WEBVTT 00:01.610 --> 00:04.840 We can do the following problem. 00:04.840 --> 00:10.360 You give me any number of charges which are fixed, 00:10.357 --> 00:12.607 anywhere you want. 00:12.610 --> 00:13.870 So here they are. 00:13.870 --> 00:16.230 Here's one guy q_1, 00:16.234 --> 00:20.224 here's q_2, here's q_3. 00:20.220 --> 00:23.890 And instead of these discrete charges, you may have a line 00:23.889 --> 00:26.979 charge too, maybe λ coulombs per meter. 00:26.980 --> 00:30.140 That guy is here. 00:30.140 --> 00:31.340 They are fixed. 00:31.340 --> 00:35.280 They are not dynamical. 00:35.280 --> 00:39.950 The object of interest to us is this charge q, 00:39.954 --> 00:43.824 which can actually move, allowed to move. 00:43.820 --> 00:48.240 The question we ask is, do we know what this charge 00:48.236 --> 00:49.116 will do? 00:49.120 --> 00:51.150 And the answer is, yes, we know what it will do, 00:51.150 --> 00:54.570 because if it's sitting here and we have already computed the 00:54.567 --> 00:57.177 electric field everywhere-- you know how to compute 00:57.175 --> 00:57.835 electric field. 00:57.840 --> 01:00.770 Add the field due to this one, that one, that one, 01:00.771 --> 01:01.431 add that. 01:01.429 --> 01:04.969 That is some electric field sitting here. 01:04.968 --> 01:08.588 Then we know, the force on this charge we're 01:08.593 --> 01:11.713 interested in, this q times the 01:11.710 --> 01:15.840 electric field at the location of the charge. 01:15.840 --> 01:18.760 So at every instant, we know what force is acting on 01:18.763 --> 01:19.053 it. 01:19.049 --> 01:24.459 Then from Newton's law, that will give me the rate of 01:24.456 --> 01:28.716 change of velocity or the acceleration. 01:28.720 --> 01:30.970 You can write it either way. 01:30.970 --> 01:33.720 That's all I need, because using that I can find 01:33.721 --> 01:35.711 out the new position, new velocity, 01:35.714 --> 01:36.304 right? 01:36.300 --> 01:39.960 The acceleration tells you, if you wait 1 nanosecond, 01:39.959 --> 01:43.519 you go to this place, because the initial velocity, 01:43.519 --> 01:45.869 I can predict where it will be, and because you know the 01:45.867 --> 01:48.727 acceleration, I know what velocity it will 01:48.732 --> 01:49.202 have. 01:49.200 --> 01:53.420 Then I keep on updating it and I get my trajectory. 01:53.420 --> 01:57.070 Student: I think it should be dt squared. 01:57.069 --> 02:02.739 Prof: Yes, thank you. 02:02.739 --> 02:06.589 So everyone understands this is how you proceed. 02:06.590 --> 02:09.130 This charge will of course push those guys, but we're not 02:09.127 --> 02:11.027 worried, because we don't let them move. 02:11.030 --> 02:13.360 They are just fixed. 02:13.360 --> 02:17.520 And if there are much such charges, we can find what they 02:17.520 --> 02:21.240 all do in this environment, which is the background 02:21.236 --> 02:22.646 electric field. 02:22.650 --> 02:24.920 That's all we can do right now, this is electrostatics. 02:24.919 --> 02:27.399 In the real world, everybody is moving and all 02:27.401 --> 02:29.661 kinds of stuff happens, but you cannot use 02:29.664 --> 02:31.214 electrostatics for that. 02:31.210 --> 02:34.730 This asymmetry between the background and the test charge 02:34.733 --> 02:37.883 that we are studying is in the static situation. 02:37.878 --> 02:41.368 Guys that move and guys that don't. 02:41.370 --> 02:43.340 Now this is something you could have done on the first day, 02:43.340 --> 02:46.380 because I think on the very first day I told you about 02:46.377 --> 02:49.297 Coulomb's law and the principle of superposition. 02:49.300 --> 02:52.670 If you put them together, you know the force on a charge 02:52.666 --> 02:53.336 q. 02:53.340 --> 02:57.210 Then you use F = ma, you predict its trajectory. 02:57.210 --> 02:59.710 But in principle, that's all you need to do 02:59.711 --> 03:02.661 electrostatic problems, but in practice you want to 03:02.655 --> 03:04.615 clean up the machinery a little bit, 03:04.620 --> 03:07.190 make it more efficient. 03:07.188 --> 03:08.898 For example, we found Gauss's law. 03:08.900 --> 03:11.160 Gauss's law follows from Coulomb's law, 03:11.157 --> 03:13.117 but we found it's quite useful. 03:13.120 --> 03:14.980 For example, if we give you a ball of 03:14.979 --> 03:17.869 charge, spherical charge, going back to Coulomb's law and 03:17.873 --> 03:19.893 first principles is very difficult. 03:19.889 --> 03:21.939 But if you use Gauss's law and the symmetries, 03:21.938 --> 03:23.958 you're able to find the electric field around the 03:23.959 --> 03:26.469 sphere, or an infinitely long cylinder, 03:26.465 --> 03:30.025 or an infinite plane or simple geometries like that. 03:30.030 --> 03:32.000 So it's not enough to say "I've got the basic 03:31.997 --> 03:32.717 equations." 03:32.720 --> 03:36.810 You always want to see if there are ways to speed up the 03:36.812 --> 03:39.792 computation to make it more efficient. 03:39.788 --> 03:49.648 So let me remind you what we did in the first part of the 03:49.645 --> 03:51.225 course. 03:51.229 --> 03:55.209 There, if you have the following question - here's a 03:55.205 --> 03:59.705 spring and here's a mass, the spring likes to have that 03:59.712 --> 04:04.602 length and I pulled it to way over there and I released it, 04:04.598 --> 04:06.538 and right now it happens to be here, 04:06.539 --> 04:08.509 going at some speed. 04:08.508 --> 04:12.868 I want to know what velocity it will have when it is here, 04:12.870 --> 04:14.880 so this position is x_1, 04:14.878 --> 04:16.318 this speed is v_1, 04:16.319 --> 04:18.029 this position is x_2. 04:18.028 --> 04:20.568 I want to know v_2. 04:20.569 --> 04:22.859 That's a reasonable thing you can ask. 04:22.860 --> 04:27.610 Unlike the electric force, the force on this mass is due 04:27.608 --> 04:28.988 to the spring. 04:28.990 --> 04:31.790 Apart from that, it's the same situation. 04:31.790 --> 04:35.380 So one way to do that is to say, I know F = ma, 04:35.379 --> 04:38.129 so the instant you start the problem here, 04:38.129 --> 04:40.979 I know the acceleration because the force of the spring is 04:40.980 --> 04:42.580 -k times x_1. 04:42.579 --> 04:45.229 Once I know the force, I know the acceleration. 04:45.230 --> 04:47.440 I can find the new velocity, new position, 04:47.440 --> 04:50.310 keep on doing it, or I can solve the equation, 04:50.310 --> 04:53.900 d^(2)x/dt^(2) is −kx/m, 04:53.899 --> 04:55.749 and I can eventually end up here. 04:55.750 --> 04:59.430 Then I can find the velocity here. 04:59.430 --> 05:05.610 But we all know that in this problem, there's a shortcut, 05:05.607 --> 05:06.487 right? 05:06.490 --> 05:07.490 Yes? 05:07.490 --> 05:13.330 Anybody in the last row want to tell me what the shortcut is? 05:13.329 --> 05:14.569 Yes? 05:14.569 --> 05:16.669 Student: You can use conservation of energy. 05:16.670 --> 05:19.390 Prof: Conservation of energy. 05:19.389 --> 05:23.979 So conservation of energy is a consequence of Newton's laws and 05:23.975 --> 05:26.485 certain properties of the force. 05:26.490 --> 05:29.950 I'm going to remind you how it's done. 05:29.949 --> 05:33.519 One attitude I could take is, you've already done this 05:33.524 --> 05:37.714 before, but usually it's better to make sure that you're all on 05:37.706 --> 05:39.996 the same page when we do this. 05:40.000 --> 05:43.480 So let me remind you how it works there. 05:43.480 --> 05:52.220 You go to Newton's laws and we write m dv/dt = 05:52.220 --> 05:53.850 F. 05:53.850 --> 05:54.990 It's all in one dimension though. 05:54.990 --> 05:58.340 I'm moving just along the line. 05:58.339 --> 06:05.609 Then I multiply both sides by v. 06:05.610 --> 06:09.810 On the left hand side, if you know basic calculus, 06:09.805 --> 06:12.795 this is really d/dt of 06:12.802 --> 06:17.172 mv^(2)/2, by the chain rule of calculus. 06:17.170 --> 06:19.810 The d/dt derivative is the v derivative, 06:19.810 --> 06:22.450 followed by the derivative of v with respect to 06:22.452 --> 06:23.102 t. 06:23.100 --> 06:25.740 v derivative will give you mv and then there is 06:25.740 --> 06:26.050 that. 06:26.050 --> 06:33.950 This becomes F dx/dt. 06:33.949 --> 06:36.969 This means in every tiny instant of time, 06:36.970 --> 06:41.480 the force times the change in the position is equal to the 06:41.475 --> 06:46.955 change in this quantity, which we know is called kinetic 06:46.964 --> 06:47.834 energy. 06:47.829 --> 06:51.449 So if you multiply both sides by dt, 06:51.449 --> 06:54.799 then you find this relation, and if you add up all the 06:54.795 --> 06:56.835 changes, we get this relation, 06:56.843 --> 07:00.783 m--so let's see what we are integrating from a starting 07:00.776 --> 07:04.126 point to an ending point and some starting time, 07:04.129 --> 07:06.149 ending time. 07:06.149 --> 07:09.659 We get mv_2^(2)/2 - 07:09.663 --> 07:13.383 mv_1^(2)/2 = integral 07:13.377 --> 07:18.897 f(x)dx from x_1 to 07:18.899 --> 07:21.609 x_2. 07:21.610 --> 07:29.370 And this is called the work energy theorem. 07:29.370 --> 07:32.760 The work energy theorem tells you something you may expect, 07:32.757 --> 07:36.257 namely, when a force acts on a body, it's going to accelerate 07:36.261 --> 07:36.671 it. 07:36.670 --> 07:38.910 It's going to accelerate it, it's going to change the 07:38.906 --> 07:39.376 velocity. 07:39.379 --> 07:41.499 You're going to ask, "How much velocity change 07:41.497 --> 07:43.657 do I get for a certain action of the force?" 07:43.660 --> 07:46.060 and the statement is, the force pushes the body from 07:46.060 --> 07:48.320 x_1 to x_2, 07:48.319 --> 07:54.219 then this is the change in this quantity called kinetic energy. 07:54.220 --> 07:57.300 This is true no matter what the force is. 07:57.300 --> 08:00.330 It's true for friction, it's true for spring, 08:00.331 --> 08:04.121 it's true for everything, because Newton's law is always 08:04.120 --> 08:04.810 valid. 08:04.810 --> 08:08.420 But if you want to extract from this a law of conservation of 08:08.418 --> 08:12.268 energy, you have to do a certain manipulation and that's what I'm 08:12.266 --> 08:13.586 going to describe. 08:13.588 --> 08:19.398 So let's go back here and try to extract from it a law of 08:19.403 --> 08:22.003 conservation of energy. 08:22.000 --> 08:25.500 So I'm going to write here k_2 - 08:25.495 --> 08:30.225 k_1 is integral F(x) dx, 08:30.230 --> 08:36.690 from x_1 to x_2. 08:36.690 --> 08:40.530 Now it's a fundamental result in calculus that integral of any 08:40.532 --> 08:44.192 function of x from the point x_1 to 08:44.186 --> 08:47.146 the point x_2 can be written as 08:47.147 --> 08:51.617 G(x_2) - G(x_1), 08:51.620 --> 08:57.890 where G is any function whose derivative is the F 08:57.889 --> 08:59.709 I'm integrating. 08:59.710 --> 09:02.260 For example, if F = x, 09:02.256 --> 09:06.256 G = x^(2)/2, because derivative 09:06.259 --> 09:09.079 x^(2)/2 is x. 09:09.080 --> 09:10.670 In fact, it's not just x^(2)/2 . 