WEBVTT 00:01.520 --> 00:04.200 Prof: When I look at you guys, I realize that I don't 00:04.197 --> 00:05.737 know what most of you look like. 00:05.740 --> 00:10.350 That's the problem in a big class like this. 00:10.350 --> 00:12.910 Usually it would be nice if you know who your students are. 00:12.910 --> 00:15.600 If they say hello, you can say hello back, 00:15.598 --> 00:19.598 but I don't know what I can do about it, because if I focus on 00:19.598 --> 00:22.678 you, I forget what I'm going to talk about. 00:22.680 --> 00:29.820 But you know what I look like, and it's kind of an unnerving 00:29.816 --> 00:36.586 asymmetry, because you might become my doctor one day. 00:36.590 --> 00:40.710 You'll be wearing a mask and carrying a knife, 00:40.712 --> 00:45.292 and you'll be thinking of problem set number three, 00:45.293 --> 00:46.213 right? 00:46.210 --> 00:50.640 And I won't even know who did it, so it's very asymmetric. 00:50.640 --> 00:59.850 All right. 00:59.850 --> 01:03.760 Okay, so anyway, maybe I'll get to know you if 01:03.755 --> 01:08.265 you ask a lot of questions, or your favorite pastime, 01:08.266 --> 01:12.256 finding something wrong on the blackboard. 01:12.260 --> 01:16.150 Look at that. 01:16.150 --> 01:16.890 I didn't do that. 01:16.890 --> 01:25.030 I cannot even reach that far. 01:25.030 --> 01:27.970 So let me now go back a little bit, 01:27.970 --> 01:32.050 because I think I rushed some of the topics near the end and I 01:32.052 --> 01:36.552 thought about it some more, so I think it will be helpful 01:36.550 --> 01:40.500 to revisit this question about image charges. 01:40.500 --> 01:42.580 I want you to think about what the point was. 01:42.580 --> 01:45.490 So here's the main thing: you all know how to think about 01:45.489 --> 01:47.359 the potential or the field, right? 01:47.360 --> 01:50.930 I give you a bunch of charges and you calculate the potential 01:50.926 --> 01:53.656 and you take derivatives, you get the field. 01:53.660 --> 01:57.590 Sometimes that's not the only kind of problem you have to 01:57.587 --> 01:58.147 solve. 01:58.150 --> 02:01.520 You may have to solve the following problem. 02:01.519 --> 02:04.489 There is some chunk, maybe a potato, 02:04.489 --> 02:09.069 and you bring some electric charge, what does it do? 02:09.068 --> 02:11.408 That's a meaningful question, because it does something, 02:11.408 --> 02:11.748 right? 02:11.750 --> 02:14.670 And nature calculates right away and does something. 02:14.669 --> 02:16.129 But if you want to figure out what happens, 02:16.128 --> 02:20.828 it's quite complicated, so a somewhat easier problem is 02:20.834 --> 02:24.934 the case of the perfectly conducting potato, 02:24.930 --> 02:28.830 which is kind of solid metal, shaped like whatever you like. 02:28.830 --> 02:32.500 You bring a charge next to it, what happens? 02:32.500 --> 02:35.510 That question occurs a lot, because we do electrical 02:35.514 --> 02:38.414 experiments in the presence of many conductors. 02:38.410 --> 02:41.020 And the simple thing about a conductor is, 02:41.022 --> 02:43.512 the whole conductor is one potential. 02:43.508 --> 02:48.558 So I want to take a simple problem and see if we can answer 02:48.555 --> 02:48.985 it. 02:48.990 --> 02:50.570 Now most problems, you cannot answer. 02:50.568 --> 02:52.638 By "cannot answer" I don't mean in principle. 02:52.639 --> 02:54.579 In principle, we know the equations, 02:54.577 --> 02:57.177 we can solve them, including the real potato. 02:57.180 --> 03:00.520 You can actually figure out everything, but it's too hard. 03:00.520 --> 03:02.600 You cannot write it in simple form. 03:02.598 --> 03:07.328 There are a few simple textbook examples where you can ask the 03:07.331 --> 03:09.661 question and you can answer. 03:09.658 --> 03:15.948 And the question you want to ask is, here is an infinite 03:15.945 --> 03:20.285 conducting plane and it is grounded. 03:20.289 --> 03:21.439 Grounded means what? 03:21.438 --> 03:25.658 You take a wire and you connect it to the earth. 03:25.658 --> 03:30.648 Now the point of connecting to the earth is that the earth is 03:30.646 --> 03:34.716 pretty close to being an object at 0 potential. 03:34.720 --> 03:36.820 In other words, if you bring a charge from 03:36.824 --> 03:39.444 infinity to the earth, the amount of work you do you 03:39.442 --> 03:40.522 can take to be 0. 03:40.520 --> 03:43.270 And on a daily basis, people are dumping charge, 03:43.270 --> 03:46.310 people are taking charge out of it, but it's so huge, 03:46.312 --> 03:47.602 it doesn't matter. 03:47.598 --> 03:51.168 It's like one of the reservoirs you studied in thermodynamics. 03:51.169 --> 03:53.009 You can take some heat, you can put some heat. 03:53.008 --> 03:54.928 It doesn't change its temperature. 03:54.930 --> 03:58.660 So the earth is a huge repository where you can take a 03:58.663 --> 04:01.273 few charges now and put a few back. 04:01.270 --> 04:02.830 It will always remain at 0. 04:02.830 --> 04:06.400 It will force you also to go to 0 potential. 04:06.400 --> 04:10.960 So that is that plane, and you have another charge, 04:10.961 --> 04:15.341 q, you slowly bring towards that plane. 04:15.340 --> 04:17.430 You may like to know what's the answer to this problem. 04:17.430 --> 04:21.560 What does the field look like everywhere? 04:21.560 --> 04:22.910 Now this plane, for convenience, 04:22.908 --> 04:24.648 I'm going to be taking to be infinite. 04:24.649 --> 04:27.829 If you want a side view of that plane, it looks like this. 04:27.829 --> 04:30.119 It divides the whole universe into two parts, 04:30.122 --> 04:33.462 where you are with your charge and the rest on the other side. 04:33.459 --> 04:36.389 So what will this plane do? 04:36.389 --> 04:39.979 You can see that if it did nothing, it's in trouble. 04:39.980 --> 04:42.490 If it did nothing, this charge at that distance 04:42.487 --> 04:44.447 will produce a positive potential. 04:44.449 --> 04:46.659 At that distance, it will produce an even bigger 04:46.656 --> 04:48.626 positive potential, because it's closer. 04:48.629 --> 04:51.049 So different parts of the sheet will be at different potentials. 04:51.050 --> 04:51.730 That's not allowed. 04:51.730 --> 04:53.060 It's an equipotential. 04:53.060 --> 04:54.120 It's a metal. 04:54.120 --> 04:57.090 It's got to arrange its potential to be constant, 04:57.093 --> 05:00.873 and because it's connected to the ground, that constant has to 05:00.872 --> 05:01.432 be 0. 05:01.430 --> 05:05.120 So the way it will lower its potential from your attempts to 05:05.115 --> 05:08.355 raise it is to suck up some negative charges from the 05:08.362 --> 05:09.052 ground. 05:09.050 --> 05:12.630 So negative charge will leave the ground and somehow come and 05:12.625 --> 05:15.125 stand in front of this positive charge, 05:15.129 --> 05:19.589 maybe something like this, over some region roughly 05:19.586 --> 05:24.666 proportional to the distance between you and the plane. 05:24.670 --> 05:26.820 Then what will happen is, the electric field lines, 05:26.819 --> 05:29.219 which normally go through the metal if it wasn't there, 05:29.220 --> 05:33.530 will now of course have to come and terminate on that metal, 05:33.529 --> 05:36.889 and they have to terminate perpendicularly because the 05:36.886 --> 05:40.556 electric field is always perpendicular to the conductor. 05:40.560 --> 05:41.980 You know why, right? 05:41.980 --> 05:44.680 Because if you move along the conductor, the potential cannot 05:44.680 --> 05:47.320 change, so the line integral of E must be 0, 05:47.315 --> 05:49.815 but basically, you should not feel any force 05:49.815 --> 05:51.615 as you move along the plane. 05:51.620 --> 05:53.350 So E is perpendicular. 05:53.350 --> 05:57.370 So some bunch of charges will be drawn from the ground, 05:57.370 --> 06:00.720 as necessary, and this is what will happen. 06:00.720 --> 06:03.870 The question is, can you say anything more in 06:03.874 --> 06:04.524 detail? 06:04.519 --> 06:09.839 Can you actually calculate the charge distribution on this 06:09.841 --> 06:11.431 infinite plane? 06:11.430 --> 06:15.190 Can you calculate the force of attraction between the charge 06:15.185 --> 06:17.345 you have and the infinite plane? 06:17.350 --> 06:19.770 You know there's going to be attraction because your charge I 06:19.771 --> 06:21.751 take to be positive, and this has got negative. 06:21.750 --> 06:22.970 They'll attract. 06:22.970 --> 06:24.610 What is the attraction? 06:24.610 --> 06:27.120 These are all well-defined questions and they exist no 06:27.119 --> 06:28.539 matter what this is made of. 06:28.540 --> 06:34.150 But for this infinite plane, we can actually answer it 06:34.149 --> 06:37.229 So we answer by the following device, which is what I was 06:37.226 --> 06:38.926 trying to tell you last time. 06:38.930 --> 06:43.090 If you take that infinite plane, your conditions are that 06:43.093 --> 06:47.703 it should be at 0 potential and to the left of it should be the 06:47.702 --> 06:50.232 charge q that I put in. 06:50.230 --> 06:52.150 These are the requirements. 06:52.149 --> 06:55.479 So you dream up another problem. 06:55.480 --> 06:59.810 That's a charge -q, same distance d from 06:59.810 --> 07:00.710 this one. 07:00.709 --> 07:01.999 Take out the plane. 07:02.000 --> 07:03.550 Forget the plane now. 07:03.550 --> 07:06.210 Think of a new problem, and -q, 07:06.211 --> 07:09.091 and you've done that many, many times. 07:09.089 --> 07:12.289 The lines go like this. 07:12.290 --> 07:15.010 And you know that on the perpendicular bisector, 07:15.009 --> 07:16.799 the lines will be perpendicular, 07:16.803 --> 07:18.943 because the potential everywhere is 0, 07:18.944 --> 07:19.644 right? 07:19.639 --> 07:22.299 Potential is a scaler and whatever this guy does, 07:22.297 --> 07:24.897 that will do - of that, because it's at the same 07:24.898 --> 07:26.668 distance, but has -q. 07:26.670 --> 07:30.130 So V is 0, so E will be 07:30.130 --> 07:31.670 perpendicular. 07:31.670 --> 07:34.670 Now if you take that arrangement and look what's 07:34.666 --> 07:38.236 happening to the left of the plane, it looks like exactly 07:38.238 --> 07:40.278 what we want in our problem. 07:40.279 --> 07:43.409 What we demand in our problem is in this part of the universe 07:43.406 --> 07:45.756 where I live, there should be charge 07:45.762 --> 07:49.712 q, there should be infinite plane at 0 potential. 07:49.709 --> 07:52.909 That condition is satisfied by this field configuration in the 07:52.913 --> 07:53.863 following sense. 07:53.860 --> 07:58.740 Start with this configuration and stick in the infinite plane, 07:58.744 --> 08:03.474 but give it the charge it needs to terminate all these field 08:03.468 --> 08:04.268 lines. 08:04.269 --> 08:05.409 How much charge will it need? 08:05.410 --> 08:07.370 You can tell right away. 08:07.370 --> 08:13.760 What's the total charge on this plane? 08:13.759 --> 08:17.539 Because all the lines leaving this have to terminate on this. 08:17.540 --> 08:21.320 You can see it'll be -q. 08:21.319 --> 08:23.039 It won't be point charge; it'll be spread out, 08:23.036 --> 08:24.356 but you need -q on this. 08:24.360 --> 08:28.330 So what we say is, if you brought the plane in and 08:28.333 --> 08:31.823 either you give it -q in advance, 08:31.819 --> 08:34.849 or you connected it to the ground, in which case it will 08:34.845 --> 08:36.325 itself suck up -q. 08:36.330 --> 08:40.520 It'll form this configuration here, so the infinite plane is 08:40.515 --> 08:44.555 able to remain where it is without disturbing the solution 08:44.561 --> 08:45.911 to the problem. 08:45.908 --> 08:48.788 Therefore the answer to the original question of what 08:48.792 --> 08:52.282 happens when you put a q in front of an infinite plane is 08:52.283 --> 08:54.893 that as far as you're on the left of this, 08:54.889 --> 08:58.299 you can compute anything you want by taking q and the 08:58.303 --> 09:01.053 image charge, -q. 09:01.048 --> 09:05.778 The field the two of them will produce will be exactly the 09:05.784 --> 09:10.774 field this guy and all these induced charges will produce. 