let us take the total 1:02:36.025 --> 1:02:40.935 E⋅dA on a surface. 1:02:40.940 --> 1:02:45.180 The total electric field is the electric field due to the first 1:02:45.181 --> 1:02:48.561 charge on that surface, the electric field due to the 1:02:48.559 --> 1:02:52.699 second charge on the surface, thanks to superposition. 1:02:52.699 --> 1:02:57.239 It's the fact that the electric field is additive over charges. 1:02:57.239 --> 1:02:59.099 But this one is q_1 1:02:59.099 --> 1:03:01.319 /ε _0. 1:03:01.320 --> 1:03:04.880 This is q_2 /ε 1:03:04.880 --> 1:03:06.530 _0. 1:03:06.530 --> 1:03:10.140 Therefore, we can now write, generalizing to any number, 1:03:10.139 --> 1:03:13.489 whether one draws a double integral with this thing, 1:03:13.487 --> 1:03:15.717 meaning it's a closed surface. 1:03:15.719 --> 1:03:18.199 You can also have an open surface, take the skin off an 1:03:18.202 --> 1:03:19.492 orange and cut it in half. 1:03:19.489 --> 1:03:24.349 That hemispherical skin is also a surface, but to signify it's a 1:03:24.353 --> 1:03:27.213 closed surface, we do it like this. 1:03:27.210 --> 1:03:32.010 On a closed surface of E⋅dA = 1:03:32.014 --> 1:03:35.574 q_enclosed /ε 1:03:35.572 --> 1:03:41.572 _0, which means sum of all the 1:03:41.565 --> 1:03:43.425 q's. 1:03:43.429 --> 1:03:46.599 And sometimes we write this as follows. 1:03:46.599 --> 1:03:50.809 Suppose inside you don't have a discrete set of charges that you 1:03:50.811 --> 1:03:53.601 can count, but a continuous blob of 1:03:53.599 --> 1:03:57.359 charge, and the charge has a certain density, 1:03:57.360 --> 1:04:05.850 ρ is the charge density. 1:04:05.849 --> 1:04:08.219 What is the q_enclosed? 1:04:08.219 --> 1:04:14.919 I claim the answer is due to a volume integral of ρ 1:04:14.923 --> 1:04:22.373 or any point r inside, times dxdydz inside that 1:04:22.373 --> 1:04:23.743 volume. 1:04:23.739 --> 1:04:27.409 You know, if you want to say how many coulombs are there, 1:04:27.409 --> 1:04:30.359 well divide the volume into tiny cubes of size 1:04:30.360 --> 1:04:31.540 dxdydz. 1:04:31.539 --> 1:04:32.789 That times the ρ of r, 1:04:32.789 --> 1:04:33.759 meaning ρ at x, 1:04:33.764 --> 1:04:35.994 y and z, is the charge inside the cube. 1:04:35.989 --> 1:04:42.999 You add it all up, you get the charge enclosed in 1:04:43.003 --> 1:04:46.953 that funny shaped object. 1:04:46.949 --> 1:04:55.239 So this is the final form of Gauss's law that we're going to 1:04:55.237 --> 1:04:56.077 use. 1:04:56.079 --> 1:04:57.889 I'm going to use the symbol, which is very useful. 1:04:57.889 --> 1:05:15.369 1:05:15.369 --> 1:05:17.309 This is the theorem of Gauss. 1:05:17.309 --> 1:05:18.599 And S = DV. 1:05:18.599 --> 1:05:24.259 DV is the boundary of V. 1:05:24.260 --> 1:05:27.310 In other words, V is like a potato, 1:05:27.309 --> 1:05:30.089 the skin of the potato is DV. 1:05:30.090 --> 1:05:32.970 Therefore if the potato is full of charge, 1:05:32.969 --> 1:05:35.989 it will emit some electric flux and the surface integral 1:05:35.994 --> 1:05:38.974 electric flux over the skin of the potato is the charge 1:05:38.965 --> 1:05:40.225 enclosed inside it. 1:05:40.230 --> 1:05:43.180 The charge enclosed inside it is not simply some constant 1:05:43.181 --> 1:05:45.761 density times volume, if the charge density varies 1:05:45.764 --> 1:05:47.034 from point to point. 1:05:47.030 --> 1:05:50.140 In each neighborhood inside their volume, 1:05:50.139 --> 1:05:53.519 you take a tiny section, a tiny little cube of size 1:05:53.519 --> 1:05:55.569 dxdydz, see how much is in there, 1:05:55.565 --> 1:05:58.585 as the function of x, y and z, 1:05:58.588 --> 1:06:02.218 and you do the integral over the volume. 