09:10.668 --> 09:14.158 You can add x^(2)/2 a constant, but the constant is 09:14.160 --> 09:16.860 not important, because they'll cancel between 09:16.856 --> 09:17.956 this and this. 09:17.960 --> 09:21.100 So you can pick the solution without a constant, 09:21.101 --> 09:21.571 okay? 09:21.570 --> 09:24.460 Now what does that mean? 09:24.460 --> 09:29.340 Let's rearrange the expression so I get K_2 09:29.336 --> 09:32.556 − G(x_2) = 09:32.559 --> 09:37.849 K_1 − G(x_1). 09:37.850 --> 09:41.250 That tells you the combination K - G does not 09:41.253 --> 09:44.483 change as the particle moves, but this doesn't look very 09:44.480 --> 09:45.010 nice. 09:45.009 --> 09:48.289 We want to get the old law of conservation of energy in the 09:48.293 --> 09:49.203 standard form. 09:49.200 --> 09:51.600 You define the function U(x), 09:51.600 --> 09:53.830 you simply - this G(x). 09:53.830 --> 09:55.440 It's a very trivial change. 09:55.440 --> 09:57.580 Then you can say K_2 09:57.578 --> 10:00.668 U(x_2) = K_1 10:00.669 --> 10:06.629 U(x_1), and that is the energy that 10:06.629 --> 10:10.449 does not change with time. 10:10.450 --> 10:15.300 So the relation between U, which is called the 10:15.298 --> 10:18.188 potential energy, and the force, 10:18.191 --> 10:20.151 is the following. 10:20.149 --> 10:26.219 The force is - the derivative of the potential energy. 10:26.220 --> 10:30.400 The - came because G is the integral of F, 10:30.403 --> 10:33.983 but U is -G, so that this formula, 10:33.979 --> 10:36.869 this means you have this relation. 10:36.870 --> 10:39.770 And you can also say U(x_2) 10:39.774 --> 10:41.934 − U(x_1) 10:41.926 --> 10:44.656 with an integral of F(x) dx 10:44.657 --> 10:47.967 from x_1 to x_2, 10:47.970 --> 10:49.620 there's a - sign here. 10:49.620 --> 10:54.850 Again, the - sign is present because F is defined to 10:54.846 --> 10:57.906 be - the derivative of U. 10:57.908 --> 11:00.438 So you can go back and forth from potential to force and 11:00.442 --> 11:01.412 force to potential. 11:01.408 --> 11:03.508 If you knew the force, you can integrate it to get the 11:03.513 --> 11:03.993 potential. 11:03.990 --> 11:06.190 If you knew the potential, you can differentiate it to get 11:06.187 --> 11:07.887 the force, and more importantly, 11:07.885 --> 11:10.105 you have this great conservation law, 11:10.110 --> 11:12.290 which is very helpful in the spring problem, 11:12.288 --> 11:14.078 because if you know the potential energy, 11:14.080 --> 11:16.530 once and for all, it's kx^(2)/2, 11:16.528 --> 11:18.418 you balance the kinetic and the potential, 11:18.418 --> 11:23.498 you can find the velocity at one other point. 11:23.500 --> 11:27.540 Now why doesn't this work for friction? 11:27.538 --> 11:30.028 Friction is the one dimensional force. 11:30.029 --> 11:31.069 Yes? 11:31.070 --> 11:32.630 Student: It's not conservative. 11:32.629 --> 11:33.869 Prof: It's not conservative. 11:33.870 --> 11:35.010 It's just a definition. 11:35.009 --> 11:39.269 But what's wrong with my argument? 11:39.269 --> 11:43.429 Where in this argument does it fail? 11:43.429 --> 11:44.969 It looks like any function--yes? 11:44.970 --> 11:47.050 Student: Because the direction of the force depends 11:47.054 --> 11:48.714 on the direction that the particle's moving. 11:48.710 --> 11:50.190 Prof: Very good. 11:50.190 --> 11:54.300 The point is that frictional force is not a function just of 11:54.296 --> 11:55.406 your location. 11:55.409 --> 11:57.959 It depends on your velocity. 11:57.960 --> 12:02.100 In other words, whereas the spring will always 12:02.101 --> 12:06.021 exert a force -kx-- if you are displaced to the 12:06.019 --> 12:08.099 right by an x, it exerts a force 12:08.100 --> 12:11.200 -kx--whether or not you're moving to the right, 12:11.200 --> 12:13.970 away from the equilibrium or towards it. 12:13.970 --> 12:16.330 Thus given x, it gives you a given force, 12:16.331 --> 12:17.941 but friction is not like that. 12:17.940 --> 12:20.250 Friction does not have a definite direction. 12:20.250 --> 12:23.450 The only thing friction wants to do is to slow you down. 12:23.450 --> 12:25.780 If you want to go right, it wants to act left. 12:25.778 --> 12:28.398 If you want to go left, it wants to act right. 12:28.399 --> 12:31.339 I will resist all political analogies. 12:31.340 --> 12:35.620 I'll just point out to you that friction does not have a 12:35.616 --> 12:37.766 definite sign, therefore it's not a function 12:37.769 --> 12:38.949 of x, but as she said, 12:38.947 --> 12:41.377 it's a function of x and of x dot, 12:41.379 --> 12:42.749 which is the velocity. 12:42.750 --> 12:45.130 So it doesn't have a unique answer. 12:45.129 --> 12:47.819 That does not mean you cannot find the work due to friction. 12:47.820 --> 12:50.160 If you know it's moving to the right, then the force is to the 12:50.160 --> 12:50.430 left. 12:50.428 --> 12:53.798 You can integrate that force, as long as it is moving to the 12:53.802 --> 12:54.262 right. 12:54.259 --> 12:55.849 But if it's going right, left, right, 12:55.851 --> 12:58.021 left and oscillating, then you cannot integrate it 12:58.019 --> 12:58.949 once and for all. 12:58.950 --> 13:02.240 That way doesn't work for friction. 13:02.240 --> 13:05.470 Now here's an even more powerful example of the law of 13:05.474 --> 13:07.004 conservation of energy. 13:07.000 --> 13:15.860 You have this famous roller coaster and you are here. 13:15.860 --> 13:18.510 I say "you are here," because you'll never get me on 13:18.506 --> 13:19.676 top of one of these guys. 13:19.678 --> 13:25.238 So you are here and you got some speed, and I want to know 13:25.236 --> 13:28.256 your speed when you are here. 13:28.259 --> 13:30.099 Again, it follows from Newton's laws. 13:30.100 --> 13:31.470 You can calculate from Newton's laws, 13:31.470 --> 13:35.540 because at every point, wherever you may be, 13:35.538 --> 13:40.318 there's the force of gravity acting this way and there is the 13:40.323 --> 13:44.633 perpendicular force from the track acting this way, 13:44.629 --> 13:47.569 and the perpendicular force is such that it cancels this part 13:47.572 --> 13:49.612 of gravity, and this part of gravity will 13:49.606 --> 13:51.926 accelerate it and you can find the velocity here. 13:51.928 --> 13:55.478 Keep doing this, you can find the velocity at 13:55.481 --> 13:56.291 the end. 13:56.289 --> 13:58.019 But that's not how we do this. 13:58.019 --> 14:00.719 We use the fact that the gravity force allows us to 14:00.716 --> 14:03.036 define an energy, and I'll come to that in a 14:03.035 --> 14:03.625 moment. 14:03.629 --> 14:07.529 Then we say kinetic potential is kinetic potential in the 14:07.528 --> 14:12.038 beginning and at the end, and we can simply find the 14:12.037 --> 14:17.247 velocity here if you knew the potential energy here. 14:17.250 --> 14:20.470 All right, so the law of conservation of energy is also a 14:20.470 --> 14:23.170 very profound result, which has survived all the 14:23.172 --> 14:25.302 revolutions of quantum mechanics. 14:25.298 --> 14:28.048 After quantum mechanics, we don't like to think in terms 14:28.048 --> 14:31.248 of force, and we don't like to think in terms of trajectories. 14:31.250 --> 14:33.810 You don't even know for sure where the particle is. 14:33.808 --> 14:37.428 But amazingly, the total energy is always 14:37.429 --> 14:40.779 conserved in the problems we study. 14:40.779 --> 14:44.379 So the law of conservation of energy is very profound and it's 14:44.383 --> 14:46.513 also very useful for computations. 14:46.509 --> 14:50.499 Now the next question I have is, if I go to two 14:50.495 --> 14:55.255 dimensions--and it's the same as in three dimensions. 14:55.259 --> 14:58.549 I'm just going to go to two dimensions--what does it take to 14:58.554 --> 15:00.794 get the law of conservation of energy? 15:00.788 --> 15:04.238 So I'm going to mimic this derivation in 2D and see if I 15:04.235 --> 15:07.175 can pull out a law of conservation of energy. 15:07.179 --> 15:09.529 So here's what I will do. 15:09.528 --> 15:13.858 So in two dimensions, let me start this time with the 15:13.855 --> 15:15.265 kinetic energy. 15:18.163 --> 15:19.933 but v has got 2 components, 15:19.926 --> 15:21.976 x and y, therefore it's 15:28.530 --> 15:30.730 v⋅v. 15:30.730 --> 15:34.470 Then let me ask, what's the rate of change of 15:34.470 --> 15:36.340 this kinetic energy? 15:36.340 --> 15:38.710 You've got to take derivatives. 15:38.710 --> 15:41.080 Now you may be a little queasy about taking derivatives from 15:41.077 --> 15:43.087 this dot product, but you can check by expanding 15:43.086 --> 15:45.246 the component, just like ordinary product, 15:45.250 --> 15:48.460 and you can take derivative of this guy times that guy, 15:48.460 --> 15:50.550 derivative of that guy times this guy, 15:50.548 --> 15:54.378 and they will both be the same, and you will find it's equal to 15:54.375 --> 16:03.845 m dv/dt, which is the force. 16:03.850 --> 16:09.950 I'm sorry. 16:09.950 --> 16:14.660 Let me erase this. 16:14.659 --> 16:20.699 It's not m dv/dt; it is mdv/dt 16:20.698 --> 16:24.788 ⋅v. 16:24.788 --> 16:27.898 So if you integrate this, cancel the dt and add 16:27.895 --> 16:30.495 all the changes, you will find 16:30.500 --> 16:36.700 K_2 - K_1 is m 16:36.695 --> 16:45.255 times dr-- let me see--I'm sorry. 16:45.259 --> 16:47.939 I want to keep the dv/dt here, 16:47.936 --> 16:51.666 then I want to write this v as dr/dt. 16:51.669 --> 16:52.099 That's right. 16:52.100 --> 16:54.520 So let me do the following, excuse me. 16:54.519 --> 16:55.419 Let me do it again. 16:55.418 --> 17:03.498 dK/dt = ma times v and v is 17:03.495 --> 17:06.405 dr/dt. 17:06.410 --> 17:10.860 Now we cancel the dts and we write the change in 17:10.855 --> 17:15.465 K as K_2 - K_1 = 17:15.465 --> 17:19.005 integral of F⋅dr from the 17:19.006 --> 17:22.376 starting point r_1 to the 17:22.382 --> 17:25.842 ending point r_2. 17:25.838 --> 17:28.588 In other words, the change in kinetic energy in 17:28.586 --> 17:31.566 one dimension is just force times distance for tiny 17:31.573 --> 17:32.413 distances. 17:32.410 --> 17:35.100 In higher dimensions, like in 2 dimensions, 17:35.098 --> 17:38.458 if the particle is moving in the 2 dimensional plane along 17:38.464 --> 17:41.244 some path and it moves a distance dr, 17:41.240 --> 17:44.450 and at that point, the force is in that direction, 17:44.450 --> 17:46.910 you should not simply take the product of the length of the 17:46.909 --> 17:48.689 force times the length of the distance, 17:48.690 --> 17:52.860 but that times the cosine of the angle. 17:52.859 --> 17:54.109 Why should it take that? 17:54.108 --> 17:56.638 Because that's what gives you the rate of change of kinetic 17:56.636 --> 17:57.026 energy. 17:57.029 --> 18:00.229 Just go to the kinetic energy and it suggests to you that you 18:00.230 --> 18:02.900 take this product and define it to be the work, 18:02.900 --> 18:05.720 because if you define it that way, just like in 1D, 18:05.720 --> 18:10.550 the work done by the force will be the change in kinetic energy. 18:10.549 --> 18:11.739 You guys with me now? 18:11.740 --> 18:15.640 In spite of this little mess up here, this ma is my force. 18:15.640 --> 18:17.180 The dts got canceled. 18:17.180 --> 18:21.070 The dK is added up to K_2 - 18:21.071 --> 18:24.811 K_1 and this is the integral. 