09:10.769 --> 09:15.359 So this is a way of taking a solution for a simple problem 09:15.364 --> 09:19.014 with an equipotential, sticking a conductor into the 09:19.009 --> 09:21.549 equipotential, and giving it the right charge. 09:21.548 --> 09:27.498 Then the problem of a charge in front of a conductor is thereby 09:27.504 --> 09:28.374 solved. 09:28.370 --> 09:31.310 Now there are some mathematical theorems--I don't have time to 09:31.306 --> 09:33.276 prove but they're not too hard in fact. 09:33.279 --> 09:36.389 I thought of lecturing on that, but there isn't enough time-- 09:36.389 --> 09:39.739 that show that if you can find one answer that has the right 09:39.743 --> 09:42.133 behavior at the boundary of the region, 09:42.129 --> 09:43.629 that has got the charges in the right place, 09:43.629 --> 09:45.609 is the only answer. 09:45.610 --> 09:46.860 Yes. 09:46.860 --> 09:49.090 Student: Is the -q evenly distributed across 09:49.085 --> 09:49.525 the plate? 09:49.529 --> 09:50.219 Prof: Let us ask. 09:50.220 --> 09:52.920 Where is the -q, right? 09:52.918 --> 09:55.968 We know that it's got to be -q because these lines 09:55.971 --> 09:57.171 are all ending here. 09:57.169 --> 09:58.629 But what's your intuition? 09:58.629 --> 10:00.949 Where do you think it will be? 10:00.950 --> 10:04.010 You think it'll be evenly distributed? 10:04.009 --> 10:05.379 Pardon me? 10:05.379 --> 10:07.479 Student: I don't think so. 10:07.480 --> 10:11.410 Prof: Where will it be? 10:11.409 --> 10:12.609 What's your feeling? 10:12.610 --> 10:13.880 Student: Probably more towards the middle. 10:13.879 --> 10:16.829 Prof: Yeah, more towards in front of this 10:16.826 --> 10:18.766 guy, and less when you go up. 10:18.769 --> 10:21.339 Now we can actually calculate that quantity. 10:21.340 --> 10:25.440 We can tell you exactly how much it will be by the following 10:25.442 --> 10:28.992 trick - we agree that to the region to the left, 10:28.990 --> 10:32.930 this charge is able to fake the effect of all these charges. 10:32.928 --> 10:34.078 That's why it's called a mirror charge. 10:34.080 --> 10:38.390 When you stand in front of a mirror, you are here and there 10:38.392 --> 10:40.922 is another person in the mirror. 10:40.918 --> 10:45.678 And the light bouncing off and coming here looks like it's 10:45.681 --> 10:47.771 coming from this image. 10:47.769 --> 10:50.809 So to the left of the mirror, either you can talk about you 10:50.807 --> 10:54.057 and the mirror and the reflected light, or forget the reflected 10:54.056 --> 10:54.576 light. 10:54.580 --> 10:58.050 It's as if there's another person behind the mirror. 10:58.048 --> 11:01.508 Of course, you know there is no other person behind the mirror, 11:01.513 --> 11:01.963 right? 11:01.960 --> 11:05.460 I don't want to be the one to break that news. 11:05.460 --> 11:06.270 There isn't. 11:06.269 --> 11:08.189 But as far as this side is concerned, 11:08.190 --> 11:10.560 it's a lot easier to draw a line from the mirror, 11:10.558 --> 11:12.838 because that will simulate the effect of what's happening 11:12.836 --> 11:13.686 through the mirror. 11:13.690 --> 11:17.180 Similarly, as long as you are interested only in the left of 11:17.182 --> 11:19.372 this region, you cannot tell the difference 11:19.370 --> 11:21.670 between the infinite plane with its induced charge, 11:21.668 --> 11:25.338 and this guy, versus this guy and just one 11:25.337 --> 11:26.677 other charge. 11:26.678 --> 11:30.218 Another question was, where is the charge and how is 11:30.220 --> 11:31.400 it distributed? 11:31.399 --> 11:34.479 You agree that if you knew the electric field here, 11:34.480 --> 11:37.830 you can find the charge density, because if you draw a 11:37.830 --> 11:40.170 little Gaussian surface like that, 11:40.168 --> 11:43.748 then we know the σ/ε_0 is 11:43.751 --> 11:45.321 the electric field. 11:45.320 --> 11:47.410 So if you know the electric field, you can find sigma, 11:47.408 --> 11:51.248 and I find the electric field very simply by saying that's the 11:51.246 --> 11:55.736 repulsion from this q, there's an attraction from that 11:55.740 --> 11:56.540 q. 11:56.538 --> 11:58.498 If you add the two, you get the sum, 11:58.504 --> 12:00.924 which I can calculate, a very simple sum. 12:00.918 --> 12:03.058 It's the contribution from two point charges. 12:03.058 --> 12:06.088 You can calculate it as a function of this angle here or 12:06.089 --> 12:09.339 as the function of the distance here, and that will tell you 12:09.337 --> 12:10.437 what σ is. 12:10.440 --> 12:12.170 And it will have the property you expect. 12:12.168 --> 12:15.578 It'll be largely in front of this guy and will vanish very 12:15.582 --> 12:17.862 quickly when you move away from him. 12:17.860 --> 12:23.520 If you integrate that surface charge density over the plane, 12:23.515 --> 12:26.675 you will in fact get -q. 12:26.678 --> 12:29.058 So you've answered some of the questions, namely, 12:29.057 --> 12:32.177 how will the charge distribute itself on this infinite plane? 12:32.178 --> 12:36.418 Next thing you can ask is, what's the force of attraction 12:36.418 --> 12:38.838 between this one and this one? 12:38.840 --> 12:42.990 Now, is it really the same as this one and this one? 12:42.990 --> 12:45.840 The answer is yes, and the reason is the following 12:45.836 --> 12:47.926 - in this region, for example, 12:47.932 --> 12:51.912 the field is due to this guy, and due to all these, 12:51.909 --> 12:55.349 but that's the same as this guy and this guy. 12:55.350 --> 12:57.850 This one of course doesn't respond to its own field. 12:57.850 --> 13:00.140 It responds to the field of all of these. 13:00.139 --> 13:03.599 But all of these precisely have the same field to the left as 13:03.602 --> 13:06.202 this one charge, therefore this will be pulled 13:06.201 --> 13:07.761 towards the one charge. 13:07.759 --> 13:09.579 It thinks it's falling towards the other charge, 13:09.576 --> 13:11.506 but it's not actually falling towards the plane. 13:11.509 --> 13:14.879 And the force will be q times -q over 13:14.878 --> 13:17.498 4Πε _0 times 13:17.504 --> 13:18.924 2d, right? 13:18.919 --> 13:21.209 That's the force of attraction. 13:21.210 --> 13:23.220 It will be −q^(2)/4 13:23.215 --> 13:25.605 Πε _0, 13:25.610 --> 13:27.410 the distance between them is 2d, 13:27.409 --> 13:29.919 so you can calculate that too. 13:29.918 --> 13:32.478 Normally it's a difficult problem, because the image 13:32.476 --> 13:35.536 charge reduces the problem of a very complex conductor to that 13:35.535 --> 13:36.635 of a point charge. 13:36.639 --> 13:37.389 Yes? 13:37.389 --> 13:38.359 Student: Shouldn't it be 2d squared? 13:38.360 --> 13:45.640 Prof: Yes, thank you. 13:45.639 --> 13:50.099 Then you can also ask, what is the energy it takes to 13:50.097 --> 13:54.397 assemble this distribution, starting from this one at 13:54.395 --> 13:59.015 infinity and this one here, and if you integrate this 13:59.024 --> 14:03.754 force, you can do the integral and you'll find 14:03.750 --> 14:07.630 −q^(2)/4 Πε 14:07.634 --> 14:12.364 _0 divided by 4d. 14:12.360 --> 14:14.410 That will be the energy. 14:14.408 --> 14:16.728 Na�vely, you may have thought it should be 14:16.730 --> 14:20.080 (q^(2)/4Πε _0)2d, 14:20.080 --> 14:24.050 because 2d is the distance between them, 14:24.049 --> 14:27.909 but it's only half as much. 14:27.908 --> 14:31.198 It's half as much because if in real life you had a and a 14:31.202 --> 14:33.732 -q and you brought them together, 14:33.730 --> 14:36.810 you will be exerting forces on both of them and calculating the 14:36.812 --> 14:37.412 work done. 14:37.408 --> 14:41.188 But in this actual example of an infinite plane, 14:41.193 --> 14:45.783 you only do work on this guy, therefore the answer will be 14:45.784 --> 14:46.514 half. 14:46.509 --> 14:48.989 So not everything is the same. 14:48.990 --> 14:50.250 For example, the energy in the 14:50.245 --> 14:52.955 electromagnetic field, which I'll tell you about later 14:52.958 --> 14:55.268 today, will be half as much as before, 14:55.269 --> 14:59.729 because in the fake problem, the energy is here and here. 14:59.730 --> 15:03.480 In the real problem, the field is non-zero only 15:03.480 --> 15:05.030 here, but 0 here. 15:05.029 --> 15:06.249 Yes? 15:06.250 --> 15:09.790 Student: For the potential energy, 15:09.788 --> 15:13.068 is it 1 over 4d or 1 over...? 15:13.070 --> 15:14.860 Prof: You would think it's 1 over 2d, 15:14.863 --> 15:15.153 right? 15:15.149 --> 15:16.039 Student: Yes. 15:16.038 --> 15:17.738 Prof: But I'm telling you it's 1 over 4d. 15:17.740 --> 15:25.240 Student: Because when you take the integral, 15:25.240 --> 15:27.590 you have 2 on... 15:27.592 --> 15:28.772 okay. 15:28.769 --> 15:30.129 Prof: All right. 15:30.129 --> 15:36.979 Now what if this plane was not grounded, it's a standalone 15:36.977 --> 15:37.937 plane? 15:37.940 --> 15:40.030 So let's not take it to be infinite in this problem. 15:40.029 --> 15:42.379 It's 1 million miles in radius, huge disc. 15:42.379 --> 15:43.859 You bring the charge in. 15:43.860 --> 15:46.690 What's it supposed to do? 15:46.690 --> 15:48.690 It cannot suddenly acquire a negative charge. 15:48.690 --> 15:52.450 Charge conservation says you only have the charge you have, 15:52.451 --> 15:53.361 which was 0. 15:53.360 --> 15:56.460 So what do you think you will do if you have that infinite 15:56.455 --> 15:56.885 plane? 15:56.889 --> 16:02.709 You have to remain at some potential. 16:02.710 --> 16:03.660 Yes? 16:03.658 --> 16:05.848 Student: Push positive charges out? 16:05.850 --> 16:08.450 Prof: It will split into positive and negative charges. 16:08.450 --> 16:10.930 Let's see what to do with the positive charges in a minute. 16:10.928 --> 16:14.298 The negative charges, it will arrange exactly the way 16:14.303 --> 16:16.643 it wants to in this problem, okay? 16:16.639 --> 16:22.009 That takes care of this being at a constant potential of 0. 16:22.009 --> 16:25.289 But it's got an equal number of positive charges, 16:25.285 --> 16:28.485 which you've got to dump on this infinite plane, 16:28.494 --> 16:31.434 without ruining the constant potential. 16:31.428 --> 16:35.678 The way to do that is to put it uniformly on this plane. 16:35.678 --> 16:38.718 If you spread it uniformly on the plane, you don't destroy the 16:38.721 --> 16:39.971 equal potential nature. 16:39.970 --> 16:43.130 But it turns out that field will be very, 16:43.129 --> 16:44.629 very, very small. 16:44.629 --> 16:48.339 It will be very small, because you're taking a charge 16:48.341 --> 16:51.551 q and you're putting it on a huge disc, 16:51.553 --> 16:53.983 like the radius of the galaxy. 16:53.980 --> 16:57.780 That's the charge density and that over ε_0 16:57.782 --> 16:59.622 will be the electric field. 16:59.620 --> 17:03.950 You can see that that can be made vanishingly small. 17:03.950 --> 17:06.790 So the infinite plane, basically you borrow from 17:06.792 --> 17:09.882 somewhere a positive charge and negative charge, 17:09.880 --> 17:12.250 and you put the negative in this particular arrangement. 17:12.250 --> 17:15.020 Positive is smeared over the whole thing uniformly. 17:15.019 --> 17:17.759 It is so dilute, because of the size of it, 17:17.756 --> 17:19.056 it doesn't matter. 17:19.058 --> 17:23.028 But it principle, that amount of positive charge 17:23.034 --> 17:25.744 will be a background constant. 17:25.740 --> 17:27.580 That way the infinite plane will be a little strange, 17:27.578 --> 17:30.708 because you don't see the richness of the phenomenon of 17:30.714 --> 17:34.084 the infinite plane because it can basically create negative 17:34.082 --> 17:37.452 charge and not fully account for the positive charge, 17:37.450 --> 17:39.170 because it's smeared over a huge distance. 17:39.170 --> 17:44.910 So a problem that's more interesting would be this one. 17:44.910 --> 17:51.830 I take a sphere of radius a, it's a conducting 17:51.830 --> 17:53.030 sphere. 