1:06:02.219 --> 1:06:05.639 Now these integrals may be hard to do, but you should know at 1:06:05.641 --> 1:06:07.011 least what I'm saying. 1:06:07.010 --> 1:06:11.420 This is just a way of counting the total charge inside a volume 1:06:11.418 --> 1:06:14.688 when the charge is continuously distributed. 1:06:14.690 --> 1:06:18.630 So I'm going to show you one application of this and we'll 1:06:18.626 --> 1:06:20.696 come back to more next time. 1:06:20.699 --> 1:06:29.159 And the one application I'm going to show you is to find the 1:06:29.159 --> 1:06:37.619 electric field due to a solid ball, solid ball of charge. 1:06:37.619 --> 1:06:41.549 So I'm going to use Gauss's theorem to do that. 1:06:41.550 --> 1:06:45.160 This is true for any surface you pick. 1:06:45.159 --> 1:06:47.169 It doesn't matter what surface you pick. 1:06:47.170 --> 1:06:51.200 So if I want to find the electric field here, 1:06:51.204 --> 1:06:56.344 I'm going to pick a surface of radius little r. 1:06:56.340 --> 1:07:01.720 Let the sphere have a radius big R. 1:07:01.719 --> 1:07:07.789 I want to find the field here and I'm going to use Gauss's 1:07:07.789 --> 1:07:08.429 law. 1:07:08.429 --> 1:07:11.969 So on the left hand side and the right hand side are two 1:07:11.974 --> 1:07:13.204 different things. 1:07:13.199 --> 1:07:17.559 On the right hand side, what is the charge enclosed? 1:07:17.559 --> 1:07:19.349 That region. 1:07:19.349 --> 1:07:24.139 Well, the charge enclosed is just some number q. 1:07:24.139 --> 1:07:27.949 This whole thing is q, q spread over a sphere 1:07:27.949 --> 1:07:29.459 of radius r, over 1:07:29.460 --> 1:07:31.630 ε_0. 1:07:31.630 --> 1:07:38.910 That's going to be the surface integral of the electric field 1:07:38.914 --> 1:07:40.984 on that sphere. 1:07:40.980 --> 1:07:48.120 E⋅dA on that sphere. 1:07:48.119 --> 1:07:52.429 Now normally, if you knew the value of an 1:07:52.431 --> 1:07:58.151 integral over a region, you cannot deduce anything but 1:07:58.146 --> 1:08:01.486 the integrand, unless what? 1:08:01.489 --> 1:08:05.929 There's one exception where if you know the integral of a 1:08:05.927 --> 1:08:09.017 function, you can find the integrand. 1:08:09.019 --> 1:08:10.409 You know when that might be? 1:08:10.409 --> 1:08:14.279 Here's a function. 1:08:14.280 --> 1:08:16.200 I tell you its integral from here to here, 1:08:16.204 --> 1:08:17.854 but I don't show you the picture. 1:08:17.850 --> 1:08:18.510 What's the integrand? 1:08:18.510 --> 1:08:20.190 You don't know. 1:08:20.189 --> 1:08:23.259 But if I also tell you the function is a constant, 1:08:23.260 --> 1:08:26.900 and I tell you the integral, and you know the width of this 1:08:26.895 --> 1:08:29.335 region, you can find the integrand. 1:08:29.340 --> 1:08:33.790 So you're going to cook this up so that this whole integral is a 1:08:33.786 --> 1:08:36.676 constant times the area of integration. 1:08:36.680 --> 1:08:41.440 And we argue that if you've got a spherical charge density, 1:08:41.435 --> 1:08:45.285 the electric field must be radial everywhere. 1:08:45.288 --> 1:08:47.778 You've not proven this, but you argue that because it's 1:08:47.777 --> 1:08:50.447 the only distribution with lines coming out everywhere, 1:08:50.448 --> 1:08:55.318 invariant under every possible rotation of the sphere. 1:08:55.319 --> 1:08:59.389 Because if you rotate the sphere, I won't know you rotated 1:08:59.385 --> 1:09:02.875 it, so the field pattern cannot look different. 