18:24.808 --> 18:29.758 All right, so now it looks pretty good, except for the 18:29.759 --> 18:31.159 complication. 18:31.160 --> 18:32.920 You have a dot product here. 18:32.920 --> 18:36.500 You have this nice result that K_2 - 18:36.502 --> 18:40.292 K_1 is the integral F⋅dr 18:40.292 --> 18:44.292 from r_1 to r_2. 18:44.288 --> 18:46.038 So let's imagine where r_1 and 18:46.035 --> 18:47.055 r_2 are. 18:47.058 --> 18:52.338 This is say where you started, and maybe you ended up here at 18:52.336 --> 18:54.356 r_2. 18:54.358 --> 18:56.918 You've just got to integrate this F from 18:56.922 --> 18:59.542 r_1 to r_2. 18:59.538 --> 19:01.618 Let's imagine there is no friction. 19:01.618 --> 19:04.688 Let's imagine F depends only on the position. 19:04.690 --> 19:08.290 What's my problem now? 19:08.289 --> 19:13.449 What problem do we have? 19:13.450 --> 19:14.750 Yes? 19:14.750 --> 19:16.500 Student: _______ going from r_2 to 19:16.502 --> 19:17.442 r_1 _______. 19:17.440 --> 19:19.100 Prof: No, this integral gives me no 19:19.104 --> 19:19.474 choice. 19:19.470 --> 19:23.420 I must go from r_1 to 19:23.422 --> 19:25.882 r_2. 19:25.880 --> 19:27.770 Anybody know what's the problem? 19:27.769 --> 19:28.739 Yes? 19:28.740 --> 19:30.970 Student: Your path. 19:30.970 --> 19:31.860 Prof: Your path. 19:31.858 --> 19:34.668 See, in 1 dimension, if I want to go from here to 19:34.673 --> 19:36.553 there, I have no choice, okay? 19:36.549 --> 19:38.919 Just move along that line. 19:38.920 --> 19:40.970 In 2 dimensions, if you want to go from one 19:40.971 --> 19:43.171 point to another, I can think of many ways. 19:43.170 --> 19:46.700 You can do that, you can take the straight line 19:46.700 --> 19:49.770 path, or you can take some other path. 19:49.769 --> 19:52.549 That's an ambiguity we have in higher dimensions. 19:52.548 --> 19:55.888 So even though the work energy theorem is valid, 19:55.890 --> 19:58.490 and no matter which path you take, the work done by the force 19:58.488 --> 20:00.218 will be the change in kinetic energy, 20:00.220 --> 20:05.730 this is not in general equal to some function at 20:05.727 --> 20:10.177 r_1 - some function at 20:10.180 --> 20:13.110 r_2. 20:13.108 --> 20:16.098 Because if it had this form, we are done. 20:16.099 --> 20:16.779 You agree? 20:16.778 --> 20:19.128 If it had this form, then again you write 20:19.133 --> 20:21.323 K_2 U_2 = 20:21.321 --> 20:23.841 K_1 U_1. 20:23.838 --> 20:26.238 But no one's telling you that is true. 20:26.240 --> 20:29.530 If it was true, you are in business. 20:29.528 --> 20:32.878 But it's a very unreasonable thing to ask of this force, 20:32.880 --> 20:35.990 that its integral from some point to another point, 20:35.990 --> 20:40.520 does not depend on what path you integrate it on. 20:40.519 --> 20:42.859 In other words, you can take that path, 20:42.859 --> 20:46.309 break it up into segments dr, and on each segment, 20:46.309 --> 20:48.219 the force is some variable. 20:48.220 --> 20:51.000 You take F⋅dr of that little segment, 20:51.003 --> 20:53.643 add it up over all the segments and make the segments 20:53.635 --> 20:54.745 vanishingly small. 20:54.750 --> 20:56.820 That's the meaning of this integral. 20:56.818 --> 21:00.378 And you're supposed to get the same answer, no matter what you 21:00.375 --> 21:03.285 do, including going like this, or anything else. 21:03.288 --> 21:06.758 How can they all give the same answer? 21:06.759 --> 21:10.799 In fact, I'll give you a simple example where it's simply not 21:10.795 --> 21:13.615 true, so the answer depends on the path. 21:13.618 --> 21:17.128 In fact, just about any force you pick randomly will not have 21:17.131 --> 21:19.591 the property, the answer does not depend on 21:19.588 --> 21:20.348 the path. 21:20.348 --> 21:23.768 Answers typically will depend on what path you take. 21:23.769 --> 21:28.809 So let's take a simple force, F = 21:28.808 --> 21:31.568 iy. 21:31.568 --> 21:34.448 I can think of more complicated ones, but I don't have time to 21:34.448 --> 21:34.918 do them. 21:34.920 --> 21:37.650 So here's a simple one - force depends on position. 21:37.650 --> 21:40.680 It points in the x direction only and it grows with 21:40.675 --> 21:41.255 y. 21:41.259 --> 21:43.479 And I'm going to do two calculations. 21:43.480 --> 21:48.070 I'm going to find the integral from the origin to the point 21:48.065 --> 21:52.725 (1,1) first by going this way and then by going that way. 21:52.730 --> 21:54.690 They are two possible paths. 21:54.690 --> 21:57.820 And you will see, when I do the calculation, 21:57.824 --> 21:59.724 the answer is different. 21:59.720 --> 22:02.920 Therefore you cannot write the answer as a difference of 22:02.922 --> 22:06.082 something here - something here, because the answer is not a 22:06.082 --> 22:08.162 function only of the starting and ending points, 22:08.160 --> 22:09.930 but how you got there. 22:09.930 --> 22:15.490 And I want to do this just to remind you how to do these 22:15.487 --> 22:16.697 integrals. 22:16.700 --> 22:22.000 So let's take this segment here. 22:22.000 --> 22:23.700 What is the work done? 22:23.700 --> 22:29.090 The work is the force which is i times y times 22:29.088 --> 22:30.458 the distance. 22:30.460 --> 22:34.430 If you move horizontally, perhaps you guys can see 22:34.428 --> 22:39.528 dr for a tiny segment simply i times dx. 22:39.529 --> 22:42.609 And x goes from 0 to 1. 22:42.609 --> 22:47.789 So how much is that? 22:47.789 --> 22:50.269 Can you do this? 22:50.269 --> 22:51.929 I want the first person who knows the answer. 22:51.930 --> 22:52.530 Yes? 22:52.529 --> 22:55.129 Student: Zero ______________. 22:55.130 --> 22:55.620 Prof: Right. 22:55.619 --> 22:58.539 On this line, y is 0. 22:58.538 --> 23:00.438 In fact, if you draw a picture of the force, 23:00.441 --> 23:01.371 it looks like this. 23:01.368 --> 23:04.498 It gets bigger and bigger and bigger as you go further off. 23:04.500 --> 23:06.980 And if you go to negative values, it looks like this. 23:06.980 --> 23:09.320 So there is no y, there is no force. 23:09.319 --> 23:11.659 That is 0. 23:11.660 --> 23:15.130 If you go to this segment, y is definitely 23:15.132 --> 23:17.452 non-zero, but notice, the force is like 23:17.454 --> 23:20.274 this and the dr is in the y direction and they are 23:20.266 --> 23:24.656 perpendicular, so the dot product is 0. 23:24.660 --> 23:25.630 Okay? 23:25.630 --> 23:28.110 dr and F are perpendicular. 23:28.108 --> 23:30.998 One is along i and one is along j dot product to 23:31.000 --> 23:31.190 0. 23:31.190 --> 23:35.130 If you take this path, the work done is 0. 23:35.130 --> 23:38.920 Now let's take the other path, go like this and then like 23:38.924 --> 23:39.404 this. 23:39.400 --> 23:41.720 On this section, we have the same problem. 23:41.720 --> 23:42.780 Force is to the right. 23:42.779 --> 23:44.139 You're moving straight up. 23:44.140 --> 23:47.270 dr and F are perpendicular. 23:47.269 --> 23:50.369 But on the last segment, from here to here, 23:50.368 --> 23:52.378 you get a non-zero contribution, 23:52.375 --> 23:54.765 because force is iy, 23:54.769 --> 23:58.929 displacement is idx and x 23:58.930 --> 24:02.850 goes from 0 to 1, and y is frozen at the 24:02.846 --> 24:05.686 value 1, because you're on this line. 24:05.690 --> 24:06.930 Frozen at the value 1. 24:06.930 --> 24:08.240 Just put 1 for it. 24:08.240 --> 24:12.340 And i⋅i is 1, integral of dx is 1. 24:12.338 --> 24:15.708 Therefore the work done by the force going up and then to the 24:15.709 --> 24:16.439 right is 1. 24:16.440 --> 24:20.320 Going to the right and then up is 0. 24:20.318 --> 24:23.598 You can take other paths and keep getting other answers. 24:23.598 --> 24:25.518 So this is a force for which it's not true, 24:25.519 --> 24:29.169 and I bet you that if you just randomly selected forces in the 24:29.172 --> 24:32.362 2 dimensional plane, they will never have the 24:32.355 --> 24:35.965 property that the integral is path independent. 24:35.970 --> 24:39.730 So it looks like we are onto a hopeless task, 24:39.733 --> 24:43.503 that the answer cannot depend on the path. 24:43.500 --> 24:47.640 So maybe there is no problem in which you can derive a law of 24:47.635 --> 24:49.355 conservation of energy. 24:49.358 --> 24:53.558 Of course, you guys know, you can derive a law of 24:53.555 --> 24:57.745 conservation of energy for a family of forces. 24:57.750 --> 25:00.290 In fact, the answer is, there's an infinite of numbers 25:00.291 --> 25:02.741 of forces you can devise for which this is true, 25:02.740 --> 25:06.910 namely, the answer does not depend on the path. 25:06.910 --> 25:10.350 Before writing those forces, let me explain to you one other 25:10.348 --> 25:12.038 way to write the condition. 25:12.038 --> 25:15.978 The condition, namely of a conservative force, 25:15.980 --> 25:18.530 is that the integral from here to here is the same as the 25:18.530 --> 25:23.210 integral from there to there, from 1 to 2 along path A 25:23.212 --> 25:26.542 and 1 to 2 along path B. 25:26.538 --> 25:30.568 They must be equal for any starting point and any ending 25:30.570 --> 25:33.430 point and any two paths joining them. 25:33.430 --> 25:39.930 So what I'm saying is, F⋅dr from 1 25:39.925 --> 25:47.415 to 2 along A = integral F⋅dr from 1 25:47.421 --> 25:50.421 to 2 along B. 25:50.420 --> 25:56.200 Let me rewrite this by putting a - sign here and equate it to 25:56.203 --> 25:56.593 0. 25:56.589 --> 25:57.319 That's fine. 25:57.318 --> 26:00.388 Shifted it to the left hand side. 26:00.390 --> 26:04.220 But then we use the fact that when you go from 1 to 2, 26:04.220 --> 26:09.030 with a - sign, this second guy is same as from 26:09.034 --> 26:14.494 2 to 1 along B of F⋅dr. 26:14.490 --> 26:16.940 In other words, the line integral going this 26:16.939 --> 26:19.559 way is - the line integral going backwards, 26:19.558 --> 26:22.138 because at every step, dr is opposite but F 26:22.141 --> 26:25.691 is the same at the point, F⋅dr changes 26:25.690 --> 26:26.070 sign. 26:26.068 --> 26:30.668 Therefore it tells you then that the line integral from here 26:30.670 --> 26:32.620 to there and back is 0. 26:32.618 --> 26:36.278 We write that by saying the line integral of 26:36.275 --> 26:41.625 F⋅dr in a closed path is 0 for all paths, 26:41.630 --> 26:43.470 for all starting points, all ending points, 26:43.470 --> 26:44.780 all paths. 26:44.779 --> 26:49.109 And that is the force that we want. 26:49.109 --> 26:52.579 That's the conservative force. 26:52.579 --> 26:53.689 I've just rewritten. 26:53.690 --> 26:56.820 I've not gotten any closer to finding conservative force, 26:56.817 --> 26:59.497 but I'm saying there are two ways to write it. 26:59.500 --> 27:02.880 Answer doesn't depend on the path, or the answer on any 27:02.878 --> 27:05.948 closed loop, the line integral on any closed loop, 27:05.945 --> 27:06.505 is 0. 27:06.509 --> 27:11.919 That means sometimes it's got to be positive, 27:11.922 --> 27:16.352 sometimes it's got to be negative. 27:16.348 --> 27:19.228 So question is, where am I going to get a force 27:19.232 --> 27:21.242 with these amazing properties? 