17:53.029 --> 17:57.859 And after distance b, I put a charge q. 17:57.859 --> 18:00.709 What will happen now? 18:00.710 --> 18:03.360 Again, now let me say this is a grounded sphere, 18:03.364 --> 18:05.514 meaning it's connected to the earth. 18:05.509 --> 18:08.639 It's maintained at 0 potential. 18:08.640 --> 18:11.060 So it's not very different from the plane, except you've got a 18:11.055 --> 18:12.795 huge sphere now instead of an infinite plane, 18:12.798 --> 18:13.588 but it's finite. 18:13.588 --> 18:20.138 You come around from infinity, you can see that some negative 18:20.142 --> 18:27.242 charges will be induced on this, and some lines will come here. 18:27.240 --> 18:30.870 And some lines may go past like this. 18:30.868 --> 18:34.598 Because this is a finite sphere, not every line has to 18:34.602 --> 18:35.802 terminate here. 18:35.798 --> 18:38.818 Now the question is, what will be the charge 18:38.824 --> 18:39.884 distribution? 18:39.880 --> 18:44.200 How will it arrange itself, and what will be the surface 18:44.199 --> 18:46.319 electric field and so on? 18:46.318 --> 18:49.918 That's what you want to calculate? 18:49.920 --> 18:54.090 So what kind of problem do you think we should try to solve? 18:54.088 --> 19:01.258 If it had an easy solution, what form do you think it will 19:01.259 --> 19:02.139 take? 19:02.140 --> 19:03.380 Yes? 19:03.380 --> 19:04.960 Student: You could put a charge on the other side of 19:04.961 --> 19:05.311 the sphere? 19:05.308 --> 19:06.958 Prof: Okay, that's not a bad guess. 19:06.960 --> 19:11.110 If you put a charge on the other side of the sphere, 19:11.108 --> 19:14.868 then in the universe in which you're living, 19:14.868 --> 19:17.288 namely outside the conductor, things have changed, 19:17.289 --> 19:19.439 right? 19:19.440 --> 19:21.620 In the image charge problem, remember, the fake charge was 19:21.622 --> 19:22.582 never in your universe. 19:22.579 --> 19:23.669 It was outside. 19:23.670 --> 19:26.270 The problem you defined yourself should remain the same 19:26.272 --> 19:28.442 where you are, but you can do stuff inside the 19:28.442 --> 19:30.182 region that's excluded from you. 19:30.180 --> 19:31.430 So you're almost right. 19:31.430 --> 19:34.660 We want to put a charge inside here. 19:34.660 --> 19:39.420 And where do you think you should put it, 19:39.420 --> 19:40.610 roughly? 19:40.608 --> 19:43.838 Can you make a guess where in the sphere it will be? 19:43.839 --> 19:46.469 Student: Center? 19:46.470 --> 19:49.680 Prof: Center is a good guess, but it turns out, 19:49.684 --> 19:53.144 if you put it in the center, this doesn't turn out to be a 19:53.142 --> 19:53.812 sphere. 19:53.808 --> 19:56.348 But you agree that it should be on this line, 19:56.347 --> 19:57.557 just from symmetry. 19:57.558 --> 19:58.998 You don't want to put it anywhere else. 19:59.000 --> 20:00.140 So I will give you the answer. 20:00.140 --> 20:01.860 It's not very hard to calculate. 20:01.858 --> 20:07.908 If you put a charge q' at a distance s, 20:07.910 --> 20:15.030 which is a^(2)/b, and the size of q' is 20:15.028 --> 20:19.498 -qa/b, a/b is less than 20:19.501 --> 20:19.681 1. 20:19.680 --> 20:24.120 So q' is less than q, and is at a distance 20:24.117 --> 20:28.877 s from the center, which is a^(2)/b. 20:28.880 --> 20:30.730 This is something you can do for fun. 20:30.730 --> 20:33.990 You can go and calculate now the potential of this negative 20:33.992 --> 20:37.422 and smaller charge together with this positive charge q 20:37.424 --> 20:40.564 and you will find, if you put q'/r 20:40.560 --> 20:42.800 for that, and q/r for this 20:42.800 --> 20:44.950 guy, it will be exactly 0 on this 20:44.948 --> 20:50.038 sphere, centered here. 20:50.038 --> 20:54.538 So it's another fortunate thing that you can get a spherical 20:54.541 --> 20:58.131 equipotential by taking two unequal charges-- 20:58.130 --> 21:00.020 not two equal, two unequal charges-- 21:00.019 --> 21:03.399 and putting it slightly off center by this amount. 21:03.400 --> 21:04.830 Let's not do the algebra now. 21:04.828 --> 21:07.878 You can all imagine doing q'/r' and you'll 21:07.875 --> 21:10.085 find it's equal to q/r, 21:10.088 --> 21:12.438 up to a sign, and therefore it will be in 21:12.440 --> 21:14.250 fact-- you can always make two things 21:14.248 --> 21:17.038 cancel at one point, but they cancel over an entire 21:17.037 --> 21:17.517 sphere. 21:17.519 --> 21:19.299 It's very amazing, but true. 21:19.299 --> 21:20.449 Yes? 21:20.450 --> 21:22.510 Student: Are the field lines always perpendicular to 21:22.510 --> 21:23.490 the surface of the sphere? 21:23.490 --> 21:25.280 Prof: They must end up perpendicular. 21:25.279 --> 21:28.959 Tell me why. 21:28.960 --> 21:32.650 Can they end up anything but perpendicular to a conductor? 21:32.650 --> 21:34.040 Student: Oh, right, no. 21:34.038 --> 21:35.668 Prof: No, because if it's got a 21:35.665 --> 21:37.595 tangential part, it will move the charges. 21:37.598 --> 21:39.968 So charges, once they've come to equilibrium, 21:39.968 --> 21:43.138 the only force you can apply to a conductor is normal to the 21:43.144 --> 21:43.794 surface. 21:43.788 --> 21:47.788 So the lines will terminate in the normal way. 21:47.788 --> 21:51.478 So the problem you want to solve, the fake problem, 21:51.477 --> 21:54.797 the image problem, is a −q' and a 21:54.796 --> 21:57.816 q so located you get a sphere. 21:57.818 --> 22:01.068 Now you go and say, let me take another problem of 22:01.069 --> 22:02.529 a conducting sphere. 22:02.529 --> 22:05.029 There's nothing in it, grounded. 22:05.028 --> 22:09.088 What it will do is suck from the ground this charge and 22:09.086 --> 22:13.066 spread it exactly the way you want in this problem. 22:13.068 --> 22:16.358 And then it will come to 0 potential. 22:16.359 --> 22:19.449 So ground is here, 0 potential. 22:19.450 --> 22:22.580 If you want to know, what is the electrostatic 22:22.577 --> 22:26.327 potential there in this problem, we can do that now. 22:26.328 --> 22:28.868 What's the electrostatic potential here? 22:28.868 --> 22:34.128 It is due to this guy at that distance and the fake charge at 22:34.134 --> 22:35.544 that distance. 22:35.538 --> 22:37.648 You should add them with the proper sign. 22:37.650 --> 22:41.180 So the problem of the field due to very complicated distribution 22:41.182 --> 22:44.212 of charges on a conducting sphere and a point charge is 22:44.210 --> 22:46.790 reduced to the problem of 2 point charges. 22:46.788 --> 22:50.798 Very easy to add their potentials, take the derivative 22:50.799 --> 22:52.389 and find the field. 22:52.390 --> 22:54.540 If you want to know the surface charge density, 22:54.542 --> 22:56.652 you find the electric field at the surface. 22:56.650 --> 22:58.610 It will come out to be perpendicular. 22:58.608 --> 23:01.558 Then σ/ε_0 = 23:01.558 --> 23:04.678 E, therefore if you knew E in magnitude, 23:04.681 --> 23:08.271 you can find sigma and you can find the charge distribution on 23:08.272 --> 23:09.512 the whole sphere. 23:09.509 --> 23:16.679 If you integrate that guy, what will you get for the 23:16.683 --> 23:21.893 charge distribution on the surface? 23:21.890 --> 23:25.660 What should it give for the total charge? 23:25.660 --> 23:27.640 Pardon me? 23:27.640 --> 23:33.380 Use Gauss's law for the real problem and the fake problem. 23:33.380 --> 23:36.790 In the real problem, surface integral of E is 23:36.792 --> 23:38.602 the charge on the sphere. 23:38.598 --> 23:41.498 But the same E is produced in this combination 23:41.499 --> 23:42.669 with this guy here. 23:42.670 --> 23:45.550 The surface integral of E is a charge enclosed, 23:45.548 --> 23:47.828 which is q', after some epsilons. 23:47.828 --> 23:50.648 Therefore the charge on the sphere will turn out to be 23:50.652 --> 23:52.572 exactly the same as the q'. 23:52.568 --> 23:55.808 Just like in the plane, there'll be some negative 23:55.810 --> 23:59.730 charge, but it's spread out in a particular way and you can 23:59.729 --> 24:00.809 verify that. 24:00.808 --> 24:05.298 Now a more interesting question is, what if this is not 24:05.296 --> 24:06.206 grounded? 24:06.210 --> 24:08.810 In other words, it's an isolated sphere and 24:08.810 --> 24:10.980 you're bringing a charge near it. 24:10.980 --> 24:13.960 It wants a certain negative charge to spread on the surface 24:13.964 --> 24:16.594 to produce a zero potential here, but you don't have 24:16.589 --> 24:17.619 negative charge. 24:17.619 --> 24:18.919 You have zero charge. 24:18.920 --> 24:21.080 So what will you do if you are that sphere? 24:21.078 --> 24:25.558 You will say 0 = q' -q'. 24:25.558 --> 24:28.238 You would split into q' and −q'. 24:28.240 --> 24:31.690 The q' will arrange itself on the sphere to exactly 24:31.692 --> 24:33.512 terminate these field lines. 24:33.509 --> 24:37.369 The only difference is, now you've got a charge, 24:37.368 --> 24:39.338 -q', left over. 24:39.338 --> 24:42.068 You have to spread it around and you don't want to screw up 24:42.073 --> 24:44.763 the equal potential nature of the sphere by doing that. 24:44.759 --> 24:47.349 You can all guess what you should do. 24:47.348 --> 24:52.248 You should spread it uniformly on that sphere. 24:52.250 --> 24:54.710 So it will contain negative charge distribution, 24:54.707 --> 24:57.577 which is biased in this direction, and a positive charge 24:57.583 --> 24:59.573 distribution that's uniform in it. 24:59.568 --> 25:02.648 Then the three of them together, this one, 25:02.650 --> 25:05.360 the positive charge, I put here of −q' 25:05.362 --> 25:08.842 and this one, three of them together will 25:08.837 --> 25:12.797 keep the sphere at a constant potential. 25:12.798 --> 25:17.128 The potential it ends up with will however not be 0, 25:17.130 --> 25:20.220 because q' and q made it 0, 25:20.220 --> 25:22.430 but you have a −q' at the center, 25:22.430 --> 25:24.280 therefore you must find the potential due to 25:24.276 --> 25:26.376 −q' on the surface of the sphere, 25:26.380 --> 25:28.830 which is that. 25:28.828 --> 25:33.438 That's the potential to which the sphere will come. 25:33.440 --> 25:35.180 So we have answered many questions. 25:35.180 --> 25:37.400 We have said if you bring a charge q near the 25:37.401 --> 25:39.121 sphere, what potential will it acquire? 25:39.118 --> 25:42.258 It will acquire exactly this potential. 25:42.259 --> 25:44.549 You can predict the charge on the sphere. 25:44.548 --> 25:50.128 You can predict the force of attraction between this and that 25:50.132 --> 25:50.972 sphere. 25:50.970 --> 25:51.770 How? 25:51.769 --> 25:54.489 Because the field that this one feels generally, 25:54.486 --> 25:57.896 the field at any point is due to this one, this one and this 25:57.896 --> 25:58.356 one. 25:58.358 --> 26:01.388 But if you want the force on this one, if you'd only find the 26:01.391 --> 26:03.111 field due to the other two guys. 26:03.108 --> 26:05.878 But the other two guys, namely the sphere with all the 26:05.875 --> 26:07.955 charges on it, is simply equal to 2 point 26:07.962 --> 26:08.592 charges. 26:08.588 --> 26:10.888 So you must find the force between these 2, 26:10.890 --> 26:13.790 add to that the force between these two and you'll get 26:13.792 --> 26:14.562 something. 26:14.559 --> 26:19.379 That will be the force. 26:19.380 --> 26:22.130 It will be a force of attraction, but you can actually 26:22.131 --> 26:22.911 calculate it. 26:22.910 --> 26:28.370 This is the trick by which you can solve a variety of problems 26:28.365 --> 26:32.385 by finding equal potentials of nice shapes. 26:32.390 --> 26:35.140 Maybe one will be a nice ellipsoid, because if it is an 26:35.137 --> 26:37.477 ellipsoid, you can stick an ellipsoid there. 26:37.480 --> 26:40.740 And inside the ellipsoid will be the image charge. 26:40.740 --> 26:43.870 And if the ellipsoid is grounded, you'll get whatever 26:43.865 --> 26:46.325 charge is needed to maintain that at 0. 