1:09:02.880 --> 1:09:05.270 If the field pattern looked different when you rotated it, 1:09:05.266 --> 1:09:07.606 then you have a problem because the cause looks the same, 1:09:07.613 --> 1:09:09.083 but the effect looks different. 1:09:09.079 --> 1:09:10.499 That's not allowed. 1:09:10.500 --> 1:09:13.770 The only allowed this is the radially outgoing electric 1:09:13.774 --> 1:09:14.264 field. 1:09:14.260 --> 1:09:16.820 Therefore the electric field is radial and also constant 1:09:16.824 --> 1:09:19.024 throughout the sphere, because all points on the 1:09:19.016 --> 1:09:20.226 sphere are equivalent. 1:09:20.229 --> 1:09:23.479 There's no reason why this is any better than that. 1:09:23.479 --> 1:09:25.679 The sphere looks the same for all directions. 1:09:25.680 --> 1:09:30.320 Therefore this whole integral is going to be 4Πr^(2) 1:09:30.317 --> 1:09:34.267 times the electric field at that radius r, 1:09:34.270 --> 1:09:39.030 because the area vector and the electric field vector are both 1:09:39.033 --> 1:09:41.253 parallel, so in this dot product, 1:09:41.247 --> 1:09:42.577 forget the dot product. 1:09:42.578 --> 1:09:43.948 It's just E times dA. 1:09:43.948 --> 1:09:45.738 There's no cosθ, just 1. 1:09:45.738 --> 1:09:48.658 You can pull the E out of the integral, 1:09:48.662 --> 1:09:52.432 because E is constant in magnitude on the sphere. 1:09:52.430 --> 1:09:54.660 And the integral of the dA is just 1:09:54.655 --> 1:09:55.765 4Πr^(2). 1:09:55.770 --> 1:09:59.150 Therefore you deduce E(r) is 1:09:59.152 --> 1:10:03.362 q/4Πε _0r^(2). 1:10:03.359 --> 1:10:04.629 It's a very profound result. 1:10:04.630 --> 1:10:07.890 It looks very simple, but it is true for not a point 1:10:07.886 --> 1:10:11.396 charge, but for the spherical distribution of charge. 1:10:11.399 --> 1:10:15.949 That the field goes like that of a point charge sitting at the 1:10:15.953 --> 1:10:19.093 origin, is a consequence of Gauss's law. 1:10:19.090 --> 1:10:23.260 If the charge inside was not uniform, 1:10:23.260 --> 1:10:25.990 suppose it's a charge q, but there's more stuff here, 1:10:25.988 --> 1:10:28.888 less stuff here, this theorem would still be 1:10:28.891 --> 1:10:32.401 true up to this point, but you can never deduce that 1:10:32.403 --> 1:10:34.733 E is a constant on a sphere, 1:10:34.729 --> 1:10:37.019 because even though a sphere is nice and symmetrical, 1:10:37.020 --> 1:10:38.620 the charge distribution is not. 1:10:38.618 --> 1:10:40.718 It could be big here, it could be small here. 1:10:40.720 --> 1:10:44.260 You know something about the integral over a surface of a 1:10:44.262 --> 1:10:47.112 varying function, then Gauss's law is no good, 1:10:47.109 --> 1:10:48.059 not useful. 1:10:48.060 --> 1:10:50.180 True, but not useful. 1:10:50.180 --> 1:10:53.350 Gauss's law is useful only when in a given problem, 1:10:53.350 --> 1:10:55.950 there's only one number you don't know. 1:10:55.948 --> 1:10:59.768 That number here happens to be, what's the strength of E 1:10:59.771 --> 1:11:01.191 at a radius r? 1:11:01.189 --> 1:11:04.149 I know it's direction is radial, I know it's magnitude is 1:11:04.152 --> 1:11:06.802 constant on the sphere by symmetry, but what is the 1:11:06.797 --> 1:11:07.377 number? 1:11:07.380 --> 1:11:10.890 You can trade that one number for this one number on the left 1:11:10.889 --> 1:11:13.289 hand side, q/ε_0, 1:11:13.287 --> 1:11:14.747 you can calculate it. 1:11:14.750 --> 1:11:19.100 I'll come back and do more examples for you guys next time. 1:11:19.100 --> 1:11:24.000