27:21.240 --> 27:25.770 And I'm just going to give you the answer. 27:25.769 --> 27:27.739 The theorem--if you want, you can call it a theorem, 27:27.739 --> 27:29.129 you can call it whatever you like. 27:29.130 --> 27:33.520 It's a fact--number 1, here is a recipe for finding 27:33.518 --> 27:35.448 conservative forces. 27:35.450 --> 27:46.360 Take any function U(x,y). 27:46.358 --> 27:47.858 I'm just going to do it in 2 dimensions. 27:47.858 --> 27:50.978 In 3 dimensions, you can make it 27:50.981 --> 27:53.501 U(x,y,z). 27:53.500 --> 27:54.990 So let's give an example here. 27:54.990 --> 27:59.970 U = x^(3)y^(2). 27:59.970 --> 28:00.800 Take that function. 28:00.799 --> 28:01.849 You can write your own ticket. 28:01.848 --> 28:04.318 Write whatever function you want, time hyperbolic 28:04.319 --> 28:06.119 sinh(y) times cosh(x), 28:06.122 --> 28:07.102 doesn't matter. 28:07.099 --> 28:08.909 Take a function. 28:08.910 --> 28:13.830 Then the force that I want is the property that 28:13.825 --> 28:18.735 F_x = -dU/dx and 28:18.740 --> 28:23.870 F_y = -dU/dy. 28:23.868 --> 28:25.658 In other words, I'm going to manufacture a 28:25.656 --> 28:28.316 force whose x component is - the x derivative of 28:28.316 --> 28:30.756 the function I picked and the y component is - the 28:30.756 --> 28:32.976 y derivative of the function I picked. 28:32.980 --> 28:35.920 So in my example, F_x will be 28:35.923 --> 28:39.323 -x derivative partial means keep y as a 28:39.324 --> 28:41.184 constant, take the derivative with 28:41.182 --> 28:42.012 respect to x. 28:42.009 --> 28:48.959 That's 3x^(2)y and F_y will be 28:48.960 --> 28:51.190 -2x^(3). 28:51.190 --> 28:58.590 3x^(2)y^(2) and -2x^(3)y. 28:58.588 --> 29:02.788 Let me see if I got that right, because you guys have been 29:02.785 --> 29:04.915 catching me too many times. 29:04.920 --> 29:09.800 Okay, I think that's all right. 29:09.798 --> 29:14.918 This force is guaranteed to be conservative. 29:14.920 --> 29:16.350 I will show you why this is true. 29:16.349 --> 29:18.689 Why is this magic working? 29:18.690 --> 29:23.180 But let me write it in another way that's more compact. 29:23.180 --> 29:26.050 So the force I want to get, written as a vector, 29:26.048 --> 29:27.338 is i times F_x, 29:27.338 --> 29:30.958 which is i times −du/dx 29:30.963 --> 29:34.223 j times −du/dy. 29:34.220 --> 29:36.910 That's the force I claim is conservative. 29:36.910 --> 29:41.660 We're going to write that as - grad U. 29:41.660 --> 29:45.370 This is called the grad and it means gradient, 29:45.365 --> 29:48.655 and this symbol is shorthand for this. 29:48.660 --> 29:53.220 There's nothing more to it. 29:53.220 --> 29:55.190 Gradient is a machine. 29:55.190 --> 29:56.350 Derivative is a machine. 29:56.348 --> 29:58.118 You give it a function, sin x, 29:58.122 --> 30:00.192 it gives you an output, cosine x. 30:00.190 --> 30:02.520 you give it x squared, it gives you 2x. 30:02.519 --> 30:04.699 Gradient is a different kind of machine. 30:04.700 --> 30:06.710 You feed into it a function U of x and 30:06.714 --> 30:08.874 y, and it gives you a vector in 30:08.866 --> 30:11.976 all of the xy plane obtained by taking the two 30:11.980 --> 30:14.050 derivatives, one along x and one 30:14.046 --> 30:17.216 along y, and assembling them into a 30:17.223 --> 30:18.643 single vector. 30:18.640 --> 30:21.910 So let's see for this particular U, 30:21.910 --> 30:29.070 the force we have = -i times 3x^(2)y^(2), 30:29.069 --> 30:34.099 -j times 2x^(3)y. 30:34.098 --> 30:36.918 Amazingly, it is claimed that this force, when you integrate 30:36.917 --> 30:39.687 it from one point to another, answer will not depend on the 30:39.690 --> 30:42.940 path; it will depend only on the end 30:42.938 --> 30:43.688 points. 30:43.690 --> 30:47.600 So we have to understand what makes this work. 30:47.598 --> 30:51.298 Before I show that, I will make another statement. 30:51.298 --> 30:58.158 I will show you why this works, but the second statement, 30:58.161 --> 31:03.921 I will not prove, which is that all conservative 31:03.921 --> 31:09.191 forces are of the form - grad U. 31:09.190 --> 31:12.090 In other words, not only is this an example of 31:12.094 --> 31:14.164 conservative force, that's it. 31:14.160 --> 31:16.160 There are no more examples. 31:16.160 --> 31:19.500 Every conservative force you cook up will always be--yes? 31:19.500 --> 31:24.740 Student: Does the j component ___________? 31:24.740 --> 31:28.640 Prof: j component should be what? 31:28.640 --> 31:29.480 Oh, that's negative. 31:29.480 --> 31:31.740 I think I erased it. 31:31.740 --> 31:32.860 How's that? 31:32.859 --> 31:33.759 Thank you. 31:33.759 --> 31:34.879 Is that what you meant? 31:34.880 --> 31:35.840 Student: Yes. 31:35.839 --> 31:38.469 Prof: Okay. 31:38.470 --> 31:42.790 So how are you guys doing with this? 31:42.789 --> 31:44.149 I'm always worried. 31:44.150 --> 31:47.890 You are inscrutable, and I don't know what's going 31:47.885 --> 31:50.365 on in there, whether silence means "I'm 31:50.365 --> 31:51.825 with you" or silence means, 31:51.828 --> 31:54.298 "I'm so far behind, I don't even know where to 31:54.300 --> 31:56.180 start telling you my problems." 31:56.180 --> 31:57.520 I don't know. 31:57.519 --> 31:58.939 But this is your class. 31:58.940 --> 32:01.790 If you don't speak up, you don't get service. 32:01.789 --> 32:03.309 You have to ask. 32:03.308 --> 32:05.168 Don't assume people next to you know anything. 32:05.170 --> 32:05.830 I've met them. 32:05.829 --> 32:08.179 They don't. 32:08.180 --> 32:11.670 So just speak up. 32:11.670 --> 32:13.210 All right. 32:13.210 --> 32:16.400 So it's like one of these psychotherapy sessions. 32:16.400 --> 32:19.750 Turn to the person next to you and insult him or her, 32:19.752 --> 32:22.142 because that's what you want to do. 32:22.140 --> 32:24.420 Don't worry about them, okay. 32:24.420 --> 32:29.150 When you go to physics seminar, you never assume the speaker 32:29.154 --> 32:30.444 has a problem. 32:30.440 --> 32:32.460 I mean, you assume that you don't have a problem. 32:32.460 --> 32:34.630 If you don't follow it, it's a problem with the 32:34.627 --> 32:35.097 speaker. 32:35.098 --> 32:38.018 That's what makes seminars exciting. 32:38.019 --> 32:41.069 It's the closest thing we have to gladiator fights in the old 32:41.073 --> 32:41.433 days. 32:41.430 --> 32:44.080 You bring a speaker from somewhere, maybe Harvard, 32:44.078 --> 32:47.378 then you put them on the stage and you just roast them for the 32:47.377 --> 32:48.187 whole hour. 32:48.190 --> 32:50.770 Now you can do that to me just fine, because this is not even 32:50.772 --> 32:51.422 my discovery. 32:51.420 --> 32:54.430 Somebody else did it hundreds of years ago. 32:54.430 --> 32:56.470 You should relish that challenge. 32:56.470 --> 32:59.890 If this was a small seminar, you will have time to do it. 32:59.890 --> 33:03.430 I don't have that much time, but I also don't have that 33:03.432 --> 33:07.632 little time that if you don't follow me, you cannot intervene. 33:07.630 --> 33:09.870 Okay, you have a question? 33:09.869 --> 33:11.319 No. 33:11.319 --> 33:12.259 Okay. 33:12.259 --> 33:17.439 Now we are going to understand why a force manufactured in this 33:17.443 --> 33:20.123 fashion is conservative, okay? 33:20.119 --> 33:24.539 So let's calculate integral F⋅dr for 33:24.535 --> 33:28.355 this force from some starting--I'm just going to call 33:28.359 --> 33:29.389 it 1 and 2. 33:29.390 --> 33:31.480 It means r_1 and r_2. 33:31.480 --> 33:34.300 What is F⋅dr? 33:34.298 --> 33:39.688 It is F_xdx F_ydy. 33:39.690 --> 33:41.180 That's the meaning of the dot product. 33:41.180 --> 33:43.400 I hope you know what I mean by this. 33:43.400 --> 33:47.880 I'm going from here to here, that vector dr is some 33:47.875 --> 33:51.875 amount of dx and some amount of dy. 33:51.880 --> 33:57.000 It's got jdy and idx. 33:57.000 --> 33:59.730 And F is some other thing, pointing in that 33:59.727 --> 34:02.397 direction, so F⋅dr is this. 34:02.400 --> 34:04.370 Just look at this part. 34:04.369 --> 34:06.149 This is what you're integrating. 34:06.150 --> 34:10.570 That becomes, except for this - sign, 34:10.574 --> 34:17.214 dU/dx times dx dU/dy 34:17.210 --> 34:19.670 times dy. 34:19.670 --> 34:22.720 Now what is that? 34:22.719 --> 34:24.229 What is that? 34:24.230 --> 34:30.000 What do you think it stands for? 34:30.000 --> 34:34.760 In 1 dimension, if I took df/dx 34:34.762 --> 34:39.202 times dx, what am I calculating? 34:39.199 --> 34:41.639 Come on guys, you know what the derivative 34:41.643 --> 34:42.123 means. 34:42.119 --> 34:47.389 When you multiply the derivative by some dx, 34:47.391 --> 34:49.291 what do you get? 34:49.289 --> 34:49.789 Pardon me? 34:49.789 --> 34:51.189 Yes, go ahead. 34:51.190 --> 34:52.100 Take a shot. 34:52.099 --> 34:53.299 Yeah. 34:53.300 --> 34:56.930 Student: You're getting what you originally ___________. 34:56.929 --> 35:00.179 Prof: You're finding the change in the function f, 35:00.177 --> 35:00.587 right? 35:00.590 --> 35:06.290 The change in the function f is df/dx 35:06.289 --> 35:08.089 times dx. 35:08.090 --> 35:11.020 There are other changes proportional to dx^(3) 35:11.019 --> 35:13.389 and dx^(2), but for small dx, 35:13.385 --> 35:14.565 that's all it is. 35:14.570 --> 35:16.330 That's the meaning of the rate of change. 35:16.329 --> 35:20.229 If you multiply the rate of change by the change you get a 35:20.228 --> 35:21.938 change in the function. 35:21.940 --> 35:25.110 If you're in 2 dimensions, if you move from one point to 35:25.106 --> 35:27.636 another point, you've got a function U 35:27.639 --> 35:30.229 that depends on x and on y. 35:30.230 --> 35:32.950 Therefore it changes, because you change x and 35:32.952 --> 35:35.102 it changes because you change y. 35:35.099 --> 35:37.259 It's changing because of two reasons. 35:37.260 --> 35:38.930 This is the change due to change in x; 35:38.929 --> 35:41.349 that's the change due to change in y. 35:41.349 --> 35:44.109 For very small, infinitesimal dx and 35:44.114 --> 35:46.224 dy, that is the change. 35:46.219 --> 35:54.409 That is just the change in the function U. 35:54.409 --> 35:57.239 It follows then that if you add up all these 35:57.239 --> 36:03.489 F⋅dr's from 1 to 2, this integral will give 36:03.489 --> 36:08.509 me U_1 - U_2. 36:08.510 --> 36:15.530 Therefore integral F⋅dr from 1 36:15.530 --> 36:23.430 to 2 is U_1 - U_2. 36:23.429 --> 36:26.499 So the magic is the following - you pick a function U, 36:26.503 --> 36:29.013 you can think of the function U as measured 36:29.014 --> 36:30.864 perpendicular to the blackboard. 36:30.860 --> 36:33.850 It's like a height or something coming out of the blackboard. 36:33.849 --> 36:37.449 You cooked up a force which is related to the rate of change of 36:37.447 --> 36:39.757 the function U, so that if you add all the 36:39.757 --> 36:42.597 F⋅dr`s, you're getting the change in 36:42.597 --> 36:44.857 the function between two points. 36:44.860 --> 36:48.410 And that change in the function is independent of how you got 36:48.414 --> 36:48.894 there. 36:48.889 --> 36:51.649 Think of U as a height of a mountain, 36:51.650 --> 36:54.220 sitting on top of the xy plane. 