26:46.328 --> 26:48.798 If it's not grounded, it will split into positive and 26:48.798 --> 26:50.798 negative charges, where the negative will 26:50.801 --> 26:52.921 distribute this way, and the positive will 26:52.923 --> 26:57.183 distribute itself in some way, so that the potential is a 26:57.182 --> 26:58.172 constant. 26:58.170 --> 27:03.040 For a sphere, we know what that some way is, 27:03.038 --> 27:05.188 which is uniform. 27:05.190 --> 27:08.700 So the reason I took some time to describe this is that 27:08.700 --> 27:12.730 problems are not always finding the potential due to a bunch of 27:12.730 --> 27:13.510 charges. 27:13.509 --> 27:16.459 That's the easiest problem, but more generally you are 27:16.461 --> 27:19.581 asked, if you're given a set of conductors and a bunch of 27:19.580 --> 27:21.030 charges, what happens? 27:21.028 --> 27:22.828 If the conductors have a nice shape, 27:22.828 --> 27:25.818 like a sphere or a plane, or certain solids of 27:25.818 --> 27:29.068 revolution, we can appeal to a different 27:29.070 --> 27:32.470 problem with image charges and solve it. 27:32.470 --> 27:36.540 Now there's a wonderful theorem that says if you get a solution 27:36.540 --> 27:38.840 this way, it's the only solution. 27:38.838 --> 27:40.458 In other words, there's a theorem, 27:40.461 --> 27:42.281 which I'm not going to write down, 27:42.279 --> 27:46.389 which says if in all of space that you're interested in, 27:46.390 --> 27:48.850 or at least in a region bounded by something, 27:48.848 --> 27:51.808 if you know the potential of the boundary and you've got a 27:51.806 --> 27:54.796 whole bunch of conductors, each one at a known potential, 27:54.797 --> 27:56.877 and you've got a whole bunch of charges, 27:56.880 --> 27:59.330 there can be only one potential function, 27:59.328 --> 28:01.648 V(r), in this whole region. 28:01.650 --> 28:03.300 You cannot have two answers. 28:03.298 --> 28:05.968 There's only one answer to that question. 28:05.970 --> 28:07.730 In other words, the potential is completely 28:07.728 --> 28:09.778 determined by knowing the charge distribution, 28:09.778 --> 28:14.148 and the potential on the various conductors you stick 28:14.151 --> 28:15.161 into that. 28:15.160 --> 28:17.100 They don't all have to be to 0. 28:17.098 --> 28:19.178 This can be at 5 volts, this can be at 9 volts, 28:19.175 --> 28:20.345 this can be at 13 volts. 28:20.348 --> 28:23.028 This can be 1 coulomb, 2 coulombs, 3 coulombs, 28:23.027 --> 28:23.917 -10 coulombs. 28:23.920 --> 28:26.980 This is say this sphere at infinity of 0 potential, 28:26.979 --> 28:28.569 there's only one answer. 28:28.568 --> 28:30.298 It's called the uniqueness theorem, 28:30.298 --> 28:32.508 which is why, if you can find some way to 28:32.509 --> 28:34.609 fudge the answer in a given region, 28:34.608 --> 28:40.168 which is completely specified, that is the answer. 28:40.170 --> 28:44.640 So that is the interlude on metallic objects called 28:44.637 --> 28:45.797 conductors. 28:45.798 --> 28:50.908 Now I'm going to go to the other problem that I did towards 28:50.907 --> 28:54.957 the end, which is the notion of a capacitor. 28:54.960 --> 29:02.900 So if you take two blobs of metal, they are both neutral. 29:02.900 --> 29:06.400 Then you grab maybe a coulomb from this and stick it there, 29:06.400 --> 29:09.600 so that becomes positive and this becomes negative. 29:09.598 --> 29:12.548 Then you want to take more, you take more stuff, 29:12.548 --> 29:14.578 you can see that you're going to run into resistance, 29:14.578 --> 29:17.138 because these guys are getting positively charged. 29:17.140 --> 29:19.040 They don't want more positive charges. 29:19.038 --> 29:21.508 Meanwhile the negative charge you leave behind wants it to 29:21.510 --> 29:23.420 come back, so you're working against that. 29:23.420 --> 29:26.750 But you do some work and you start pumping charge into this. 29:26.750 --> 29:29.910 So in the end, suppose you have a charge 29:29.906 --> 29:33.466 Q there and charge -Q there. 29:33.470 --> 29:36.430 Then there will be a potential energy difference between the 29:36.432 --> 29:39.552 two, because there's a certain amount of work needed to go from 29:39.546 --> 29:40.446 here to there. 29:40.450 --> 29:42.780 The potential difference is a unique number, 29:42.782 --> 29:45.882 because the whole solid has one potential, other solid has 29:45.875 --> 29:47.065 another potential. 29:47.068 --> 29:49.298 No matter where you start and where you end, 29:49.296 --> 29:52.086 if you find the work done, that potential difference we 29:52.093 --> 29:53.443 like to call V. 29:53.440 --> 29:58.170 Therefore V is always going to be proportional to 29:58.173 --> 30:03.423 Q and the constants of proportionality we like to write 30:03.423 --> 30:06.783 downstairs and call it capacitance. 30:06.778 --> 30:10.888 So that's the ability of the system to hold charge. 30:10.890 --> 30:15.490 If you've got any two metal containers, you can store energy 30:15.488 --> 30:18.918 by drawing the charges from one to the other, 30:18.916 --> 30:20.706 putting them there. 30:20.710 --> 30:24.900 I did calculate the capacitance for a very simple system and I'm 30:24.896 --> 30:28.816 going to stick to the simple system because we don't want to 30:28.817 --> 30:30.677 get lost in the details. 30:30.680 --> 30:35.390 It's the parallel plate capacitor in which I put some 30:35.394 --> 30:40.834 charge Q on the upper plate and -Q on the lower 30:40.833 --> 30:41.743 plate. 30:41.740 --> 30:44.910 Someone says, what's the capacitance of the 30:44.910 --> 30:45.590 system? 30:45.588 --> 30:49.238 For that, you must find the voltage difference between the 30:49.238 --> 30:52.948 two plates, and you want to take Q over V. 30:52.950 --> 30:56.230 Now there's going to be an electric field here, 30:56.228 --> 30:59.578 E, which is σ/ε_0 30:59.577 --> 31:00.857 pointing down. 31:00.858 --> 31:03.758 So the voltages difference will be the electric field times the 31:03.762 --> 31:06.482 distance, because that's what the line integral will be. 31:06.480 --> 31:10.780 That's the distance between the plates. 31:10.778 --> 31:18.108 And σ is Q/A. 31:18.108 --> 31:21.208 Now in a real capacitor, near the ends there'll be some 31:21.212 --> 31:22.192 funny business. 31:22.190 --> 31:25.260 It's not going to be simply like this, but we take it to be 31:25.262 --> 31:28.552 so large in this extent that the edge effects are neglected. 31:28.549 --> 31:30.649 Then this formula is valid. 31:30.650 --> 31:33.980 Then you can compare it to Q/C and you can 31:33.979 --> 31:36.159 see C is ε_0 31:36.159 --> 31:39.849 a/d. So this is what I have done towards the end 31:39.851 --> 31:40.761 of class. 31:40.759 --> 31:44.479 That's the capacitance of that. 31:44.480 --> 31:47.210 Once again, you can talk about capacitance, but you cannot 31:47.213 --> 31:48.273 always calculate it. 31:48.269 --> 31:50.689 You can take two irregular metallic objects and in 31:50.689 --> 31:52.959 principle for a transfer of charge q. 31:52.960 --> 31:55.560 There will be a voltage v, and the ratio is the 31:55.558 --> 31:57.568 capacitance, but you cannot compute it. 31:57.568 --> 32:01.268 But for a parallel plate geometry, or another one I did 32:01.273 --> 32:05.463 in class near the end, you take two concentric spheres 32:05.460 --> 32:09.930 and put some charge on this one, maybe and - on that one, 32:09.933 --> 32:14.423 you all know how to find the field in the region in between. 32:14.420 --> 32:17.140 You can integrate it and you can find the potential 32:17.142 --> 32:17.852 difference. 32:17.848 --> 32:20.888 And then you can see it is proportional to Q, 32:20.891 --> 32:23.991 and you take the ratio, you'll get the capacitance of 32:23.992 --> 32:24.532 that. 32:24.528 --> 32:28.258 Now the point, one thing I want to calculate, 32:28.259 --> 32:30.749 which also I think I started doing is, 32:30.750 --> 32:37.370 if I take a capacitor and I charge it, 32:37.368 --> 32:40.788 from 0 to some charge Q_0, 32:40.788 --> 32:47.048 what's the total amount of energy stored in the capacitor? 32:47.048 --> 32:50.608 So the way I think about it is, I take some intermediate 32:50.608 --> 32:54.358 situation, when there's a charge Q on the capacitor, 32:54.362 --> 32:55.982 not the final amount. 32:55.980 --> 33:01.930 Then I want to take a little amount of charge dQ and I 33:01.925 --> 33:07.075 want to move it from the negative to the positive. 33:07.078 --> 33:10.408 Since at that stage the potential difference is 33:10.412 --> 33:14.832 Q/C that potential difference times dQ is 33:14.830 --> 33:16.280 the work you do. 33:16.278 --> 33:18.558 That's the definition of potential difference, 33:18.557 --> 33:21.637 how much work it takes to move a coulomb from one plate to the 33:21.643 --> 33:22.153 other. 33:22.150 --> 33:24.810 If you're moving dQ coulombs, that's the work you 33:24.807 --> 33:25.047 do. 33:25.048 --> 33:29.008 You integrate that from 0 to some maximum value 33:29.011 --> 33:32.111 Q_0, you see you get 33:32.114 --> 33:35.564 Q_0 ^(2)/2C. 33:35.558 --> 33:38.648 So the energy in a capacitor is q^(2)/2C. 33:38.650 --> 33:40.450 Forget the 0, subscript 0. 33:40.450 --> 33:45.560 Usually Q stands for the charge of a capacitor. 33:45.558 --> 33:47.738 But you can also write it in another way. 33:47.740 --> 33:52.370 Since Q = CV, you can write it as 33:54.880 --> 34:03.870 There are two ways to write the energy. 34:09.099 --> 34:11.569 but I'm going to play with that expression and get a very 34:11.568 --> 34:16.028 interesting result, very profound. 34:16.030 --> 34:18.960 Capacitance is ε_0 34:18.961 --> 34:23.271 A/d, what about the voltage? 34:23.269 --> 34:26.929 It is E/d. 34:26.929 --> 34:29.589 Sorry, voltage E times d. 34:29.590 --> 34:34.880 E squared, d squared. 34:34.880 --> 34:35.870 So what do you get? 34:35.869 --> 34:43.759 You get ε_0 E^(2)/2 times a 34:43.757 --> 34:46.067 times d. 34:46.070 --> 34:49.650 What is A times d? 34:49.650 --> 34:54.920 It's a volume of the region that contains the electric 34:54.922 --> 34:55.722 field. 34:55.719 --> 34:59.409 Before you charge a capacitor, there is no energy and there is 34:59.405 --> 35:00.065 no field. 35:00.070 --> 35:03.730 Once you charge the capacitor, you've got a field. 35:03.730 --> 35:06.420 And if you wish, you can say I've got to ascribe 35:06.418 --> 35:09.508 that energy to the fact that it's a non-zero field, 35:09.510 --> 35:12.410 in which case, this is the energy per unit 35:12.411 --> 35:15.301 volume, due to an electric field. 35:15.300 --> 35:16.530 That's a very interesting notion. 35:16.530 --> 35:22.660 Little u is the energy per unit volume and that happens 35:22.664 --> 35:27.194 to be ε_0 E^(2)/2. 35:27.190 --> 35:29.730 In other words, it takes energy to establish 35:29.728 --> 35:30.968 the electric field. 35:30.969 --> 35:33.279 So in this room, if you've go to a tiny region 35:33.282 --> 35:36.472 where E is essentially constant over the tiny region. 35:36.469 --> 35:40.519 ε_0 E^(2)/2 times the tiny 35:40.523 --> 35:44.013 volume is the energy in that tiny region. 35:44.010 --> 35:46.030 So once you've got an electric field, it just cannot just 35:46.027 --> 35:46.457 disappear. 35:46.460 --> 35:49.430 Law of conservation of energy will require that you account 35:49.429 --> 35:51.939 for it, and this is the energy per unit volume. 35:51.940 --> 35:54.570 It turns out to work even if you've got radio waves, 35:54.570 --> 35:56.870 electromagnetic waves, going anywhere, 35:56.869 --> 35:59.279 to take that energy, electric field squared times 35:59.284 --> 36:03.514 epsilon over 2, that's the energy density. 36:03.510 --> 36:07.970 All right, so this is really the end of what I wanted to 36:07.974 --> 36:11.794 finish last time, but I wanted to go back to the 36:11.789 --> 36:15.279 study of equipotentials and conductors. 36:15.280 --> 36:26.220 But now I am setting the stage for electrical circuits. 36:26.219 --> 36:30.099 Now this is the kind of thing some of you probably did 36:30.101 --> 36:32.081 somewhere in high school. 36:32.079 --> 36:36.299 So how many people have done basic circuits in high school? 