WEBVTT 00:01.890 --> 00:07.210 Prof: Let's start with a brief recall of Gauss's Law. 00:07.210 --> 00:11.260 It is so important I wanted to make sure you guys understand 00:11.256 --> 00:14.476 not only how it is used but how it's derived. 00:14.480 --> 00:18.170 I derived it for you last time, but I thought the presentation 00:18.165 --> 00:22.205 could be improved a little bit, so I'm going to do it one more 00:22.214 --> 00:27.524 time, just this time only hitting the 00:27.517 --> 00:29.387 highlights. 00:29.390 --> 00:33.450 The content of Gauss's Law is very easy to visualize, 00:33.450 --> 00:35.940 so I'm going to draw you a picture that'll tell you the 00:35.938 --> 00:37.778 whole story, then there are all the 00:37.776 --> 00:38.346 equations. 00:38.350 --> 00:40.420 The picture is the following. 00:40.420 --> 00:45.900 You take one charge q and you draw the lines of force 00:45.904 --> 00:48.604 coming from the one charge. 00:48.600 --> 00:50.330 It's a point charge. 00:50.330 --> 01:01.430 Therefore, you have no freedom but to draw lines that look like 01:01.426 --> 01:02.676 this. 01:02.679 --> 01:07.309 Now, how many lines you draw is a matter of convention. 01:07.310 --> 01:10.430 You can draw eight or twelve, whatever it is, 01:10.427 --> 01:14.677 but you've got to agree that the number per coulomb should be 01:14.680 --> 01:15.390 fixed. 01:15.390 --> 01:17.040 So we'll fix some number. 01:17.040 --> 01:21.040 So I'm going to pick for whatever charge is here these 01:21.036 --> 01:21.636 lines. 01:21.640 --> 01:26.100 You cannot have a distribution of lines in which it's suddenly 01:26.104 --> 01:29.694 more dense in just this direction, for example. 01:29.688 --> 01:33.808 That's not allowed, because if I take this point 01:33.810 --> 01:37.270 charge and I rotate it, it looks the same, 01:37.274 --> 01:40.574 but if I took the line distribution with more dense 01:40.565 --> 01:43.325 here than other places if I rotate it, 01:43.330 --> 01:46.040 it looks different. 01:46.040 --> 01:50.210 So if the cause of the field lines looks the same after 01:50.209 --> 01:53.759 rotation, the effect, which is the field lines, 01:53.760 --> 01:55.460 should not change. 01:55.459 --> 01:58.389 You can only have a distribution of lines with 01:58.394 --> 02:02.184 respect to symmetry of the source of the cause of the field 02:02.177 --> 02:02.827 lines. 02:02.829 --> 02:04.319 You understand that? 02:04.319 --> 02:07.729 It cannot be biased one way or the other, whereas this 02:07.730 --> 02:09.980 distribution, if you turn it around, 02:09.982 --> 02:11.722 looks exactly the same. 02:11.718 --> 02:17.378 So we draw those lines, and the content of Gauss's Law 02:17.375 --> 02:20.145 is really the following. 02:20.150 --> 02:27.530 We are going to count the lines crossing a sphere, 02:27.527 --> 02:36.257 S, and we're going to count the lines crossing some generic 02:36.260 --> 02:38.520 surface S'. 02:38.520 --> 02:45.110 You can see at a glance without doing any calculations that the 02:45.113 --> 02:47.243 following is true. 02:47.240 --> 02:59.400 Lines crossing S = lines crossing S', right? 02:59.400 --> 03:00.410 That's clear to everybody? 03:00.408 --> 03:02.588 You don't need to do any equation for that. 03:02.590 --> 03:05.140 That's because the lines have nowhere to go but out, 03:05.139 --> 03:07.039 and you can count them on this circle, 03:07.038 --> 03:09.988 that circle as long as you enclose the whole thing, 03:09.990 --> 03:12.100 the solid surface. 03:12.098 --> 03:14.828 Enclosing this charge you're going to get the same lines. 03:14.830 --> 03:18.600 That's the heart of Gauss's Law, but that's not the final 03:18.603 --> 03:22.043 form, but that's the part that starts everything. 03:22.038 --> 03:24.318 Now I'm going to calculate the left hand side, 03:24.323 --> 03:26.813 and I'm going to calculate the right hand side. 03:26.810 --> 03:28.450 I'm going to equate them. 03:28.449 --> 03:32.319 From that I will get almost everything I want. 03:32.318 --> 03:35.628 Now, how many lines are coming out of S? 03:35.628 --> 03:39.338 We know--first the notion of line density. 