36:54.219 --> 36:56.519 Then F is really proportional to the rate of 36:56.516 --> 36:58.306 change of the height of the mountain. 36:58.309 --> 37:00.639 And F⋅dr is the change when you move a 37:00.644 --> 37:02.734 distance dr, a vector distance dr. 37:02.730 --> 37:05.250 You add all the changes, what are you adding? 37:05.250 --> 37:07.680 You're adding the height, your elevation. 37:07.679 --> 37:10.199 And when you're done, the total change is the height 37:10.197 --> 37:13.157 difference between the final point and the starting point. 37:13.159 --> 37:15.659 It doesn't depend on how you climb that mountain. 37:15.659 --> 37:18.019 You can take a long path, you can take a short path. 37:18.018 --> 37:22.298 As long as you're keeping track of how many feet have I climbed, 37:22.304 --> 37:24.894 you're going to get only one answer. 37:24.889 --> 37:28.169 So not every F has this property, but an F 37:28.172 --> 37:31.822 derived from U by that trick of taking the gradient has 37:31.815 --> 37:32.885 this property. 37:32.889 --> 37:38.059 Therefore for that F, you might say this is for 37:38.061 --> 37:40.111 conservative force. 37:40.110 --> 37:41.840 But we always knew F⋅dr is 37:41.840 --> 37:43.690 K_2 - K_1, 37:43.690 --> 37:46.820 therefore we get K_1 37:46.818 --> 37:50.558 U_1 = K_2 37:50.556 --> 37:52.726 U_2. 37:52.730 --> 37:56.150 In other words, the U that you began 37:56.150 --> 38:00.060 with is the potential energy for that problem. 38:00.059 --> 38:04.669 So what you're manufacturing is pairs of potentials and forces. 38:04.670 --> 38:07.280 For every U that you dream up, there's a force, 38:07.280 --> 38:10.040 which is obtained by gradient, - gradient of U. 38:10.039 --> 38:13.029 And in that problem, where that force is acting on 38:13.034 --> 38:15.074 particles, the energy that is conserved 38:15.067 --> 38:18.147 will be the kinetic, the function U at the 38:18.148 --> 38:22.388 starting point, will be equal to kinetic plus 38:22.385 --> 38:25.235 potential at the end point. 38:25.239 --> 38:29.059 And the interesting thing is, there are no other conservative 38:29.061 --> 38:32.691 forces, except forces you can get by taking a U and 38:32.690 --> 38:34.220 taking its gradient. 38:34.219 --> 38:38.119 So all conservative forces fall under this category, 38:38.119 --> 38:42.389 and they all have a conserved quantity where the potential 38:42.387 --> 38:46.877 energy is the U from which the F is derived. 38:46.880 --> 38:51.960 So we can say it as follows - F = - gradient of 38:51.956 --> 38:53.006 U. 38:53.010 --> 38:55.540 The gradient is the combination of i times the x 38:55.543 --> 38:57.363 derivative and j times the y 38:57.364 --> 38:58.214 derivative. 38:58.210 --> 39:03.810 And U at 2 - U at 1 is - integral 39:03.811 --> 39:08.441 F⋅dr from 1 to 2. 39:08.440 --> 39:13.470 This is the 2 dimensional generalization of something I 39:13.472 --> 39:16.552 wrote in 1 dimension somewhere. 39:16.550 --> 39:18.450 Oh, way on the top. 39:18.449 --> 39:20.359 Here, F = - dU /dx, 39:20.356 --> 39:22.166 U(x_2) − 39:22.166 --> 39:25.146 U(x_1) is - integral of F. 39:25.150 --> 39:28.840 The only difference is the integrand has become this 39:28.840 --> 39:31.660 F⋅dr, rather than the 39:31.663 --> 39:33.693 F_x dx. 39:33.690 --> 39:37.000 So you can get conservation of energy in higher dimensions for 39:37.000 --> 39:39.280 those forces that come from a potential. 39:39.280 --> 39:43.950 U is called a potential. 39:43.949 --> 39:53.199 U is called a potential energy. 39:53.199 --> 39:59.359 So now we can ask the following question - suppose I give you a 39:59.355 --> 40:00.145 force. 40:00.150 --> 40:02.330 I know if it's conservative or not. 40:02.329 --> 40:06.089 How are you going to find out? 40:06.090 --> 40:06.940 Now there are two options. 40:06.940 --> 40:09.620 One is, you are so smart, you can look at the function 40:09.623 --> 40:12.113 and say, "Hey, this is the gradient 40:12.114 --> 40:14.554 of some other function U." 40:14.550 --> 40:15.710 If you do that, then you are done, 40:15.713 --> 40:17.303 because if it's the gradient of some function, 40:17.300 --> 40:18.360 we know it's conservative. 40:18.360 --> 40:21.600 But what if you cannot see that right away? 40:21.599 --> 40:23.819 With simple polynomials, you can easily guess. 40:23.820 --> 40:26.400 Like the example I gave you, if F_x is 40:26.403 --> 40:28.903 -3x^(2)y^(2), F_y is 40:28.898 --> 40:31.568 -2x^(3)y, maybe you can guess that U 40:31.574 --> 40:33.314 is x^(3)y^(2). 40:33.309 --> 40:36.599 But for more complicated functions, you won't be able to 40:36.601 --> 40:37.081 do it. 40:37.079 --> 40:40.669 So there's a process for testing any force to see if it's 40:40.668 --> 40:41.628 conservative. 40:41.630 --> 40:43.430 And the process is the following. 40:43.429 --> 40:46.549 It argues as follows - F_x is 40:46.554 --> 40:48.574 −dU/dx. 40:48.570 --> 40:53.850 F_y is -dU/dy. 40:53.849 --> 40:56.969 Then consider dF_x / 40:56.967 --> 41:01.207 dy, which is -d^(2)U/dydx. 41:01.210 --> 41:03.950 Then consider dF_y / 41:03.945 --> 41:07.665 dx, which is -d^(2)U/dxdy. 41:07.670 --> 41:10.420 And it's the property of partial derivatives that the 41:10.420 --> 41:13.910 cross derivative, second cross derivative, 41:13.914 --> 41:20.314 is independent of whether you take dydx or dxdy. 41:20.309 --> 41:28.419 So let's go to our test case, U = x^(3)y^(2). 41:28.420 --> 41:34.080 dU/dx = 3x^(2)y^(2) and 41:34.083 --> 41:39.223 dU/dy = 2x^(3)y. 41:39.219 --> 41:41.239 Now I'm saying, take the x derivative of 41:41.242 --> 41:42.432 this y derivative. 41:42.429 --> 41:46.039 That's written as d^(2)U/dydx. 41:46.039 --> 41:49.209 y derivative of the x derivative gives me 41:49.210 --> 41:50.210 6x^(2)y. 41:50.210 --> 41:52.860 Then I say take the x derivative of the y 41:52.860 --> 41:53.500 derivative. 41:53.500 --> 41:55.940 This gives me 3x^(2). 41:55.940 --> 42:01.110 That means 6x^(2)y. 42:01.110 --> 42:03.280 So cross derivatives are always equal. 42:03.280 --> 42:04.270 Yes? 42:04.268 --> 42:05.848 Student: Is this _______ if we're going ________ 42:05.846 --> 42:06.416 taking __________? 42:06.420 --> 42:08.840 Prof: Yes, that's correct. 42:08.840 --> 42:12.150 So in 2 dimensions, this is the only test you have 42:12.146 --> 42:12.886 to apply. 42:12.889 --> 42:16.469 The test you have to apply is, are the cross derivatives of 42:16.469 --> 42:17.579 the force equal? 42:17.579 --> 42:19.819 Namely, is the y derivative of the force 42:19.824 --> 42:22.124 F_x = to x derivative of 42:22.119 --> 42:23.339 F_y? 42:23.340 --> 42:28.970 You want to see if dF_x/dy 42:28.967 --> 42:32.297 = dF_y/dx. 42:32.300 --> 42:34.630 If that is true, it is conservative. 42:34.630 --> 42:37.890 And it comes from the fact that if F is the derivative of 42:37.893 --> 42:41.053 some function U, then this mix derivative being 42:41.054 --> 42:42.994 equal is a diagnostic for that. 42:42.989 --> 42:46.029 In 3 dimensions, there are 2 more equations. 42:46.030 --> 42:48.530 We're not going to use them, but I'll just mention what they 42:48.525 --> 42:48.775 are. 42:48.780 --> 42:51.690 You get that by moving every index 1 notch. 42:51.690 --> 42:54.400 So this x goes to y, this y goes to 42:54.396 --> 42:57.446 z, this y goes to z, this x goes to 42:57.449 --> 42:58.089 y. 42:58.090 --> 42:59.800 There's one more condition. 42:59.800 --> 43:03.830 And another condition, z goes back to x, 43:03.829 --> 43:07.239 x goes z--let's see, y goes to z, 43:07.239 --> 43:11.099 z goes to x = dF_x/ 43:11.103 --> 43:12.053 dz. 43:12.050 --> 43:15.740 I'm not worried about whether you write this down or not, 43:15.744 --> 43:19.844 but I do want you to know in 2 dimensions, you have to remember 43:19.835 --> 43:21.085 this condition. 43:21.090 --> 43:25.800 So this mathematic extends to all dimensions, 43:25.795 --> 43:29.855 but I'm doing it for the case of 2D. 43:29.860 --> 43:33.810 So let's take stock of what has been done. 43:33.809 --> 43:38.069 What has been done is to think from scratch on how to get the 43:38.068 --> 43:41.688 law of conservation of energy in high dimensions. 43:41.690 --> 43:43.680 You start with the kinetic energy and you ask, 43:43.679 --> 43:44.609 why does it change? 43:44.610 --> 43:47.550 And you find it changes due to the force, and you find the 43:47.554 --> 43:50.194 change in kinetic energy is the line integral of 43:50.190 --> 43:53.120 F⋅dr from start to finish. 43:53.119 --> 43:56.079 But that doesn't give you a work energy-- 43:56.079 --> 43:57.909 it gives you a work energy theorem, 43:57.909 --> 43:59.589 but not a law of conservation of energy, 43:59.590 --> 44:02.760 because this integral can depend on the path. 44:02.760 --> 44:04.860 Then you say, maybe there are some functions 44:04.864 --> 44:07.514 for which the answer depends only on the end points. 44:07.510 --> 44:09.470 Then we get a law of conservation of energy, 44:09.469 --> 44:13.629 and we found out that we can manufacture such functions at 44:13.626 --> 44:15.406 will, by taking any function of 44:15.411 --> 44:18.881 U, taking its gradient, 44:18.878 --> 44:21.668 or - the gradient. 44:21.670 --> 44:25.220 Therefore that's a way to generate conservative forces and 44:25.222 --> 44:28.342 they are the only conservative forces there are. 44:28.340 --> 44:30.960 There's no other conservative forces which are not gradients 44:30.963 --> 44:31.633 of something. 44:31.630 --> 44:34.480 And the relation between the potential and the force, 44:34.481 --> 44:35.691 you should know now. 44:35.690 --> 44:39.120 It's what you learned in 1D, suitably generalized to higher 44:39.119 --> 44:39.889 dimensions. 44:39.889 --> 44:43.519 Change in potential is this and the forces obtained from the 44:43.523 --> 44:45.743 potential by taking the gradient, 44:45.739 --> 44:50.449 the change in potential is obtained by integrating the 44:50.449 --> 44:51.159 force. 44:51.159 --> 44:53.779 If you ever get confused about sign and - sign and so on, 44:53.780 --> 44:55.760 go back to the harmonic oscillator, 44:55.760 --> 45:06.710 for which the thing is 1 over--is kx^(2)/2. 45:06.710 --> 45:13.660 So let me give you one simple example that we all know, 45:13.655 --> 45:19.695 which is the force of gravity near the earth. 45:19.699 --> 45:25.409 So here is the earth, so there is some x and 45:25.414 --> 45:30.564 y coordinates, and here is z. 45:30.559 --> 45:33.889 So I might as well write down the answer, because you guys all 45:33.893 --> 45:34.553 know this. 45:34.550 --> 45:41.640 U for gravity is mg times z. And 45:41.637 --> 45:45.677 the force of gravity, which is 45:45.677 --> 45:51.537 -idu/dx - jdu/dy, 45:51.539 --> 45:56.559 - kdu/dz, all you got here is 45:56.556 --> 45:59.726 -kmg, because it doesn't depend on 45:59.728 --> 46:01.398 x, it doesn't depend on y. 46:01.400 --> 46:03.670 It depends only on z and the z derivative is 46:03.670 --> 46:04.070 trivial. 46:04.070 --> 46:11.400 Force is -mg pointing down. 46:11.400 --> 46:14.600 So the force is the gradient of the potential and you can ask, 46:14.599 --> 46:17.779 is the potential difference between two points, 46:17.780 --> 46:20.610 U_2 - U_1, 46:20.610 --> 46:26.280 is it equal to integral of F⋅dr from 1 46:26.280 --> 46:26.990 to 2? 