36:36.300 --> 36:38.860 So I'm going to assume you've done some of it, 36:38.856 --> 36:41.926 so I won't do it in that detail, but I will mention all 36:41.925 --> 36:43.285 the essential facts. 36:43.289 --> 36:47.069 For the few of you that didn't do it, you have a chance to keep 36:47.070 --> 36:48.290 up with the class. 36:48.289 --> 36:52.039 The first thing in electrical circuits is, you've got some 36:52.041 --> 36:55.731 wire and you've got an electric current flowing in it. 36:55.730 --> 36:58.910 We need a description of the current. 36:58.909 --> 37:01.159 The current is defined as follows. 37:01.159 --> 37:05.069 Imagine this is the perfect cylinder, cross section 37:05.068 --> 37:05.928 A. 37:05.929 --> 37:09.539 You cut it somewhere and you watch all the charges go by, 37:09.538 --> 37:13.338 and you see the number of coulombs that go by per second. 37:13.340 --> 37:19.880 That's called the electric current and is measured in 37:19.880 --> 37:21.140 amperes. 37:21.139 --> 37:27.369 So 1 coulomb per second is 1 amp. 37:27.369 --> 37:31.269 All right, now let us ask, what's the connection between 37:31.268 --> 37:35.448 the electric current and what's going on microscopically? 37:35.449 --> 37:46.929 So let n = number of carriers per unit volume. 37:46.929 --> 37:53.949 Let e be the charge of a carrier. 37:53.949 --> 37:56.849 Now here is one of the biggest nuisances in life. 37:56.849 --> 38:00.709 As you know, in a wire, the current is 38:00.710 --> 38:03.110 carried by electrons. 38:03.110 --> 38:07.630 Because the charge is negative, when you draw a picture like 38:07.628 --> 38:12.218 this, the current to the right, electrons are actually moving 38:12.224 --> 38:13.454 to the left. 38:13.449 --> 38:16.919 What we will instead do is to just keep an eye on the 38:16.918 --> 38:18.718 direction of the current. 38:18.719 --> 38:20.759 You imagine there are positively charged objects 38:20.759 --> 38:23.189 carrying the current in the direction of the current, 38:23.190 --> 38:26.180 whereas in reality, it's negatively charged objects 38:26.184 --> 38:30.084 moving in the opposite direction that produce the same current. 38:30.079 --> 38:35.279 So you can ask yourself, if you wait 1 second, 38:35.282 --> 38:40.952 how many coulombs will go past this checkpoint? 38:40.949 --> 38:44.179 You can see that it's a cylinder whose length is the 38:44.179 --> 38:47.729 velocity, because in 1 second, it will have gone v 38:47.726 --> 38:48.926 times 1 second. 38:48.929 --> 38:52.529 All those guys have crossed the finish line. 38:52.530 --> 38:57.710 Therefore the current will be A times v (is the 38:57.708 --> 39:02.358 volume of stuff that's gone), that's the number of carriers 39:02.358 --> 39:04.968 in the volume, that's the charge of each 39:04.972 --> 39:05.542 carrier. 39:05.539 --> 39:13.859 That's the total current. 39:13.860 --> 39:19.510 So we like to define a quantity called the current density, 39:19.512 --> 39:23.122 which is the current per unit area. 39:23.119 --> 39:30.429 That will be equal to nev. 39:30.429 --> 39:34.349 And actually, current density is a vector so 39:34.351 --> 39:39.731 if you want, because velocity is a vector, you can make it a 39:39.733 --> 39:41.653 vector like that. 39:41.650 --> 39:44.490 In this problem, the current density is uniform. 39:44.489 --> 39:46.779 In fact, in any wire, the current here and the 39:46.784 --> 39:49.494 current there and the current there do not change, 39:49.489 --> 39:52.399 because if it changed, charges will pile up in some 39:52.400 --> 39:54.030 places like a traffic jam. 39:54.030 --> 39:56.960 Then it will resist it till this thing evens out. 39:56.960 --> 39:58.850 Current is constant on a wire. 39:58.849 --> 40:01.169 Need not be uniform across a cross section of the wire, 40:01.168 --> 40:02.928 but I'm going to take it to be uniform. 40:02.929 --> 40:07.869 But if it's not constant, then you can have a J 40:07.867 --> 40:12.657 that's varying with space, so the current crossing that 40:12.664 --> 40:15.514 surface will be the surface integral of 40:15.505 --> 40:18.045 J⋅dA. 40:18.050 --> 40:20.070 In other words, divide that area into tiny 40:20.065 --> 40:23.005 little patches, dA, then as we have seen 40:23.014 --> 40:25.464 many times, whenever something is flowing, 40:25.456 --> 40:28.346 it's the dot product of the flow rate with the area that 40:28.347 --> 40:29.817 measures the actual flow. 40:29.820 --> 40:33.080 You add it over the surface, that's the current crossing a 40:33.077 --> 40:33.647 surface. 40:33.650 --> 40:35.220 We won't use this formula very much. 40:35.219 --> 40:41.189 I'm just mentioning it for completeness. 40:41.190 --> 40:46.690 All right, so the next question is, I've told you there is no 40:46.690 --> 40:49.990 electric field inside a conductor. 40:49.989 --> 40:53.109 But if you take a resistor, even a nice thing like copper, 40:53.105 --> 40:54.795 it's not a perfect conductor. 40:54.800 --> 40:59.490 Actually, there's an electric field inside a conductor. 40:59.489 --> 41:02.269 That's what makes a current flow. 41:02.268 --> 41:06.608 If you want any wire to carry current, except for ideal wires 41:06.614 --> 41:10.604 that have no resistance, any realistic conductor needs a 41:10.597 --> 41:12.187 field to drive it. 41:12.190 --> 41:16.020 The reason is that if you look microscopically at these 41:16.021 --> 41:20.211 carriers, they're all going like crazy in all directions. 41:20.210 --> 41:22.180 That's precisely why they don't carry a current. 41:22.179 --> 41:23.579 They don't have a common direction. 41:23.579 --> 41:26.379 They're going very fast but they're going nowhere. 41:26.380 --> 41:27.310 One of these guys is going to the right. 41:27.309 --> 41:29.909 For every one going to the right, there's one to the left, 41:29.909 --> 41:32.009 one going northeast and one going southwest. 41:32.010 --> 41:35.820 The velocity all averages out to 0. 41:35.820 --> 41:38.990 But if you apply an electric field to the right, 41:38.989 --> 41:42.189 you can imagine somehow in the middle of all this chaos, 41:42.190 --> 41:47.140 there'll be an overall tendency to drift to the right. 41:47.139 --> 41:50.129 So let's see how much there is. 41:50.130 --> 41:53.480 I found that the current density was nev. 41:53.480 --> 41:54.990 Forget about the vector sign now. 41:54.989 --> 41:58.179 Just everything is along x. 41:58.179 --> 42:03.899 I want to find here an average velocity. 42:03.900 --> 42:08.140 What's going to be the average velocity over all the particles 42:08.139 --> 42:09.389 at a given time? 42:09.389 --> 42:11.579 Now here's the picture of conduction. 42:11.579 --> 42:15.489 You should in fact find it very paradoxical that when you have 42:15.492 --> 42:18.252 an electric field, the electric field produces a 42:18.251 --> 42:21.821 force e times E, has an acceleration a, 42:21.824 --> 42:24.944 so the charges should accelerate. 42:24.940 --> 42:28.160 If they accelerate more and more, the current should keep on 42:28.163 --> 42:30.243 growing because velocity is growing. 42:30.239 --> 42:34.129 But you get a steady current, in spite of a force acting to 42:34.134 --> 42:34.944 the right. 42:34.940 --> 42:38.900 Do you know why that happens? 42:38.900 --> 42:43.800 Why doesn't everything accelerate and pick up more and 42:43.797 --> 42:45.827 more and more speed? 42:45.829 --> 42:47.479 Student: Resistance. 42:47.480 --> 42:50.720 Prof: It is resistance, but microscopically, 42:50.719 --> 42:53.959 why don't these particles pick up speed forever? 42:53.960 --> 42:56.160 They collide. 42:56.159 --> 43:00.939 They collide with basically the impurities in the solid. 43:00.940 --> 43:03.500 And every time they hit their head on one of the impurities, 43:03.500 --> 43:06.510 they don't know what happened, and they bounce off the 43:06.512 --> 43:08.902 collision in a totally random direction. 43:08.900 --> 43:12.120 So they may go in this direction, hit an impurity. 43:12.119 --> 43:14.259 There's no telling which way they'll come out. 43:14.260 --> 43:16.720 They can come out in any direction with equal 43:16.717 --> 43:17.497 probability. 43:17.500 --> 43:22.450 So they'll lose all memory of what they were doing after each 43:22.449 --> 43:23.439 collision. 43:23.440 --> 43:28.400 Now if I look at my clock, and I look at the entire set of 43:28.400 --> 43:32.840 electrons and I say, what's the average velocity? 43:32.840 --> 43:36.180 The average velocity, which I denote with a bar, 43:36.175 --> 43:39.505 is obtained by averaging individual velocity. 43:39.510 --> 43:44.260 Individual velocity is the velocity at time 0 since the 43:44.260 --> 43:48.130 last collision, plus eE/m times 43:48.130 --> 43:50.330 t_i. 43:50.329 --> 43:52.429 In other words, let me explain to you slowly 43:52.427 --> 43:53.107 what I mean. 43:53.110 --> 43:56.540 Take particle number i. 43:56.539 --> 43:59.739 Let us say it has been t_i seconds 43:59.735 --> 44:01.905 since it had its last collision. 44:01.909 --> 44:04.869 So just after the collision, it has got initial velocity 44:04.867 --> 44:07.687 which is completely random, but since that thing, 44:07.686 --> 44:10.786 because the electric field it has been accelerating, 44:10.789 --> 44:12.349 it has not collided with anything yet, 44:12.349 --> 44:15.269 so its velocity will be eE/m times 44:15.269 --> 44:16.949 t_i. 44:16.949 --> 44:20.489 Therefore the average velocity, which you find by the averaging 44:20.494 --> 44:22.904 of everything, has the following property - 44:22.896 --> 44:24.836 the first one will give you 0. 44:24.840 --> 44:27.390 If you average over all possible initial velocities, 44:27.393 --> 44:29.403 they'll be 0, because they're pointing in 44:29.398 --> 44:30.498 random directions. 44:30.500 --> 44:34.460 So what you really need is eE/m times some 44:34.460 --> 44:38.590 τ, where τ is the name for the 44:38.585 --> 44:42.955 average time since the last collision. 44:42.960 --> 44:44.660 That time will have a range of values. 44:44.659 --> 44:46.639 Some guys will have just collided; 44:46.639 --> 44:49.159 some won't have collided for a long time. 44:49.159 --> 44:51.669 So you have to find that average, and we don't know how 44:51.672 --> 44:53.912 to compute it right now, but that is some average 44:53.907 --> 44:55.767 t_i for a material. 44:55.768 --> 44:57.608 The larger the t_i is, 44:57.608 --> 45:00.458 the better the conductor it is, because it can go for a long 45:00.460 --> 45:02.300 time on average without colliding. 45:02.300 --> 45:04.810 So the whole idea is, the minute you collide, 45:04.813 --> 45:07.333 you lose everything, because you scatter off, 45:07.327 --> 45:08.867 forgetting your memory. 45:08.869 --> 45:12.409 So any coherent motion you have in the direction of the field is 45:12.414 --> 45:15.064 there only because you have not yet collided. 45:15.059 --> 45:18.089 So the net current is a function of how many seconds 45:18.085 --> 45:21.225 have elapsed since the last collision for each guy. 45:21.230 --> 45:23.440 On average, it gives you this. 45:23.440 --> 45:28.500 So if you use the symbol tau for that, therefore the current 45:28.501 --> 45:33.221 density will be ne times the average v. 45:33.219 --> 45:41.909 The average v is eE/mτ. 45:41.909 --> 45:47.679 So if you write it as ne^(2)τ/m 45:47.681 --> 45:50.931 times the electric field. 45:50.929 --> 45:52.779 So this is the very, very important result. 45:52.780 --> 46:03.140 So I'm going to go back and study this result. 46:03.139 --> 46:06.939 It tells you that the current density in a wire is there 46:06.943 --> 46:10.893 thanks to the electric field and the number in front of it 46:10.885 --> 46:13.305 happens to be the number of carriers 46:13.306 --> 46:15.586 e^(2)τ/m. 46:15.590 --> 46:17.240 Does that formula make sense? 46:17.239 --> 46:19.829 Take a minute to look at that formula. 46:19.829 --> 46:21.459 You apply a field. 