03:39.340 --> 03:42.410 I've got to remind you what a line density is. 03:42.410 --> 03:47.960 Line density is the lines per area, but it's not enough to say 03:47.961 --> 03:48.601 area. 03:48.598 --> 03:52.258 I'm going to put a perpendicular symbol there. 03:52.258 --> 03:56.838 That means the lines are going, say, from left to right like 03:56.836 --> 03:57.376 this. 03:57.378 --> 04:03.158 You want to put an area that intercepts it perpendicularly, 04:03.158 --> 04:05.708 or if you like, the area vector which is always 04:05.710 --> 04:08.650 normal to the area should be parallel to the lines. 04:08.650 --> 04:11.910 So it's the number of lines crossing unit area perpendicular 04:11.909 --> 04:12.739 to the lines. 04:12.740 --> 04:15.230 That's called the line density. 04:15.229 --> 04:17.929 If you tilted this area; the same area, 04:17.932 --> 04:21.762 but at a bad angle, it'll intersect fewer lines. 04:21.759 --> 04:23.449 That you know. 04:23.449 --> 04:25.849 Okay, so this is the meaning of line density. 04:25.850 --> 04:29.190 What's the line density going to look like in this problem? 04:29.189 --> 04:33.789 Line density is going to be proportional to the charge you 04:33.786 --> 04:38.296 have because we agreed we'll put some number of lines per 04:38.300 --> 04:42.070 coulomb, and it's inversely proportional 04:42.074 --> 04:47.354 to the radius because the area of a sphere of radius r 04:47.350 --> 04:49.460 is 4Πr^(2). 04:49.459 --> 04:52.509 So if you cut it with a bigger sphere of bigger radius the 04:52.507 --> 04:55.447 density, lines per area, will go like 1/r^(2). 04:55.449 --> 04:57.209 So it's proportional to q. 04:57.209 --> 05:00.159 It's proportional to 1/r^(2), 05:00.160 --> 05:00.750 okay? 05:00.750 --> 05:05.790 So let me write not equal to but proportional to. 05:05.790 --> 05:08.880 You have to stop me if you don't follow anything I'm 05:08.884 --> 05:09.434 saying. 05:09.430 --> 05:12.910 There's no reason why you shouldn't follow this argument. 05:12.910 --> 05:20.520 Then I say how about the electric field? 05:20.519 --> 05:21.629 The strength of the electric field, 05:21.629 --> 05:25.809 not the direction, the electric field E at 05:25.810 --> 05:30.250 radius r is also proportional to q and 05:30.254 --> 05:33.744 also proportional to 1/r^(2). 05:33.740 --> 05:36.780 Because from Coulomb's Law the field is the force on a unit 05:36.781 --> 05:38.621 test charge, it is q_1q 05:38.617 --> 05:39.717 _2. 05:39.720 --> 05:42.190 q_2 is 1 because the test charge is 1 and 05:42.190 --> 05:43.620 there's q/4Πε 05:43.622 --> 05:44.752 _0r^(2). 05:44.750 --> 05:45.940 Forget the 4Π. 05:45.940 --> 05:49.630 It's proportional to q/r^(2). 05:49.629 --> 05:53.189 From this we may conclude that the line density is proportional 05:53.187 --> 05:56.517 to the electric field because they are both proportional to 05:56.516 --> 05:58.006 q/r^(2). 05:58.009 --> 06:05.409 Therefore, we can say line density is equal to some 06:05.406 --> 06:12.946 constant c times the electric field in magnitude; 06:12.949 --> 06:17.349 direction of the lines is along E, but the magnitude. 06:17.350 --> 06:18.420 Do you understand that? 06:18.420 --> 06:20.590 This guy is proportional to this. 06:20.589 --> 06:22.309 This may have some proportionality constant, 06:22.307 --> 06:24.457 this may have another one, but nothing changes the fact 06:24.463 --> 06:26.543 that this expression's proportional to that one. 06:26.540 --> 06:28.450 The constant will depend on details. 06:28.449 --> 06:31.879 Let me just call it some number c. 06:31.879 --> 06:32.949 c is up to you. 06:32.949 --> 06:35.829 It really depends on how many lines you want to draw per 06:35.834 --> 06:36.364 coulomb. 06:36.360 --> 06:37.980 I made a particular choice last time, 06:37.976 --> 06:40.126 1/ε_0 lines per coulomb, 06:40.129 --> 06:42.509 but you don't have to make that to prove the law. 06:42.509 --> 06:46.259 So this time I'm just assuming the lines is proportional to 06:46.262 --> 06:49.952 charge, but you pick your own proportionality constant. 06:49.949 --> 06:53.679 So I've got this line density that's proportional to electric 06:53.682 --> 06:54.182 field. 