46:26.989 --> 46:28.569 You can easily check that. 46:28.570 --> 46:34.260 If you go from here to here, straight up, 46:34.255 --> 46:42.775 then the integral of -mg times dz from start to 46:42.784 --> 46:49.754 finish, with a - sign is just mgz - 0. 46:49.750 --> 46:51.440 So indeed, it is true. 46:51.440 --> 46:53.610 And if you took two points, not on top of each other, 46:53.610 --> 46:56.280 but something like this, you can always do the integral 46:56.280 --> 46:58.520 from here to here, where there's no change in 46:58.523 --> 47:00.723 potential, and do the integral there, 47:00.722 --> 47:02.812 which will give the same answer. 47:02.809 --> 47:06.579 So gravity's a very trivial example, a very easy case, 47:06.581 --> 47:10.071 but you can see that gravitation is a conservative 47:10.068 --> 47:10.778 force. 47:10.780 --> 47:12.630 And you can see why it's a conservative force, 47:12.630 --> 47:16.660 because if you took a path where you went like this, 47:16.659 --> 47:20.349 let's say, and you went like this, where you get the same 47:20.347 --> 47:22.057 answer as here and here. 47:22.059 --> 47:26.069 Because in this path and this path, you don't do any work, 47:26.074 --> 47:29.454 because F and dr are perpendicular, 47:29.454 --> 47:30.164 right? 47:30.159 --> 47:32.189 F is like this, and the displacement is like 47:32.193 --> 47:34.103 that, and it's the same thing, 47:34.101 --> 47:36.541 whereas here, if you go like this, 47:36.541 --> 47:39.521 F is down and dr is down, 47:39.518 --> 47:41.208 F is down, dr is down, 47:41.210 --> 47:43.970 and it's the same F here and here; 47:43.969 --> 47:48.399 are the same height so you get the same answer. 47:48.400 --> 47:52.480 Okay, but now, why did we spend quite a bit of 47:52.483 --> 47:57.473 time on conservative forces, because that's not what the 47:57.474 --> 47:59.384 primary focus is? 47:59.380 --> 48:01.860 It was electricity and electrostatics. 48:01.860 --> 48:06.030 So we've got to ask ourselves, does the electrostatic force 48:06.025 --> 48:08.895 meet the test of a conservative force? 48:08.900 --> 48:11.620 Well, we know it will, because otherwise I wouldn't 48:11.619 --> 48:14.339 have spent this much time building up the stuff. 48:14.340 --> 48:17.810 But let's just verify that. 48:17.809 --> 48:20.469 It looks like a very formidable task. 48:20.469 --> 48:23.269 Let's see why it's a formidable task to verify it's 48:23.268 --> 48:24.108 conservative. 48:24.110 --> 48:27.800 I'm claiming that you've got all kinds of these charges 48:27.797 --> 48:31.557 producing a field here and I want to prove that the line 48:31.556 --> 48:35.036 integral of this force on any closed loop is 0. 48:35.039 --> 48:39.369 Or I want to prove the integral on one path is the integral on 48:39.373 --> 48:42.503 any other path, for any 2 paths between any 2 48:42.498 --> 48:43.278 points. 48:43.280 --> 48:48.650 I don't have time to show all that, right? 48:48.650 --> 48:51.750 If the theorem is wrong, do you agree it may not be so 48:51.750 --> 48:52.920 hard to show that? 48:52.920 --> 48:56.040 What will it take to show it is not conservative? 48:56.039 --> 48:57.159 Yes? 48:57.159 --> 48:59.119 Student: Just 2 paths for which the integral is not 48:59.119 --> 48:59.389 equal. 48:59.389 --> 48:59.909 Prof: Yes. 48:59.909 --> 49:03.539 If you pick any 2 paths, should be the same end points, 49:03.543 --> 49:06.843 so always the integral is not equal, that's it. 49:06.840 --> 49:09.820 But the fact that they were equal doesn't mean that somebody 49:09.818 --> 49:12.548 else will not find some other two paths with some other 49:12.547 --> 49:14.817 starting points which they are not equal. 49:14.820 --> 49:16.250 So how are we going to do this? 49:16.250 --> 49:20.230 How are we going to test for every configuration of electric 49:20.233 --> 49:23.613 field, every element of charges is conservative? 49:23.610 --> 49:31.910 So what do you think a strategy will be? 49:31.909 --> 49:32.899 Yes? 49:32.900 --> 49:34.310 Any ideas? 49:34.309 --> 49:36.829 How am I going to take all possible charges and work it out 49:36.827 --> 49:37.347 each time? 49:37.349 --> 49:38.179 Yes? 49:38.179 --> 49:40.709 Student: I guess you could find the electric field 49:40.710 --> 49:42.490 and use that test to see-- Prof: Right, 49:42.489 --> 49:44.409 but to find the electric field and see if it works, 49:44.409 --> 49:46.589 you have to calculate the electric field as a function of 49:46.592 --> 49:48.732 x, y and z for every 49:48.733 --> 49:50.033 arrangement of charges. 49:50.030 --> 49:51.850 Then you can ask, is that true, 49:51.847 --> 49:53.057 those derivatives? 49:53.059 --> 49:53.649 Yes? 49:53.650 --> 49:55.140 Student: But what works for one charge should work for 49:55.144 --> 49:56.154 all charges, because of supersposition. 49:56.150 --> 49:57.270 Prof: There we go. 49:57.268 --> 50:01.628 If it works for one charge, it will work for any 50:01.630 --> 50:03.950 arrangement of charges. 50:03.949 --> 50:05.909 So you may not have appreciated fully the power of 50:05.909 --> 50:07.389 superposition, but you can see now. 50:07.389 --> 50:09.699 If you did not realize that, the problem looks 50:09.704 --> 50:10.584 insurmountable. 50:10.579 --> 50:13.009 Who can handle every arrangement of charges? 50:13.010 --> 50:16.470 But if charge by charge it is true that the electric field due 50:16.467 --> 50:19.757 to charge 1 is conservative, electric field due to charge 2 50:19.755 --> 50:20.885 is conservative. 50:20.889 --> 50:22.509 When I add them up, the net field is 50:22.510 --> 50:24.550 E_1 E_2. 50:24.550 --> 50:26.330 The line integral of E_1 E_2 50:26.327 --> 50:28.347 is the line integral of E_1_ 50:28.351 --> 50:30.201 the line integral of E_2. 50:30.199 --> 50:32.539 E_1 gives 0 on any closed loop, 50:32.543 --> 50:34.493 other will give 0 on any closed loop. 50:34.489 --> 50:36.909 The trick is going to be just do it for one charge. 50:36.909 --> 50:38.119 That's it. 50:38.119 --> 50:41.159 If you can see it for one charge, you get by superposition 50:41.159 --> 50:45.139 that it's true for any charge, because the field is additive 50:45.143 --> 50:48.123 and the integral is additive, okay. 50:48.119 --> 50:49.469 That's the two parts. 50:49.469 --> 50:51.329 If you've got charges, the net e is 50:51.327 --> 50:53.317 E_1 E_2. 50:53.320 --> 50:54.940 The integral of E_1 E_2 50:54.943 --> 50:56.783 is the integral of E_1 the 50:56.777 --> 50:58.117 integral of E_2. 50:58.119 --> 50:59.529 That's the way integrals work. 50:59.530 --> 51:02.580 If each one vanishes on a closed loop, you're done. 51:02.579 --> 51:04.179 So I'm just going to take one charge. 51:04.179 --> 51:10.609 Here's one charge and let me draw for convenience all the 51:10.610 --> 51:14.170 field lines coming out of it. 51:14.170 --> 51:17.390 Then I want to see, if I do a line integral, 51:17.391 --> 51:21.211 say from here to here, will I get the same answer on 51:21.210 --> 51:22.710 some other path? 51:22.710 --> 51:26.330 Let's test that. 51:26.329 --> 51:29.779 So the first integral is from 1 to 2 on this path. 51:29.780 --> 51:31.670 Then I'm going to take another one, 51:31.670 --> 51:34.220 which is different from it, where you can see the answer is 51:34.222 --> 51:36.712 the same, then go on making more and more 51:36.708 --> 51:39.988 changes, and the answer will not change. 51:39.989 --> 51:45.279 So I'm going to take another path where I go tangent to the 51:45.280 --> 51:47.470 lines, along a circle. 51:47.469 --> 51:52.299 Then I go here, till I get to that radius, 51:52.295 --> 51:56.175 then I turn around and do that. 51:56.179 --> 52:01.529 I claim the integral will be the same on this path, 52:01.525 --> 52:05.155 this one, that one and that one. 52:05.159 --> 52:09.259 So I want you to think about why it's going to be true, 52:09.257 --> 52:09.787 okay? 52:09.789 --> 52:12.429 Think of it, why it's okay to do that. 52:12.429 --> 52:17.469 You've got a new path but it doesn't make a difference. 52:17.469 --> 52:18.029 Ready? 52:18.030 --> 52:21.290 Tell you the answer now? 52:21.289 --> 52:26.189 How many people don't know why it's true? 52:26.190 --> 52:28.480 So other people know why it's true. 52:28.480 --> 52:31.310 I'm going to ask the other people. 52:31.309 --> 52:33.349 Okay, why is it true? 52:33.349 --> 52:35.139 Student: I raised my hand. 52:35.139 --> 52:37.329 Prof: You raised your hand. 52:37.329 --> 52:39.119 All right. 52:39.119 --> 52:39.639 Yes? 52:39.639 --> 52:41.759 Student: It's true, because when you're moving on 52:41.757 --> 52:43.517 the paths this way, they're radial, 52:43.516 --> 52:46.296 so the radius is constant, which would mean that there's 52:46.302 --> 52:46.742 no _______. 52:46.739 --> 52:47.969 Prof: There's no what? 52:47.969 --> 52:49.899 Student: There's no change in force because the 52:49.898 --> 52:50.648 radius is constant. 52:50.650 --> 52:52.650 Prof: That's not a good reason. 52:52.650 --> 52:53.740 The force is changing. 52:53.739 --> 52:57.709 It's that way there, it's that way there. 52:57.710 --> 52:58.910 Why is it not? 52:58.909 --> 53:00.789 Student: It's always perpendicular to the force. 53:00.789 --> 53:04.529 Prof: The force and the displacement are perpendicular 53:04.525 --> 53:07.315 in this section, because force is radial, 53:07.317 --> 53:11.267 displacement by definition is tangential along the circle. 53:11.268 --> 53:17.638 So dr is perpendicular to F. 53:17.639 --> 53:19.409 Oh, by the way, I should tell you something. 53:19.409 --> 53:22.139 I usually forget this. 53:22.139 --> 53:25.739 We want the electric force to be conservative, 53:25.739 --> 53:26.379 right? 53:26.380 --> 53:30.090 The electric force in any situation on a charge is 53:30.085 --> 53:32.805 q times the electric field. 53:32.809 --> 53:35.499 You will agree that if the electric field is conservative, 53:35.498 --> 53:38.048 multiplying by a number q is not going to change 53:38.045 --> 53:38.655 the fact. 53:38.659 --> 53:40.929 So I'm going to prove the electric field is conservative, 53:40.931 --> 53:43.371 then the electric force on any charge is just proportional to 53:43.365 --> 53:43.645 it. 53:43.650 --> 53:44.740 It will also be conservative. 53:44.739 --> 53:47.739 So really taking force on a unit charge and seeing what 53:47.735 --> 53:48.285 happens. 53:48.289 --> 53:51.339 So this section is for free. 53:51.340 --> 53:54.870 This section--we'll come to that--this section is also for 53:54.865 --> 53:58.385 free, because F looks like that and dr looks 53:58.393 --> 53:59.263 like that. 53:59.260 --> 54:04.950 Here you can pair the sections, so that that section is paired 54:04.954 --> 54:09.434 with that section, because you're moving the same 54:09.434 --> 54:13.174 radial distance with a radial force. 54:13.170 --> 54:15.090 F⋅dr, the dot product, 54:15.094 --> 54:16.974 is the same here, section by section. 