46:21.460 --> 46:23.640 That's why they even know which way to flow. 46:23.639 --> 46:25.419 If you don't have a field in a wire, the current doesn't know 46:25.416 --> 46:26.006 which way to flow. 46:26.010 --> 46:28.760 It's just random motion, not going anywhere, 46:28.760 --> 46:30.680 like molecules in this room. 46:30.679 --> 46:33.709 They're not going anywhere in particular, even though they're 46:33.706 --> 46:34.156 moving. 46:34.159 --> 46:37.089 You are saying that the current that you're going to get for 46:37.085 --> 46:40.205 unit area is bigger if you have a bigger density of carriers. 46:40.210 --> 46:42.770 That makes sense, because they're the ones 46:42.773 --> 46:44.153 carrying the charge. 46:44.150 --> 46:47.130 This is inversely proportional to the mass. 46:47.130 --> 46:48.380 You can understand why. 46:48.380 --> 46:50.900 The electric field, whatever force it produces, 46:50.902 --> 46:54.032 the acceleration is inversely proportional to the mass. 46:54.030 --> 46:56.730 The bigger the τ, the bigger the response, 46:56.730 --> 46:59.740 because they can go for a longer time on average before 46:59.744 --> 47:01.814 colliding, therefore they have more time 47:01.806 --> 47:04.016 to pick up speed in the direction of the field. 47:04.019 --> 47:06.779 e^(2) is interesting. 47:06.780 --> 47:09.870 One e comes because the force on the carrier is little 47:09.873 --> 47:11.423 e times big E. 47:11.420 --> 47:14.460 The second e comes because the current it carries 47:14.461 --> 47:16.511 is itself proportional to e. 47:16.510 --> 47:17.280 You understand? 47:17.280 --> 47:20.050 The charge of the carrier affects it in two ways. 47:20.050 --> 47:22.560 Any time it moves, it carries a charge e. 47:22.559 --> 47:24.979 How much it moves depends on the force the electric field 47:24.981 --> 47:25.631 exerts on it. 47:25.630 --> 47:26.970 That's another e. 47:26.969 --> 47:29.369 So this is our expression. 47:29.369 --> 47:34.199 And we write it as σ times E, 47:34.204 --> 47:39.044 where sigma is called the conductivity. 47:39.039 --> 47:42.309 You notice that this argument doesn't care what material it 47:42.307 --> 47:42.587 is. 47:42.590 --> 47:44.580 It could be copper, it could be aluminum, 47:44.583 --> 47:45.833 it could be some alloy. 47:45.829 --> 47:47.979 For all of them, the answer depends on 47:47.978 --> 47:49.718 ne^(2)τ/m. 47:49.719 --> 47:53.679 What depends on the actual carrier, the mass is just the 47:53.675 --> 47:55.325 mass of the electron. 47:55.329 --> 47:58.179 τ is what varies from problem to problem. 47:58.179 --> 48:00.719 Some materials have very large τ, some materials are very 48:00.717 --> 48:01.307 small τ. 48:01.309 --> 48:06.059 That's what decides how good or bad a conductor it is 48:06.059 --> 48:07.889 So this is the formula. 48:07.889 --> 48:11.519 Now sometimes you can write it in another way. 48:11.519 --> 48:12.179 Let's see now. 48:12.179 --> 48:17.059 Suppose you have a wire. 48:17.059 --> 48:19.159 What is the total current in the wire? 48:19.159 --> 48:23.999 The total current in the wire is the current density times 48:24.001 --> 48:26.891 area, which is σAE. 48:26.889 --> 48:31.139 Now let us say this E was obtained by applying the 48:31.144 --> 48:35.554 voltage difference V between the two end points of a 48:35.552 --> 48:37.682 wire of length L. 48:37.679 --> 48:48.589 There is σAV/L. 48:48.590 --> 48:49.230 Do you understand? 48:49.230 --> 48:51.640 The electric field times the length of the wire is the 48:51.643 --> 48:53.833 voltage difference between the two end points. 48:53.829 --> 48:56.179 So rather than saying the current is driven by the 48:56.184 --> 48:58.564 electric field, let's say the current is driven 48:58.561 --> 49:01.101 by the voltage difference between the two ends of the 49:01.099 --> 49:01.489 wire. 49:01.489 --> 49:05.549 That's why I wrote E as V/L. 49:05.550 --> 49:10.180 But now you see it looks like V/R that R 49:10.184 --> 49:14.594 is by definition is called the resistance of that wire. 49:14.590 --> 49:21.040 It's measured in ohms, as denoted by this symbol. 49:21.039 --> 49:24.459 This is how you get ohm's law, because just by going through 49:24.463 --> 49:28.353 the microscopic equation, and applying it for a wire of 49:28.353 --> 49:31.283 length L, I'm able to find that the 49:31.278 --> 49:33.588 current is proportional to V, 49:33.590 --> 49:35.350 divided by some number. 49:35.349 --> 49:41.759 That number looks like L/Aσ, 49:41.760 --> 49:44.440 also written as Lρ/A, 49:44.440 --> 49:49.460 where ρ = 1/σ is called the resistivity. 49:49.460 --> 49:55.650 It's just the inverse of the conductivity. 49:55.650 --> 49:59.040 So the bigger the resistivity, the bigger the resistance, 49:59.039 --> 50:01.459 but notice, if the wire is twice as long, 50:01.460 --> 50:04.610 the resistance of the wire will be twice as big. 50:04.610 --> 50:06.810 Does that make sense to you? 50:06.809 --> 50:09.269 For a given material, if you double the length of the 50:09.268 --> 50:10.638 wire, resistance is double. 50:10.639 --> 50:13.799 If you double the area, resistance is turned into half 50:13.795 --> 50:16.325 its value, because if you've got a big 50:16.333 --> 50:20.193 wire, you can think of it as two wires that are carrying the 50:20.188 --> 50:23.398 current together, therefore the resistance is 50:23.396 --> 50:24.346 half as much. 50:24.349 --> 50:29.319 So this is the relation between resistivity and the resistance. 50:29.320 --> 50:32.250 As far as we are concerned, the main thing for us is just 50:32.248 --> 50:36.748 ohm's law, which we are going to use - 50:36.753 --> 50:41.263 V = IR, and this is telling you roughly 50:41.259 --> 50:42.559 where you get it. 50:42.559 --> 50:45.279 So what's the summary of all this talk? 50:45.280 --> 50:50.070 In a wire, unlike in a perfect conductor, there is an electric 50:50.070 --> 50:50.700 field. 50:50.699 --> 50:55.219 And it's the electric field that keeps the charges moving. 50:55.219 --> 50:57.499 But whereas an electron, for example, 50:57.503 --> 51:00.553 in the vacuum electric field that will accelerate 51:00.550 --> 51:01.630 indefinitely. 51:01.630 --> 51:04.530 The carriers in the wire do not accelerate indefinitely because 51:04.532 --> 51:06.972 they keep bumping into stuff, and every time they bump into 51:06.969 --> 51:08.989 stuff, they lose the gain they had. 51:08.989 --> 51:10.439 They start all over again. 51:10.440 --> 51:13.200 So at any given time, the activity I have or the 51:13.197 --> 51:16.717 motion I have depends on how many guys are still around since 51:16.719 --> 51:18.069 the last collision. 51:18.070 --> 51:21.120 They're the ones who have been picking up speed, 51:21.123 --> 51:24.633 and that's how you get conductivity proportional to the 51:24.630 --> 51:25.800 applied field. 51:25.800 --> 51:28.450 So it's hard to get a velocity proportional to force. 51:28.449 --> 51:31.379 You always get acceleration proportional to force, 51:31.382 --> 51:34.322 but when you've got random motion, the velocity is 51:34.315 --> 51:36.585 proportional to the applied field. 51:36.590 --> 51:39.720 By the way, I should tell you, in a real solid, 51:39.719 --> 51:43.529 if you ask what do they collide into, can you imagine? 51:43.530 --> 51:48.600 You have a mental picture of a solid where the atoms form a 51:48.601 --> 51:49.741 nice array. 51:49.739 --> 51:51.879 The nuclei form a nice periodic array. 51:51.880 --> 51:56.850 The electrons in a metal are free to travel the length and 51:56.849 --> 51:58.679 width of the solid. 51:58.679 --> 52:02.329 But who do you think I'm talking about when I say 52:02.331 --> 52:03.321 collisions? 52:03.320 --> 52:04.790 Pardon me? 52:04.789 --> 52:05.639 Student: Other electrons. 52:05.639 --> 52:07.259 Prof: Other electrons is possible. 52:07.260 --> 52:10.380 How about with the nuclei? 52:10.380 --> 52:12.240 If you're an electron, you're going through, 52:12.239 --> 52:15.099 you see a nuclei every whatever, 10 to the -8 52:15.096 --> 52:18.126 centimeters, there's another nucleus, 52:18.125 --> 52:20.605 but that's not what matters. 52:20.610 --> 52:23.210 This is more advanced theory, when you try to find the 52:23.213 --> 52:24.593 conductivity of materials. 52:24.590 --> 52:29.350 A perfectly periodic lattice of nuclei, electrons have a way to 52:29.351 --> 52:32.811 travel through them without ever colliding. 52:32.809 --> 52:36.169 It's like, if you cannot see and the furniture is all in a 52:36.166 --> 52:38.756 certain place, you can navigate freely around 52:38.759 --> 52:39.289 them. 52:39.289 --> 52:42.319 But if somebody moves something and you're not expecting it, 52:42.320 --> 52:44.120 that's when you have a collision. 52:44.119 --> 52:46.769 And that motion comes about when you heat the solid. 52:46.768 --> 52:49.388 When you heat the solid, the nuclei start vibrating, 52:49.391 --> 52:51.911 so you don't know quite where somebody will be. 52:51.909 --> 52:54.809 That gives a small probability for the electron to collide with 52:54.811 --> 52:55.141 them. 52:55.139 --> 52:57.939 Therefore the conductivity will quite often depend on the 52:57.943 --> 53:00.453 temperature, but it can also depend on the force of 53:00.447 --> 53:02.197 interactions between electrons. 53:02.199 --> 53:05.859 But even if electrons don't interact with each other in a 53:05.864 --> 53:09.724 serious way, the collision with the nucleus is controlled by 53:09.724 --> 53:11.234 lattice vibrations. 53:11.230 --> 53:13.570 But no matter how sophisticated the calculation is, 53:13.568 --> 53:16.228 you can go to my office and ask, what are people doing for 53:16.233 --> 53:17.033 conductivity? 53:17.030 --> 53:20.480 Everyone is trying to compute this quantity τ 53:20.483 --> 53:22.933 and ne^(2)τ/m. 53:22.929 --> 53:24.529 The n, the e^(2), 53:24.534 --> 53:26.624 they're all what you think they mean. 53:26.619 --> 53:29.239 τ is more sophisticated, and you have to calculate it in 53:29.244 --> 53:30.094 a quantum theory. 53:30.090 --> 53:31.650 But in the end, after all the work, 53:31.650 --> 53:34.400 you get a number τ, you put it in the same formula, 53:34.400 --> 53:40.120 ne^(2)τ/m to get the conductivity of a 53:40.115 --> 53:41.255 material. 53:41.260 --> 53:44.970 So now I want to do a little electric circuits for you. 53:44.969 --> 53:48.079 So here's one simple circuit I'm going to do. 53:48.079 --> 53:55.789 I'm going to take a capacitor, charge it up to some amount 53:55.786 --> 54:00.936 Q, then I'm going to put my 54:00.938 --> 54:09.908 resistor here like this and ask what happens when I close that 54:09.909 --> 54:11.379 switch. 54:11.380 --> 54:14.300 So when I close the switch, we can all imagine what will 54:14.300 --> 54:14.780 happen. 54:14.780 --> 54:17.450 These positive guys are dying to get over to the negative 54:17.454 --> 54:20.564 side, but they cannot jump the gap here, because it's a vacuum. 54:20.559 --> 54:23.639 But if you give them a path, they will go through that and 54:23.639 --> 54:25.259 come back to the other side. 54:25.260 --> 54:28.770 So charges will start going round if only I close the 54:28.771 --> 54:29.381 switch. 54:29.380 --> 54:32.350 So when I close the switch, let me write down an equation 54:32.349 --> 54:33.569 for what will happen. 54:33.570 --> 54:35.910 There's a resistance here. 54:35.909 --> 54:39.249 This is the current I flowing here. 54:39.250 --> 54:43.050 The fundamental equation you write down in any circuit is if 54:43.045 --> 54:46.315 you start at any point, you take any closed path and 54:46.315 --> 54:48.365 find all the changes in potential, 54:48.369 --> 54:51.179 the change has to be 0. 54:51.179 --> 54:53.219 Because it's coming from a conservative field, 54:53.221 --> 54:55.581 the integral of the electric field on a loop is 0. 54:55.579 --> 54:59.709 That means the total change in potential from anywhere back to 54:59.711 --> 55:03.101 the same place is 0, because it's like a height. 55:03.099 --> 55:06.809 So when I start here and I go through the capacitor, 55:06.809 --> 55:11.569 I go up in electrical height by an amount Q/C. 55:11.570 --> 55:16.010 This conductor is assumed to be a perfect conductor, 55:16.010 --> 55:18.690 so there is no electric field inside here, 55:18.690 --> 55:23.150 there's no change in potential until I come to this resistor. 