06:54.180 --> 06:58.710 Now I'm going to go back and I'm going to calculate the left 06:58.708 --> 06:59.628 hand side. 06:59.629 --> 07:03.639 Now, let's give this equation a number, equation number 1. 07:03.639 --> 07:11.549 I'm going to calculate the left hand side of 1. 07:11.550 --> 07:17.480 Left hand side of 1 is the number of lines crossing this 07:17.476 --> 07:18.766 sphere, S. 07:18.769 --> 07:24.999 So that's going to be equal to--let me call it 07:25.000 --> 07:31.510 Φ_S is equal to number crossing S, 07:31.509 --> 07:35.249 crossing this sphere S. 07:35.250 --> 07:38.560 Φ is the standard symbol for flux. 07:38.560 --> 07:41.290 So what is it? 07:41.290 --> 08:04.810 Well, Φ_S = line density times area of sphere, 08:04.810 --> 08:07.950 because on a sphere the lines are uniform in density so the 08:07.949 --> 08:11.309 total number crossing the sphere is the line density times area 08:11.305 --> 08:12.275 of the sphere. 08:12.278 --> 08:17.638 But the line density is a constant times the electric 08:17.637 --> 08:24.017 field at radius r (if you like) times area of the sphere 08:24.024 --> 08:27.224 which is 4Πr^(2). 08:27.220 --> 08:32.090 That's the lines crossing the inner surface which is this 08:32.086 --> 08:34.516 sphere of radius r. 08:34.519 --> 08:46.349 Now, Φ_S' is lines crossing S'. 08:46.350 --> 08:50.130 Now you've got to be a little careful when you find the lines 08:50.128 --> 08:51.008 crossing S'. 08:51.009 --> 08:54.159 You understand why that's more difficult? 08:54.158 --> 08:59.768 What are the two complications in finding the lines crossing S' 08:59.767 --> 09:04.197 compared to this one which I did in one second? 09:04.200 --> 09:04.900 Any idea? 09:04.899 --> 09:05.949 Yes? 09:05.950 --> 09:07.810 Student: The normals are all parallel to the field 09:07.808 --> 09:08.068 lines? 09:08.070 --> 09:09.140 Prof: Which is not parallel? 09:09.139 --> 09:10.589 Student: The normal to the surface isn't parallel to 09:10.591 --> 09:10.961 the ________. 09:10.960 --> 09:13.600 Prof: Okay, the surface is not--surface 09:13.600 --> 09:17.180 vector at each point is not parallel to the electric field. 09:17.178 --> 09:19.478 For example, if you've got a crazy surface 09:19.480 --> 09:22.070 like this-- I'll try to replicate that 09:22.066 --> 09:24.416 surface, but I don't remember what I 09:24.417 --> 09:26.537 drew, some weird object like this. 09:26.538 --> 09:31.748 Then the electric field point this way, but the normal to the 09:31.745 --> 09:34.345 surface will point that way. 09:34.350 --> 09:36.160 See, E points that way. 09:36.158 --> 09:38.338 Tiny area there, which I'm calling vector 09:38.342 --> 09:40.802 dA, points in a different direction. 09:40.798 --> 09:43.798 So each region has a different direction. 09:43.798 --> 09:48.268 Secondly, the strength of the field itself is varying on the 09:48.272 --> 09:49.032 surface. 09:49.029 --> 09:51.469 Unlike on a sphere where everything is at the same 09:51.470 --> 09:54.160 distance r some are further, some are closer. 09:54.158 --> 09:57.788 But it is still true that if you want the number of lines 09:57.788 --> 10:00.968 cross a patch, if you pick a little area here, 10:00.970 --> 10:04.800 if you want to know how many lines cross that you take the 10:04.796 --> 10:08.676 normal to the area, call it a vector dA, 10:08.677 --> 10:11.617 and you multiply by line density. 10:11.620 --> 10:17.040 Therefore, this is equal to the integral of the electric field 10:17.038 --> 10:22.008 at some point r dot product with a tiny area which 10:22.014 --> 10:23.174 is there. 10:23.168 --> 10:28.528 At each region you've got to draw a different dA. 10:28.528 --> 10:32.028 So you take the surface, you chop it up into tiny little 10:32.029 --> 10:32.539 tiles. 10:32.538 --> 10:35.388 Each tile is small enough to be considered a plane. 10:35.389 --> 10:38.539 Then the area vector is perpendicular to the plane, 10:38.543 --> 10:41.953 outward from the surface, and of magnitude equal to the 10:41.951 --> 10:43.971 numerical area of the plane. 