54:16.969 --> 54:19.189 Any work you're doing here, you're doing here, 54:19.186 --> 54:20.906 therefore the answer is the same. 54:20.909 --> 54:24.269 Then you can see that you can go on adding all kinds of 54:24.273 --> 54:26.103 changes, radial and angle, 54:26.099 --> 54:28.449 radial or angular, and when you're done, 54:28.452 --> 54:30.912 all the radial integrals will give you that part, 54:30.909 --> 54:33.789 and the angulars won't count. 54:33.789 --> 54:36.309 So as you move around, it's only when you're moving 54:36.313 --> 54:38.943 radially you're even aware that you're doing work. 54:38.940 --> 54:40.850 Going in the angle doesn't matter. 54:40.849 --> 54:45.379 And the net radial change in going from here to here is that 54:45.380 --> 54:48.070 distance, no matter how you move. 54:48.070 --> 54:51.600 Now you might say, prove to me that that integral 54:51.599 --> 54:55.719 is the same as that integral, because this was made up of 54:55.717 --> 54:58.067 radial and angular sections. 54:58.070 --> 55:01.910 This doesn't seem that way, but we can always make a very 55:01.909 --> 55:05.679 fine mesh in which every displacement is approximated by 55:05.679 --> 55:08.079 radial angle or radial angular. 55:08.079 --> 55:11.359 So you can approximate any contour by sequence of radial 55:11.364 --> 55:12.564 and angular moves. 55:12.559 --> 55:14.239 Angular moves are free. 55:14.239 --> 55:17.819 Radial moves keep track of the change in the distance times the 55:17.820 --> 55:20.710 force and you can pair them with this reference. 55:20.710 --> 55:25.610 Every dr we move there we have a dr here. 55:25.610 --> 55:30.460 So that is the pictorial proof of this. 55:30.460 --> 55:37.920 So let's now calculate what is the potential due to that 55:37.922 --> 55:40.232 electric field. 55:40.230 --> 55:42.210 So we're going to do the calculation. 55:42.210 --> 55:48.620 So we use the formula that U at r_2 55:48.617 --> 55:53.257 - U at r_1 is - the 55:53.255 --> 55:58.665 integral e⋅dr from 1 to 2. 55:58.670 --> 56:00.350 You see, we know it is conservative. 56:00.349 --> 56:02.189 I've shown you with this argument. 56:02.190 --> 56:05.930 So if you want to find the potential energy between 1 and 56:05.927 --> 56:09.397 2, you can take any path you like between 1 and 2. 56:09.400 --> 56:11.720 So let's pick the points. 56:11.719 --> 56:17.579 Here is point 1 and here is point 2. 56:17.579 --> 56:20.969 You can pick any path you like, so here is what I'm going to 56:20.972 --> 56:21.262 do. 56:21.260 --> 56:25.810 I'm going to go radially out and then I'm going to go along 56:25.813 --> 56:26.603 the arc. 56:26.599 --> 56:31.119 So this is 1, this is 2, but I'm going to 56:31.117 --> 56:36.987 start at some intermediate point 3, then go right. 56:36.989 --> 56:41.139 So work is done here; no work is done there. 56:41.139 --> 56:43.809 So let's just find the work in going from 1 to 3, 56:43.809 --> 56:48.999 which is -e_r (is a vector in the radial 56:49.003 --> 56:52.163 direction) ¼Πε 56:52.157 --> 56:54.567 _0r^(2). 56:54.570 --> 56:56.020 That is F. 56:56.018 --> 56:58.328 The displacement is e_r times the 56:58.331 --> 57:00.221 little length of e_r. 57:00.219 --> 57:01.839 Do you understand that? 57:01.840 --> 57:05.590 A tiny section here, dr, has a magnitude 57:05.592 --> 57:08.612 equal to the change in the radius, 57:08.610 --> 57:11.060 and a direction equal to the radius unit rate-- 57:11.059 --> 57:13.999 the radial direction. 57:14.000 --> 57:18.140 And you want to do this from r_1 to 57:18.141 --> 57:20.011 r_2. 57:20.010 --> 57:24.260 Okay, e_r ⋅e_r 57:24.257 --> 57:27.197 is 1 so I get--I forgot one thing. 57:27.199 --> 57:32.159 You guys know what I forgot here? 57:32.159 --> 57:33.199 One thing's missing. 57:33.199 --> 57:33.929 Yes? 57:33.929 --> 57:35.009 Student: Q. 57:35.010 --> 57:37.250 Prof: When did you notice that? 57:37.250 --> 57:38.110 Just now? 57:38.110 --> 57:40.650 Student: Little bit. 57:40.650 --> 57:42.160 Prof: Don't wait. 57:42.159 --> 57:43.759 Don't wait even one second. 57:43.760 --> 57:45.330 I told you what happens to me, right? 57:45.329 --> 57:48.529 When I go home, they tie me to a stool and they 57:48.530 --> 57:50.620 force me to watch this tape. 57:50.619 --> 57:53.229 And it's just awful when you think you dropped a symbol, 57:53.230 --> 57:54.630 and you're waiting for somebody to say, 57:54.630 --> 57:56.590 "Hey, what about q?" 57:56.590 --> 58:00.200 So don't have any--if you hesitate, it's only to wait, 58:00.201 --> 58:01.361 not to stop me. 58:01.360 --> 58:03.990 All right, so look, I'll get the q because I 58:03.992 --> 58:06.732 know the formula from the time I was in a sandbox. 58:06.730 --> 58:09.720 I know something is wrong, but I don't want to get to that 58:09.719 --> 58:12.289 last stage, then go back and change everything. 58:12.289 --> 58:16.579 So I forgot the q sitting at the origin here. 58:16.579 --> 58:19.879 Okay, so this is q/4Πε 58:19.878 --> 58:21.368 _0. 58:21.369 --> 58:26.079 Integral of -1/r^(2) is just 1/r, so that's 58:26.077 --> 58:31.457 1/r_2 − 1/r_1. 58:31.460 --> 58:35.530 Again, sorry, I made a mistake now. 58:35.530 --> 58:40.010 The change in potential energy is really q times 58:40.014 --> 58:44.754 e, but q is the test charge, so I'm going to 58:44.748 --> 58:46.658 call this V. 58:46.659 --> 58:49.789 So it's hard for you guys to do it on the notebook, 58:49.791 --> 58:53.361 because I want to distinguish between potential energy and 58:53.362 --> 58:54.242 potential. 58:54.239 --> 58:57.979 In other words, the force--now some other test 58:57.978 --> 59:02.548 charge q is the electric field times q. 59:02.550 --> 59:06.470 And the force is going to be - the gradient of the potential 59:06.471 --> 59:07.071 energy. 59:07.070 --> 59:13.830 The electric field is going to be - the gradient of the 59:13.827 --> 59:15.327 potential. 59:15.329 --> 59:16.079 Yes? 59:16.079 --> 59:19.409 Student: In that last line, should that be a negative? 59:19.409 --> 59:20.179 Prof: A negative? 59:20.179 --> 59:21.149 Student: Yeah. 59:21.150 --> 59:22.520 Prof: Let's see. 59:22.518 --> 59:28.888 There's a negative here, and integral of -1/r^(2) 59:28.887 --> 59:30.737 is 1/r. 59:30.739 --> 59:31.609 Student: Oh, okay. 59:31.610 --> 59:34.690 Prof: So the upper limit, then there's lower limit. 59:34.690 --> 59:36.830 So let me correct what I said. 59:36.829 --> 59:38.839 I'm trying to find the difference in what's called the 59:38.844 --> 59:39.304 potential. 59:39.300 --> 59:41.980 It's not the potential energy. 59:41.980 --> 59:46.540 Potential difference between some point and some other point 59:46.543 --> 59:49.333 is the work done on a unit charge. 59:49.329 --> 59:51.019 If you've got some other charge, you should multiply it 59:51.018 --> 59:51.518 by the charge. 59:51.518 --> 59:54.148 So the energy of some other charge in this field, 59:54.150 --> 59:56.450 the potential energy, is q times the 59:56.452 --> 59:57.222 potential. 59:57.219 --> 1:00:03.249 So U is q times this thing that I'm calling 1:00:03.248 --> 1:00:04.538 potential. 1:00:04.539 --> 1:00:07.439 So we're comparing this expression to this expression, 1:00:07.440 --> 1:00:12.180 we can conclude that V at any point r is 1:00:12.184 --> 1:00:17.024 q/4Πε _0(1/r) 1:00:17.018 --> 1:00:18.538 any constant. 1:00:18.539 --> 1:00:20.569 Any constant, because you cannot immediately 1:00:20.568 --> 1:00:23.258 say this guy is equal to this guy and that guy is equal to 1:00:23.259 --> 1:00:25.059 that guy, because even if you add a 1:00:25.061 --> 1:00:27.761 constant to both, it will still be true. 1:00:27.760 --> 1:00:29.860 And we can pick the constant any way we like. 1:00:29.860 --> 1:00:34.970 It is common to pick the constant so that V goes 1:00:34.965 --> 1:00:38.365 to 0 as r goes to infinity. 1:00:38.369 --> 1:00:39.559 That's what we will do. 1:00:39.559 --> 1:00:42.609 That's the convention. 1:00:42.610 --> 1:00:52.810 Then we can write here, what is the V(r). 1:00:52.809 --> 1:00:55.509 This is the V due to a single charge, 1:00:55.514 --> 1:00:59.294 and the U(r) = to some q_0. 1:00:59.289 --> 1:01:01.619 I don't want to call q again, because q is the 1:01:01.615 --> 1:01:02.795 guy producing the potential. 1:01:02.800 --> 1:01:05.200 q_0 is the guy experiencing the force. 1:01:05.199 --> 1:01:15.099 So q_0 times V is the potential energy 1:01:15.099 --> 1:01:23.049 of charge q, of charge q_0 1:01:23.050 --> 1:01:27.270 sitting at that point. 1:01:27.268 --> 1:01:29.488 Now let's also verify, if you like, 1:01:29.492 --> 1:01:33.422 that this V(r) does produce the electric 1:01:33.416 --> 1:01:34.066 field. 1:01:34.070 --> 1:01:36.010 In other words, I got the V by 1:01:36.007 --> 1:01:37.567 integrating electric field. 1:01:37.570 --> 1:01:40.000 I want to go back, just to get you familiar with 1:01:39.998 --> 1:01:40.358 this. 1:01:40.360 --> 1:01:46.710 Getting the electric field from the potential. 1:01:46.710 --> 1:01:52.170 Let's try to do that. 1:01:52.170 --> 1:01:56.450 So E_x--let me just do 1:01:56.451 --> 1:02:02.421 E_x--it's −dV/dx, 1:02:02.422 --> 1:02:03.552 right? 1:02:03.550 --> 1:02:08.420 -dV /dx = -q/4Πε 1:02:08.418 --> 1:02:10.268 _0. 1:02:10.268 --> 1:02:16.038 Derivative of 1/r is -1/r^(2) times 1:02:16.043 --> 1:02:18.693 dr/dx. 1:02:18.690 --> 1:02:20.990 What is dr/dx? 1:02:20.989 --> 1:02:24.939 r is (x^(2) y^(2) 1:02:24.940 --> 1:02:27.290 z^(2))^(1/2). 1:02:27.289 --> 1:02:31.029 dr/dx is just x divided by this whole 1:02:31.028 --> 1:02:33.718 thing, is x divided by r. 1:02:33.719 --> 1:02:37.959 So this becomes q/4Πε 1:02:37.958 --> 1:02:41.268 _0 (1/r^(2)) 1:02:41.268 --> 1:02:43.748 (x/r). 1:02:43.750 --> 1:02:47.970 This is E_x. 1:02:47.969 --> 1:02:50.829 So what will be the vector E? 1:02:50.829 --> 1:02:52.759 Can you do that in your head? 1:02:52.760 --> 1:02:56.190 Multiply this by i, and do similarly for y 1:02:56.193 --> 1:02:58.133 and z and add them up. 1:02:58.130 --> 1:02:59.290 What will you find? 1:02:59.289 --> 1:03:03.599 You will find E will equal q/4Πε 1:03:03.603 --> 1:03:06.773 _0 (1/r^(2)), 1:03:06.766 --> 1:03:09.926 times r over r. 1:03:09.929 --> 1:03:13.379 Put r over r, position vector over r, 1:03:13.376 --> 1:03:16.096 is the unit vector in the radial direction. 1:03:16.099 --> 1:03:18.829 I'm doing this a little fast, because this is something you 1:03:18.829 --> 1:03:19.959 can go home and check. 1:03:19.960 --> 1:03:21.540 I don't want to spend too much time doing it. 1:03:21.539 --> 1:03:24.419 I claim that if you took this potential and took its x 1:03:24.422 --> 1:03:26.902 derivative or y derivative or z 1:03:26.896 --> 1:03:29.126 derivative, you get the x, 1:03:29.128 --> 1:03:33.438 y or z components of the electric field. 1:03:33.440 --> 1:03:39.450 Okay, if you didn't follow this, you should go home and 1:03:39.454 --> 1:03:42.354 check that this is true. 1:03:42.349 --> 1:03:45.269 All right, so this is not that important for me, 1:03:45.273 --> 1:03:47.953 this checking, because we know it's going to 1:03:47.949 --> 1:03:48.509 work. 1:03:48.510 --> 1:03:53.010 What is important for me is for you to know that we've now found 1:03:53.010 --> 1:03:56.660 that the potential due to a single charge leads to a 1:03:56.655 --> 1:03:58.295 conservative force. 