55:23.150 --> 55:25.630 A resistor will not carry current unless there's a voltage 55:25.632 --> 55:26.332 applied to it. 55:26.329 --> 55:28.869 You've seen there, and this is the higher end of 55:28.867 --> 55:31.507 the voltage, this is the lower end of the voltage, 55:31.514 --> 55:33.354 because it's flowing downhill. 55:33.349 --> 55:36.579 So you have a drop in voltage by an amount RI, 55:36.583 --> 55:39.263 then you come back to the starting point. 55:39.260 --> 55:43.580 All the changes you had better add up to 0. 55:43.579 --> 55:45.569 That's the fundamental statement that in any electrical 55:45.565 --> 55:47.055 circuit, if you start anywhere and go 55:47.063 --> 55:49.333 through any path in the circuit and come back to a starting 55:49.329 --> 55:51.189 point, the net change in voltage has 55:51.192 --> 55:53.242 to be 0, because every point has an 55:53.242 --> 55:54.372 electrical height. 55:54.369 --> 55:56.379 So this is our equation. 55:56.380 --> 56:02.920 Now we can write it as RI = Q/C. 56:02.920 --> 56:10.350 But what's the relation between I and Q? 56:10.349 --> 56:14.479 Can you find a relation between I and Q? 56:14.480 --> 56:19.430 Suppose a current flows for a short time. 56:19.429 --> 56:22.109 Who's paying for it? 56:22.110 --> 56:26.100 Where is the charge coming from? 56:26.099 --> 56:27.229 Yes? 56:27.230 --> 56:34.450 Student: Isn't I Q divided by time? 56:34.449 --> 56:37.859 Prof: It's not necessarily divided by time, 56:37.858 --> 56:39.528 but it's a derivative. 56:39.530 --> 56:43.720 I is dQ /dt, because in a small time 56:43.717 --> 56:45.997 dt, when the current I 56:45.998 --> 56:48.838 flows, Idt coulombs flow through here, 56:48.840 --> 56:50.740 but they've got to come from there. 56:50.739 --> 56:54.729 But the only thing missing is the - sign. 56:54.730 --> 56:57.530 Because if I is defined this way and is positive, 56:57.529 --> 57:00.269 a positive I depletes the charge, so I is 57:00.269 --> 57:01.669 −dQ/dt. 57:01.670 --> 57:08.890 So I have the equation here that looks like--so go back to 57:08.887 --> 57:16.737 RI = Q/C and write it as RdQ/dt. 57:16.739 --> 57:23.879 This is my equation. 57:23.880 --> 57:32.730 That can be written as dQ/Q = 57:32.731 --> 57:37.581 -dt/RC. 57:37.579 --> 57:41.259 Now if you integrate that from start to finish and start to 57:41.255 --> 57:43.795 finish, this will give you log of 57:43.802 --> 57:48.392 Q over the initial Q will be - the time divided 57:48.394 --> 57:49.624 by RC. 57:49.619 --> 57:58.129 Or Q of time t will be Q_0e^(-t) 57:58.126 --> 58:01.056 ^(/RC). 58:01.059 --> 58:04.019 That means after you close the switch, 58:04.018 --> 58:07.128 if you measure the charge and the capacitor as a function of 58:07.128 --> 58:09.488 time, it starts with some 58:09.494 --> 58:14.614 Q_0 and it decays exponentially. 58:14.610 --> 58:17.450 How long does it take for it to completely discharge? 58:17.449 --> 58:22.039 The answer is infinite amount of time. 58:22.039 --> 58:26.189 Why is the capacitor not able to discharge it completely? 58:26.190 --> 58:28.450 Can you think about that? 58:28.449 --> 58:31.269 Why doesn't it just get it over with, right? 58:31.268 --> 58:33.818 Just dump all the charge to the other plate? 58:33.820 --> 58:34.750 Why is it taking forever? 58:34.750 --> 58:37.580 Student: There's always resistance. 58:37.579 --> 58:39.039 Prof: There is always resistance. 58:39.039 --> 58:42.109 There has always been resistance, so what's happening 58:42.108 --> 58:43.168 as time goes by? 58:43.170 --> 58:50.090 Student: > 58:50.090 --> 58:53.500 Prof: The voltage on the capacitor plate is decreasing 58:53.500 --> 58:54.580 with time, right? 58:54.579 --> 58:57.719 That's the voltage driving the current through the resistor. 58:57.719 --> 59:01.189 So as it drives current through the resistor and it begins to 59:01.188 --> 59:03.728 become empty, it's able to drive less current 59:03.733 --> 59:05.183 through the resistor. 59:05.179 --> 59:08.009 So it's trying to work against itself, but it doesn't have 59:08.009 --> 59:10.589 enough Q on it to drive more Q down. 59:10.590 --> 59:11.300 That's why. 59:11.300 --> 59:13.810 That's the meaning of the equation. 59:13.809 --> 59:17.149 The rate of flow of Q is proportional to Q itself. 59:17.150 --> 59:18.700 So as long as there is some Q left, 59:18.699 --> 59:21.169 it will be decaying, but it will never come to 0, 59:21.170 --> 59:24.860 because the driving force for the decay is Q itself. 59:24.860 --> 59:26.910 And if you say how long should I wait? 59:26.909 --> 59:31.759 There's a certain time called 1/RC, which is called the 59:31.757 --> 59:33.027 time constant. 59:33.030 --> 59:38.300 But any exponential function, e^(-t)/t_0, 59:38.300 --> 59:40.280 if t is much bigger than t_0, 59:40.280 --> 59:43.000 then you've got e to the minus huge number and that's a 59:42.998 --> 59:43.888 negligible number. 59:43.889 --> 59:46.339 So whenever you say something's falling exponentially, 59:46.338 --> 59:48.188 it doesn't mean that it may take forever, 59:48.188 --> 59:49.758 or It may happen very quickly. 59:49.760 --> 59:52.040 It depends what's up in the exponent. 59:52.039 --> 59:55.189 All exponential functions will look like e to the minus 59:55.186 --> 59:57.916 time over some other number with dimension of time. 59:57.920 --> 1:00:00.080 That's the unit in which you measure time. 1:00:00.079 --> 1:00:02.949 If little t is many times big t_0, 1:00:02.947 --> 1:00:06.267 then it's e to the minus big number, which is negligible. 1:00:06.268 --> 1:00:10.488 So you've got to wait many time constants before a capacitor 1:00:10.485 --> 1:00:11.695 will discharge. 1:00:11.699 --> 1:00:14.299 So capacitors are pretty dangerous. 1:00:14.300 --> 1:00:16.990 In fact, your computer, if you open it, 1:00:16.990 --> 1:00:20.390 there can be capacitors inside which are charged, 1:00:20.389 --> 1:00:23.789 even though it's not plugged into the mains. 1:00:23.789 --> 1:00:26.339 So don't think it's safe to open an electrical device, 1:00:26.344 --> 1:00:28.614 just because it's not plugged into the mains. 1:00:28.610 --> 1:00:30.770 Because people tell you, "Hey, pull the plug, 1:00:30.766 --> 1:00:32.436 then you can do what you want." 1:00:32.440 --> 1:00:35.580 Not quite, because if you do that, that's a big fat 1:00:35.581 --> 1:00:38.221 capacitor, that R stands for you. 1:00:38.219 --> 1:00:42.159 That current is going to go right through you. 1:00:42.159 --> 1:00:44.679 That's why they always tell you, "Do not take this 1:00:44.679 --> 1:00:46.779 computer to your bathtub" for example. 1:00:46.780 --> 1:00:48.980 So these warnings, even though they sound 1:00:48.976 --> 1:00:50.786 ridiculous, part of it is true. 1:00:50.789 --> 1:00:52.179 Capacitors are very dangerous. 1:00:52.179 --> 1:00:55.239 What you want to do is discharge all your capacitors 1:00:55.239 --> 1:00:55.719 first. 1:00:55.719 --> 1:00:58.639 Even in flash bulbs, that's what happens. 1:00:58.639 --> 1:01:00.809 You charge up a lot of charge in a capacitor, 1:01:00.813 --> 1:01:03.883 and the resistance there is the bulb itself, the little coil in 1:01:03.876 --> 1:01:04.516 the bulb. 1:01:04.518 --> 1:01:07.398 When we close the circuit, the capacitor dumps all its 1:01:07.400 --> 1:01:09.520 charge and then in the brief moment, 1:01:09.518 --> 1:01:12.928 the coil heats up, namely R heats up and 1:01:12.929 --> 1:01:15.079 glows and you have a flash. 1:01:15.079 --> 1:01:17.489 But there you want time constant to be very small, 1:01:17.489 --> 1:01:20.639 because how long are you going to tell people to keep smiling? 1:01:20.639 --> 1:01:24.219 So you've got to get on with it, so you put in a very quick, 1:01:24.217 --> 1:01:27.607 very rapid time constant, so R is going to be very 1:01:27.612 --> 1:01:28.222 small. 1:01:28.219 --> 1:01:31.109 Sometimes you want the decay to be very, very slow. 1:01:31.110 --> 1:01:34.320 People are even thinking of getting rid of batteries and 1:01:34.320 --> 1:01:35.780 just buying capacitors. 1:01:35.780 --> 1:01:38.340 With a capacitor, you can use it to drive 1:01:38.336 --> 1:01:41.466 something, but slowly the voltage will go down. 1:01:41.469 --> 1:01:44.209 So if your device can operate over a range of voltage, 1:01:44.211 --> 1:01:46.801 the voltage is just Q(t)/C. 1:01:46.800 --> 1:01:48.610 That's the voltage at a given time. 1:01:48.610 --> 1:01:52.130 If it can operate up to that voltage, between that and that 1:01:52.130 --> 1:01:54.560 voltage, you can run it for that time. 1:01:54.559 --> 1:02:01.619 Okay, so this is the fate of the current. 1:02:01.619 --> 1:02:04.129 Now what is Q_0? 1:02:04.130 --> 1:02:07.800 At t = 0, what is the charge in the 1:02:07.795 --> 1:02:08.865 capacitor? 1:02:08.869 --> 1:02:11.989 Well, that's the initial charge in the capacitor. 1:02:11.989 --> 1:02:14.939 What is the current in this problem, I? 1:02:14.940 --> 1:02:16.150 It's just dQ/dt. 1:02:16.150 --> 1:02:20.880 It's −dQ/dt, and if you do 1:02:20.876 --> 1:02:25.726 −dQ/dt, you'll find there's 1:02:25.726 --> 1:02:29.206 Q_0/RC. 1:02:29.210 --> 1:02:32.820 Just take the derivative of this function times 1:02:32.824 --> 1:02:34.794 e^(−t/RC). 1:02:34.789 --> 1:02:37.159 At t = 0, Q_0/C is the 1:02:37.157 --> 1:02:39.737 voltage in the capacitor, divided by R as a 1:02:39.737 --> 1:02:40.367 current. 1:02:40.369 --> 1:02:46.529 That's a current it starts with, but the current also 1:02:46.532 --> 1:02:52.462 decays exponentially, so with another prefactor. 1:02:52.460 --> 1:02:55.900 But there's one calculation I wanted to do, 1:02:55.896 --> 1:02:58.676 which is the energy calculation. 1:02:58.679 --> 1:03:05.019 If I had a capacitor that was charged, I had an energy 1:03:09.440 --> 1:03:12.670 Q^(2)/2C. 1:03:12.670 --> 1:03:15.290 At the end of the day, my capacitor's completely 1:03:15.286 --> 1:03:16.006 discharged. 1:03:16.010 --> 1:03:19.160 You want to ask yourself, what happened to the energy I 1:03:19.164 --> 1:03:20.514 had in the capacitor? 1:03:20.510 --> 1:03:23.380 And we all know the answer is that it went through the 1:03:23.378 --> 1:03:26.298 resistor and heated it up, and the heat energy or light 1:03:26.300 --> 1:03:28.520 energy is where you got your rewards. 1:03:28.518 --> 1:03:32.498 But we've got to make sure that the energy deposited in the 1:03:32.500 --> 1:03:36.550 resistor over all time is equal to the energy you had in the 1:03:36.548 --> 1:03:37.508 capacitor. 1:03:37.510 --> 1:03:41.790 That's the last one thing I wanted you to check. 1:03:41.789 --> 1:03:46.239 So when current goes through a resistor--here's a current going 1:03:46.244 --> 1:03:50.554 through a resistor--what is the rate at which energy is being 1:03:50.554 --> 1:03:51.494 consumed? 1:03:51.489 --> 1:03:53.409 So you must think, let me draw the resistor this 1:03:53.414 --> 1:03:53.664 way. 1:03:53.659 --> 1:03:55.169 Its' very suggestive. 1:03:55.170 --> 1:03:58.430 You're flowing downhill electrically through voltage 1:03:58.434 --> 1:03:59.144 V. 1:03:59.139 --> 1:04:04.189 Every coulomb that falls down loses an energy Q times 1:04:04.190 --> 1:04:06.160 V as it falls. 1:04:06.159 --> 1:04:09.469 And that's the energy that's given to the wire by colliding 1:04:09.469 --> 1:04:11.809 with the stuff in it and heating it up. 1:04:11.809 --> 1:04:15.089 Therefore in 1 second, the number of coulombs falling 1:04:15.092 --> 1:04:17.682 down is VI and that's the power. 1:04:17.679 --> 1:04:22.799 So power in the resistor is VI. 1:04:22.800 --> 1:04:26.000 That's the rate of energy consumption. 1:04:26.000 --> 1:04:33.630 So in this problem, V is Q/C and I, 1:04:33.632 --> 1:04:43.182 I got somewhere here--I'm too close to the board to see where 1:04:43.175 --> 1:04:45.715 anything is. 1:04:45.719 --> 1:04:47.169 I want to first find the current. 1:04:47.170 --> 1:04:48.550 Here is my current. 