1:03:58.300 --> 1:04:00.610 The field due to a single charge leads to a conservative 1:04:00.606 --> 1:04:02.316 one, therefore by superposition, 1:04:02.315 --> 1:04:04.145 if you add any number of charges, 1:04:04.150 --> 1:04:06.730 if you find the potential due to all of them, 1:04:06.730 --> 1:04:13.580 then you take its gradient that will give you a conservative 1:04:13.583 --> 1:04:15.563 electric field. 1:04:15.559 --> 1:04:18.459 So what have you got so far? 1:04:18.460 --> 1:04:29.020 Our gain is the following - we have a law of conservation of 1:04:29.019 --> 1:04:32.529 energy, which is the following - 1:04:36.815 --> 1:04:41.105 of the body you're interested in) times V at the point 1:04:45.181 --> 1:04:49.831 q times v at the point r_2. 1:04:49.829 --> 1:04:54.269 And what is V at any point r? 1:04:54.268 --> 1:04:57.928 It's the sum 1/4Πε 1:04:57.927 --> 1:05:00.167 _0. 1:05:00.170 --> 1:05:03.090 Let's imagine there are many, many charges, 1:05:03.090 --> 1:05:04.360 q_1, q_2, 1:05:04.356 --> 1:05:05.416 q_3, each one at 1:05:05.422 --> 1:05:06.722 r_1, r_2, 1:05:06.722 --> 1:05:07.792 r_3, etc. 1:05:07.789 --> 1:05:11.019 This equals q_i which is 1:05:11.016 --> 1:05:15.446 charge number I divided by the length of the vector from 1:05:15.452 --> 1:05:20.052 r_i to the r where I'm finding the 1:05:20.048 --> 1:05:21.338 potential. 1:05:21.340 --> 1:05:25.070 In other words, this is where I want the 1:05:25.070 --> 1:05:27.080 potential r. 1:05:27.079 --> 1:05:28.549 This is r_1_ 1:05:28.550 --> 1:05:30.850 and q_1 is sitting there. 1:05:30.849 --> 1:05:33.379 This is r_2 and q_2_ 1:05:33.376 --> 1:05:34.926 is sitting there and so on. 1:05:34.929 --> 1:05:37.959 And the potential at the point that I'm interested in is 1:05:37.960 --> 1:05:41.270 obtained by taking each q divided by the distance from 1:05:41.268 --> 1:05:43.878 that point, and this is the usual 1:05:43.882 --> 1:05:47.022 1/4Πε _0. 1:05:47.018 --> 1:05:53.148 That is the complete story, and the charges are not 1:05:53.152 --> 1:05:59.532 discrete but continuous you can write an integral. 1:05:59.530 --> 1:06:02.710 So every problem, you first find the potential by 1:06:02.706 --> 1:06:05.616 adding the potential from all the charges. 1:06:05.619 --> 1:06:09.409 Then its gradient will give it the electric field. 1:06:09.409 --> 1:06:13.819 But we have now a law of conservation of energy with this 1:06:13.820 --> 1:06:15.870 as the conserved energy. 1:06:15.869 --> 1:06:29.629 The second advantage is that it is easier to work with V 1:06:29.628 --> 1:06:34.288 than with E. 1:06:34.289 --> 1:06:39.579 So why do you think it's easier to find V and then find 1:06:39.581 --> 1:06:43.661 E from it, by taking derivatives than the 1:06:43.657 --> 1:06:45.477 other way around? 1:06:45.480 --> 1:06:46.480 Yes? 1:06:46.480 --> 1:06:48.440 Student: V is a scalar. 1:06:48.440 --> 1:06:49.180 Prof: Right. 1:06:49.179 --> 1:06:54.749 Student: > 1:06:54.750 --> 1:06:56.340 Prof: I'm sorry, I didn't hear the last part. 1:06:56.340 --> 1:06:58.030 Student: It's always easier to take derivatives than 1:06:58.032 --> 1:06:59.772 to integrate-- Prof: Very good. 1:06:59.769 --> 1:07:00.409 Yes. 1:07:00.409 --> 1:07:03.189 Let me repeat that. 1:07:03.190 --> 1:07:06.600 If you want to find the potential due to many charges, 1:07:06.601 --> 1:07:09.821 you simply add the numbers coming from each one. 1:07:09.820 --> 1:07:13.400 There are no arrows, there are no vectors. 1:07:13.400 --> 1:07:17.050 Notice the formula for V has no vectors in it. 1:07:17.050 --> 1:07:18.540 Each one contributes a number. 1:07:18.539 --> 1:07:20.659 Each charge contributes a number to the point they're 1:07:20.664 --> 1:07:21.324 interested in. 1:07:21.320 --> 1:07:23.210 Add them all up, find the number here, 1:07:23.210 --> 1:07:24.490 there, everywhere else. 1:07:24.489 --> 1:07:27.639 Then take derivatives of your answer to get the field. 1:07:27.639 --> 1:07:30.159 That's going to be a lot easier than adding the vectors. 1:07:30.159 --> 1:07:35.469 So I'm going to illustrate that with one simple problem and 1:07:35.474 --> 1:07:38.594 that's the last thing for today. 1:07:38.590 --> 1:07:40.770 That's the dipole. 1:07:40.769 --> 1:07:42.909 So here is our dipole. 1:07:42.909 --> 1:07:47.249 Let's take a charge q, -q at -a and 1:07:47.253 --> 1:07:49.073 q at a. 1:07:49.070 --> 1:07:55.930 And we want to find the field there. 1:07:55.929 --> 1:08:00.819 So let's call that distance r_ and let's 1:08:00.818 --> 1:08:04.418 call that distance r_-. 1:08:04.420 --> 1:08:08.450 These are not vectors; these are just lengths. 1:08:08.449 --> 1:08:11.879 If you want to find the electric field directly, 1:08:11.878 --> 1:08:14.138 you know what you have to do. 1:08:14.139 --> 1:08:17.899 You have to find the vector here and you've got to find the 1:08:17.900 --> 1:08:21.660 vector there due to the second guy, add the two vectors and 1:08:21.662 --> 1:08:22.962 find the result. 1:08:22.960 --> 1:08:24.990 Of course, you can do it, but it's going to be very 1:08:24.992 --> 1:08:26.712 tedious, because you have to find the 1:08:26.711 --> 1:08:29.391 direction of the vector here and the direction of the vector 1:08:29.390 --> 1:08:32.040 here, in terms of all these 1:08:32.042 --> 1:08:33.432 coordinates. 1:08:33.430 --> 1:08:37.080 What we will do instead is to find the potential everywhere, 1:08:37.078 --> 1:08:38.968 and take the x and y derivatives of the 1:08:38.970 --> 1:08:40.570 potential to get the E_x and 1:08:40.569 --> 1:08:41.479 E_y. 1:08:41.479 --> 1:08:42.689 Well, let's do that. 1:08:42.689 --> 1:08:46.439 So let's find the total potential due to these two guys. 1:08:46.439 --> 1:08:50.999 The first one is q/4Πε 1:08:50.997 --> 1:08:57.637 _0 divided by 1/r_ - 1:08:57.640 --> 1:09:02.590 1/r_-. That's it. 1:09:02.590 --> 1:09:04.270 Just wanted to simplify the expression a little bit. 1:09:04.270 --> 1:09:07.150 So this is true for any separation, 1:09:07.149 --> 1:09:09.829 but I want the limit in which r_ and 1:09:09.828 --> 1:09:12.608 r_- are much bigger than the separation 1:09:12.609 --> 1:09:13.469 between them. 1:09:13.470 --> 1:09:19.190 So let's define an r which is the position vector of 1:09:19.193 --> 1:09:25.413 the point you're interested in from the center of the dipole. 1:09:25.408 --> 1:09:26.578 So we've got to find r_ and 1:09:26.578 --> 1:09:27.188 r_-. 1:09:27.189 --> 1:09:30.299 You get that, you're done. 1:09:30.300 --> 1:09:36.640 So vector r_ ^( )you can see = 1:09:36.635 --> 1:09:42.965 r - ai/2 and vector 1:09:42.970 --> 1:09:51.510 r_- = vector r ai/2. 1:09:51.510 --> 1:09:54.910 Because the vector ai/2 looks like 1:09:54.905 --> 1:09:59.165 this, so ai/2 r_ should give 1:09:59.171 --> 1:10:00.401 you r. 1:10:00.399 --> 1:10:01.839 You can check that. 1:10:01.840 --> 1:10:05.460 And you can check that ai/2 r 1:10:05.461 --> 1:10:07.901 gives you r_-. 1:10:07.899 --> 1:10:10.309 so these are the two expressions. 1:10:10.310 --> 1:10:12.900 So what is r_ ? 1:10:12.899 --> 1:10:18.709 r^( ) is the length of the vector r_ . 1:10:18.710 --> 1:10:23.000 That's equal to the square root of the length squared. 1:10:23.000 --> 1:10:29.340 The length squared of r_ squared = root of 1:10:29.341 --> 1:10:34.611 (r - ai/2) ⋅ (r - 1:10:34.609 --> 1:10:37.189 ai/2). 1:10:37.189 --> 1:10:39.389 Student: Why is it a i/2? 1:10:39.390 --> 1:10:42.540 Prof: Oh, I'm sorry, it's not. 1:10:42.538 --> 1:10:45.068 It is half the distance, but it's really only a. 1:10:45.069 --> 1:10:50.449 Thank you. 1:10:50.448 --> 1:10:52.428 Yeah, in fact, one puts the 2a as a 1:10:52.432 --> 1:10:54.612 separation to avoid all these factors of 2. 1:10:54.609 --> 1:10:56.809 So now you take the length squared. 1:10:56.810 --> 1:11:01.990 It's equal to r^(2) a^(2) minus twice 1:11:01.985 --> 1:11:05.295 r⋅a. 1:11:05.300 --> 1:11:08.860 The other one would be r^(2) a^(2) twice 1:11:08.859 --> 1:11:10.969 r⋅a. 1:11:10.970 --> 1:11:16.780 And that we can approximate as r times 1 - twice 1:11:16.775 --> 1:11:21.935 r⋅ a/r^(2)^( )to 1:11:21.936 --> 1:11:23.546 the 1 half. 1:11:23.550 --> 1:11:29.750 I'll tell you what I did here. 1:11:29.750 --> 1:11:31.780 I'm going to neglect a squared compared to r 1:11:31.780 --> 1:11:33.600 squared, because r is much bigger than it. 1:11:33.600 --> 1:11:36.080 But I'm going to keep this r⋅a 1:11:36.082 --> 1:11:38.252 term because this has got two powers of a, 1:11:38.248 --> 1:11:39.918 where a is a small number. 1:11:39.920 --> 1:11:42.420 It's got 1 power of a and 1 power of r, 1:11:42.421 --> 1:11:45.071 so it's much stronger than this term, so I keep that. 1:11:45.069 --> 1:11:47.599 Keeping the second term, so I've forgotten the 1:11:47.603 --> 1:11:48.733 a^(2) term. 1:11:48.729 --> 1:11:51.989 In this one, from the theorem I've been 1:11:51.985 --> 1:11:56.695 giving you guys all the time, it's 1 - r⋅ 1:11:56.698 --> 1:11:58.668 a/r^(2). 1:11:58.670 --> 1:12:03.670 Remember 1 x^(n) is roughly 1 nx dot dot dot 1:12:03.666 --> 1:12:07.316 if x is small, and x is indeed small, 1:12:07.319 --> 1:12:09.509 because it's r⋅a 1:12:09.512 --> 1:12:10.852 over r squared. 1:12:10.850 --> 1:12:18.620 So now we can see V = q/4Πε 1:12:18.618 --> 1:12:24.518 _0{1/[r(1- r⋅ 1:12:24.518 --> 1:12:31.138 a/r^(2))] - 1/[r(1 r⋅ 1:12:31.136 --> 1:12:35.306 a/r^(2))]} . 1:12:35.310 --> 1:12:40.350 This is going to be found in every textbook in the planet, 1:12:40.350 --> 1:12:42.650 so don't worry about it. 1:12:42.649 --> 1:12:46.929 If you simplify this again by taking it upstairs, 1:12:46.932 --> 1:12:49.972 you'll get twice a⋅r 1:12:49.966 --> 1:12:51.926 /r^(2). 1:12:51.930 --> 1:12:57.690 That's the potential. 1:12:57.689 --> 1:13:00.469 Now twice a times q is the dipole moment, 1:13:00.470 --> 1:13:08.340 so it's p⋅r /4Πε 1:13:08.342 --> 1:13:12.282 _0r^(3). 1:13:12.279 --> 1:13:15.309 That's the final piece, the dipole moment. 1:13:15.310 --> 1:13:21.260 Dipole moment is a charge times the separation between the 1:13:21.255 --> 1:13:25.945 charges, which is 2a times q. 1:13:25.948 --> 1:13:28.558 Okay, so what one should do now is, 1:13:28.560 --> 1:13:31.270 having taken this V, one can take its x and 1:13:31.274 --> 1:13:34.334 y derivatives very easily to calculate the electric field 1:13:34.327 --> 1:13:35.197 at each point. 1:13:35.198 --> 1:13:37.308 It's very easy to take the derivative of this, 1:13:37.306 --> 1:13:39.036 just pick the x or y. 1:13:39.038 --> 1:13:42.058 It's one of the homework problems I've assigned to you, 1:13:42.056 --> 1:13:44.066 where I show you the dipole field. 1:13:44.069 --> 1:13:47.089 So the moral of the story is, add the potential due to all 1:13:47.094 --> 1:13:50.124 the charges, then take the derivative, because derivatives 1:13:50.118 --> 1:13:51.338 are easier to take. 1:13:51.340 --> 1:13:52.180 Yes? 1:13:52.180 --> 1:13:53.760 Student: The condition for that result, 1:13:53.762 --> 1:13:55.142 is r much bigger than a? 1:13:55.140 --> 1:13:58.440 Prof: Yes. 1:13:58.439 --> 1:14:01.429 Up till some point, everything is exact, 1:14:01.434 --> 1:14:04.664 but then this is long distance property. 1:14:04.659 --> 1:14:08.719 Any questions? 1:14:08.720 --> 1:14:10.430 Okay, thank you. 1:14:10.430 --> 1:14:16.000