1:04:48.550 --> 1:04:50.400 I'm sorry, there's an easier way to do this. 1:04:50.400 --> 1:04:53.820 Instead of writing VI, let me write it as 1:04:53.822 --> 1:04:56.812 I^(2)R, because 1V is IR. 1:04:56.809 --> 1:05:01.349 So I want to integrate from 0 to infinity the quantity 1:05:01.349 --> 1:05:02.719 I^(2)R. 1:05:02.719 --> 1:05:05.409 Now I'm not going to waste your time, guys. 1:05:05.409 --> 1:05:06.589 First of all, you know how to do this 1:05:06.588 --> 1:05:06.948 integral. 1:05:06.949 --> 1:05:09.299 Square this, multiply by R and do the 1:05:09.298 --> 1:05:11.588 integral, and believe me, you will just get 1:05:11.594 --> 1:05:13.784 Q_0 ^(2)/2C. 1:05:13.780 --> 1:05:17.520 I don't want to spend time doing that. 1:05:17.519 --> 1:05:18.749 So that's where the energy goes. 1:05:18.750 --> 1:05:30.980 1:05:30.980 --> 1:05:32.510 You follow that? 1:05:32.510 --> 1:05:35.810 The capacitor discharges through the resistor and the 1:05:35.813 --> 1:05:39.313 energy you pumped into it comes out in the form of heat, 1:05:39.306 --> 1:05:41.146 or light if it's glowing. 1:05:41.150 --> 1:05:44.070 And we have seen the balance of energy. 1:05:44.070 --> 1:05:48.100 But now the trouble with this circuit is that it doesn't last 1:05:48.103 --> 1:05:48.913 very long. 1:05:48.909 --> 1:05:50.549 You've done it once, you're finished. 1:05:50.550 --> 1:05:52.160 The capacitor discharges. 1:05:52.159 --> 1:05:57.629 If you want an experiment where the current can keep on running, 1:05:57.634 --> 1:06:01.984 then you need what's called a cell or a battery. 1:06:01.980 --> 1:06:04.030 I've got to explain to you a little bit about the cell. 1:06:04.030 --> 1:06:07.980 It's something I did not fully understand the first time 1:06:07.976 --> 1:06:12.206 around, so I want to share with you whatever understanding I 1:06:12.208 --> 1:06:14.288 have, how the cell works. 1:06:14.289 --> 1:06:17.959 People generally tell you a cell provides a certain voltage 1:06:17.963 --> 1:06:21.573 difference between the end points, maybe 1.2 volts between 1:06:21.572 --> 1:06:23.222 this and the terminal. 1:06:23.219 --> 1:06:27.019 That's certainly true, but in light of what we have 1:06:27.016 --> 1:06:31.646 learned, here is what is the correct way to think about it. 1:06:31.650 --> 1:06:33.880 The analogy is with the ski resort. 1:06:33.880 --> 1:06:36.740 So here is the ski resort. 1:06:36.739 --> 1:06:39.169 All these guys are coming down. 1:06:39.170 --> 1:06:41.810 Let's follow one person coming down the ski. 1:06:41.809 --> 1:06:45.329 Gravity's acting down here, then you sort of loop around 1:06:45.331 --> 1:06:47.191 and you come here for free. 1:06:47.190 --> 1:06:52.390 Then there is a lift that takes you to the top. 1:06:52.389 --> 1:06:58.949 So let's call it the force of the lift. 1:06:58.949 --> 1:07:01.139 The force of the lift has nothing to do with gravity, 1:07:01.141 --> 1:07:02.871 driven by some other engines and so on. 1:07:02.869 --> 1:07:10.219 And we define force of the lift on the closed loop to be 1:07:10.217 --> 1:07:14.757 something called curly E. 1:07:14.760 --> 1:07:20.490 This is the mechanical analog of what's called electromotive 1:07:20.492 --> 1:07:21.272 force. 1:07:21.268 --> 1:07:26.298 Now this line integral of this force is not 0 on a closed loop, 1:07:26.300 --> 1:07:29.460 because on the way up, it is doing some work, 1:07:29.460 --> 1:07:32.750 but it doesn't do anything the rest of the circuit. 1:07:32.750 --> 1:07:35.790 It's always acting up, moving a distance h. 1:07:35.789 --> 1:07:40.239 Maybe that force times h is the electromotor force. 1:07:40.239 --> 1:07:43.369 So you go to the top, you come down, 1:07:43.367 --> 1:07:48.637 and the electromotive force comes and pushes you back to the 1:07:48.637 --> 1:07:52.567 top, and again, gravity brings you down. 1:07:52.570 --> 1:07:54.470 So in a battery, what you have is the following 1:07:54.465 --> 1:07:56.685 - that's where you have to follow this very closely. 1:07:56.690 --> 1:08:01.580 Here is an electrical cell. 1:08:01.579 --> 1:08:06.599 In an electrical cell, there are a lot of positive 1:08:06.603 --> 1:08:11.423 charges and there are a lot of negative charges, 1:08:11.422 --> 1:08:15.732 and the field in fact looks like this. 1:08:15.730 --> 1:08:17.980 So when a current flows, it goes like that, 1:08:17.979 --> 1:08:19.909 goes through resistor, comes down. 1:08:19.908 --> 1:08:22.988 But the current, you realize, 1:08:22.994 --> 1:08:25.424 has to flow up here. 1:08:25.420 --> 1:08:28.670 Inside the cell, if the current flows this way, 1:08:28.673 --> 1:08:31.083 the current has to go like this. 1:08:31.078 --> 1:08:34.668 The electric field is actually pointing down between the 1:08:34.671 --> 1:08:35.261 plates. 1:08:35.260 --> 1:08:39.350 That's not what makes it move, because there's an extra force. 1:08:39.350 --> 1:08:41.180 This is your electric field. 1:08:41.180 --> 1:08:43.860 There's an extra force called E', 1:08:43.859 --> 1:08:48.249 which is of chemical origin, which pushes your charges 1:08:48.247 --> 1:08:53.377 against their will from lower potential to higher potential. 1:08:53.380 --> 1:08:55.310 That is like the ski lift. 1:08:55.310 --> 1:08:58.630 Electric field everywhere is like gravity. 1:08:58.630 --> 1:09:01.800 Its line integral that way and line integral that way are all 1:09:01.798 --> 1:09:02.378 the same. 1:09:02.380 --> 1:09:05.390 But this E' has a line integral, 1:09:05.390 --> 1:09:07.790 E'⋅dr on a closed loop, 1:09:07.788 --> 1:09:12.728 which is not equal to 0 and in fact is called the emf. 1:09:12.729 --> 1:09:16.289 You can see it's not 0, because it's non-zero here. 1:09:16.289 --> 1:09:18.339 Everywhere else it is 0. 1:09:18.340 --> 1:09:20.650 And in the region where it's non-zero, there's no 1:09:20.648 --> 1:09:21.368 cancellation. 1:09:21.368 --> 1:09:25.518 The displacement and the force are all in the same direction. 1:09:25.520 --> 1:09:28.980 So it's a non-conservative force sitting inside a battery. 1:09:28.979 --> 1:09:30.709 It's a chemical force. 1:09:30.710 --> 1:09:34.660 The usual electric field that we all like, 1:09:34.662 --> 1:09:40.642 E⋅dr, in fact has line integral equal 1:09:40.641 --> 1:09:41.511 to 0. 1:09:41.510 --> 1:09:45.480 But when you're inside this battery, 1:09:45.479 --> 1:09:48.559 if you ask how big is this E' and how big is this 1:09:48.557 --> 1:09:51.237 E, what happens inside the battery 1:09:51.235 --> 1:09:53.135 is, once you start piling up 1:09:53.144 --> 1:09:55.784 charges there, this electric field opposed to 1:09:55.783 --> 1:09:59.343 the charge is coming, the chemical force will exactly 1:09:59.342 --> 1:10:02.142 balance the electrical force inside. 1:10:02.140 --> 1:10:07.030 So inside this region, the electric field will be - 1:10:07.034 --> 1:10:09.194 that chemical force. 1:10:09.189 --> 1:10:18.689 E' is the chemical force per charge. 1:10:18.689 --> 1:10:22.009 Inside the battery, there is something that takes 1:10:22.011 --> 1:10:26.301 the charge against its natural instinct, against gravity if you 1:10:26.301 --> 1:10:28.101 like, and takes it up. 1:10:28.100 --> 1:10:30.530 You need something like that to make this thing run, 1:10:30.525 --> 1:10:32.805 otherwise charges will just sit in the bottom. 1:10:32.810 --> 1:10:34.040 They won't go up. 1:10:34.039 --> 1:10:36.069 So you need an external agency. 1:10:36.069 --> 1:10:37.939 Sometimes you can have a belt. 1:10:37.939 --> 1:10:41.339 In a Van der Graaf generator, you have a belt that runs up 1:10:41.337 --> 1:10:44.377 and down and carries positive charge to the top, 1:10:44.380 --> 1:10:48.120 even though the top is already positively charged. 1:10:48.118 --> 1:10:49.778 Left to itself, a positive charge would never 1:10:49.775 --> 1:10:51.805 like to go to the dome which is positively charged, 1:10:51.810 --> 1:10:54.970 but you drag it on that belt, kicking and screaming. 1:10:54.970 --> 1:10:57.630 You've got to do that to charge it up against its will. 1:10:57.630 --> 1:10:59.660 That's what provides the force there. 1:10:59.658 --> 1:11:02.988 You can connect that guy to the resistor and the current will 1:11:02.988 --> 1:11:03.818 keep flowing. 1:11:03.819 --> 1:11:06.569 Now here is the interesting thing. 1:11:06.569 --> 1:11:10.569 Since inside this loop, the electric field and the 1:11:10.573 --> 1:11:13.683 chemical force are exactly balanced. 1:11:13.680 --> 1:11:17.560 Now let's look at E⋅dr 1:11:17.561 --> 1:11:21.361 between here and here, between the positive and 1:11:21.362 --> 1:11:23.182 negative terminal. 1:11:23.180 --> 1:11:26.620 That's the same as E⋅dr 1:11:26.623 --> 1:11:27.273 inside. 1:11:27.270 --> 1:11:32.640 So E⋅dr 1:11:32.641 --> 1:11:40.031 from the positive to the negative terminal is 1:11:40.029 --> 1:11:47.919 −E' ⋅dr from the 1:11:47.921 --> 1:11:57.321 positive to the negative terminal is V_- 1:11:57.323 --> 1:12:02.533 - V_ . 1:12:02.529 --> 1:12:11.409 Therefore V_ - V_- is the 1:12:11.407 --> 1:12:12.307 emf. 1:12:12.310 --> 1:12:14.380 So if the chemical force has an emf, 1:12:14.380 --> 1:12:16.350 which is the line integral of the chemical force around a 1:12:16.347 --> 1:12:17.957 loop, which is basically inside a 1:12:17.957 --> 1:12:20.187 battery, that will be also equal to the 1:12:20.189 --> 1:12:23.669 voltage difference for a person living outside the battery. 1:12:23.670 --> 1:12:26.120 If a person outside the battery went on a loop like this from 1:12:26.117 --> 1:12:28.027 here to here, the line integral done by the 1:12:28.029 --> 1:12:30.739 electric field, which is the voltage dropped, 1:12:30.738 --> 1:12:33.248 if you want, the voltage gain between this 1:12:33.253 --> 1:12:37.213 one and this one, will be exactly equal to the 1:12:37.207 --> 1:12:37.727 emf. 1:12:37.729 --> 1:12:43.189 So V_ and this is V_-. 1:12:43.189 --> 1:12:45.409 Let me just make sure this part is correct. 1:12:45.408 --> 1:12:48.958 The line integral is 0, or if you like, 1:12:48.960 --> 1:12:52.450 the integral of the electric field from here to here, 1:12:52.448 --> 1:12:53.988 doesn't matter which path you take, 1:12:53.988 --> 1:12:55.808 but the electric field is conservative. 1:12:55.810 --> 1:13:01.780 That's equal to the integral of -E from here to here, 1:13:01.779 --> 1:13:05.229 but that's the same as being the polarities reversed in the 1:13:05.234 --> 1:13:08.844 integral, and that will be the emf. 1:13:08.840 --> 1:13:16.560 So that = integral E⋅dr 1:13:16.559 --> 1:13:18.649 from - to . 1:13:18.649 --> 1:13:22.259 And that will = V_ - 1:13:22.259 --> 1:13:27.039 V_-, if you write it that way. 1:13:27.038 --> 1:13:29.278 Okay, so the point is, when you have a battery, 1:13:29.277 --> 1:13:31.607 if you don't want to get into any of the details, 1:13:31.612 --> 1:13:33.512 this is what you guys have to know. 1:13:33.510 --> 1:13:36.580 If you don't look under the hood, just look at those two 1:13:36.579 --> 1:13:37.249 terminals. 1:13:37.250 --> 1:13:40.840 There'll be a voltage difference between them equal to 1:13:40.844 --> 1:13:43.494 what's called the emf of the battery. 1:13:43.488 --> 1:13:46.538 The subtle thing to remember is, there are non conservative 1:13:46.536 --> 1:13:48.476 forces at work inside the battery, 1:13:48.479 --> 1:13:51.179 this is what you may not realize, whose line integral is 1:13:51.184 --> 1:13:53.544 in fact not 0, but you don't have to know all 1:13:53.536 --> 1:13:53.836 that. 1:13:53.840 --> 1:13:56.390 You just have to know the integral, you have to know what 1:13:56.390 --> 1:13:59.170 happens if I go from here to here in an electrical circuit. 1:13:59.170 --> 1:14:01.890 I go up in voltage equal to V . 1:14:01.890 --> 1:14:03.530 That's what I want you to remember. 1:14:03.529 --> 1:14:06.159 For those of you who want to know the details, 1:14:06.161 --> 1:14:08.151 this is the reason that happens. 1:14:08.149 --> 1:14:10.419 From now on, we won't look inside the 1:14:10.417 --> 1:14:11.107 terminal. 1:14:11.109 --> 1:14:14.129 We'll take every battery to have an emf E, 1:14:14.127 --> 1:14:17.077 meaning if you jump from a negative to positive, 1:14:17.082 --> 1:14:18.532 you gain a voltage. 1:14:18.529 --> 1:14:23.999