10:43.970 --> 10:47.170 So the lines crossing every tiny little square on the big 10:47.172 --> 10:50.492 surface is the number of lines per unit area times the area 10:50.490 --> 10:53.350 times the cosine of the angle which you need, 10:53.350 --> 10:58.000 because the area vector may not be parallel to the flow. 10:58.000 --> 11:01.710 I have explained many times why the flow there, 11:01.710 --> 11:04.070 the flow vector and the area vector are not parallel you've 11:04.067 --> 11:05.527 got to put in a cosθ. 11:05.528 --> 11:10.138 That's what comes into this dot product. 11:10.139 --> 11:12.649 And we write the symbol like this to tell you not only is it 11:12.653 --> 11:14.683 a double integral, meaning integral over a 11:14.676 --> 11:17.406 two-dimensional region, but it's over a closed 11:17.405 --> 11:19.045 two-dimensional region. 11:19.049 --> 11:22.759 That symbol is to remind you. 11:22.759 --> 11:26.189 So now, Φ_S equal to Φ_S' is what 11:26.188 --> 11:27.388 I'm going to write. 11:27.389 --> 11:29.419 So I'm going to write--oh, I made one mistake. 11:29.418 --> 11:36.818 Can anybody see one thing missing in this equation here? 11:36.820 --> 11:39.390 This equation's not correct. 11:39.389 --> 11:44.329 To find the lines crossing S' I took E times dA. 11:44.330 --> 11:48.040 It's a constant c because the line density is not 11:48.038 --> 11:51.138 simply the electric field it's the constant. 11:51.139 --> 11:56.349 So I forgot the constant, but I knew that I would need it 11:56.351 --> 12:00.541 because when I now equate Φ_S to 12:00.538 --> 12:05.278 Φ_S' I get constant times the electric 12:05.284 --> 12:10.124 field at r times 4Πr^(2) is equal to 12:10.124 --> 12:13.574 constant times the surface integral of 12:13.567 --> 12:18.497 E⋅dA on some surface S'. 12:18.500 --> 12:21.480 But E(r) times 4Πr^(2) is c 12:21.476 --> 12:23.306 times q/ε_0 12:23.306 --> 12:25.496 because E is q/4Πε 12:25.500 --> 12:27.120 _0r^(2). 12:27.120 --> 12:28.140 You see that? 12:28.139 --> 12:29.089 This much you should know. 12:29.090 --> 12:30.730 I'm not going to write the details. 12:30.730 --> 12:38.160 So that is equal to c times surface integral of 12:38.155 --> 12:41.795 E⋅dA. 12:41.798 --> 12:45.328 Now you see the constant does not matter, 12:45.330 --> 12:49.560 so whatever the constant is I can make the following statement 12:49.562 --> 12:53.312 and that's going to be the beginning of Gauss's Law. 12:53.308 --> 12:57.528 So Gauss's Law tells me that if you took a charge and you sat 12:57.529 --> 13:01.539 under it with any surface whatsoever--I'm not going to put 13:01.539 --> 13:03.509 a prime on this anymore. 13:03.509 --> 13:05.779 S is going to be the surface. 13:05.778 --> 13:11.678 Then the surface integral of the electric field due to that 13:11.682 --> 13:17.692 charge over a surface is equal to the charge that is sitting 13:17.688 --> 13:22.268 inside divided by ε_0, 13:22.269 --> 13:25.379 and it does not depend on the details of the surface. 13:25.379 --> 13:35.259 13:35.259 --> 13:39.809 Okay, everybody should know in principle how to do the surface 13:39.808 --> 13:40.628 integral. 13:40.629 --> 13:42.999 If I give you a surface, and at the surface at every 13:42.995 --> 13:44.755 point I give you a vector E, 13:44.759 --> 13:47.389 which is electric field, you should know how to 13:47.393 --> 13:48.943 numerically calculate it. 13:48.940 --> 13:52.070 You may not know how to do the integral analytically, 13:52.072 --> 13:55.272 but operationally you should know the meaning of this, 13:55.267 --> 13:55.807 okay? 13:55.808 --> 13:58.668 If E were actually the velocity of some stuff running 13:58.672 --> 14:01.392 out of the volume you're counting in each region how many 14:01.389 --> 14:04.349 guys are escaping per second and you're adding it all up. 14:04.350 --> 14:06.820 And to find how many guys are leaving a tiny area, 14:06.820 --> 14:09.650 you multiply the area by the velocity of flow times the 14:09.653 --> 14:12.543 cosine of the angle between the velocity in the area. 14:12.539 --> 14:14.609 That's it. 14:14.610 --> 14:18.040 Okay, now we do the next thing. 14:18.038 --> 14:24.568 Let's call this charge q_1 because I'm 14:24.572 --> 14:31.592 now going to put a second guy q_2 here. 14:31.590 --> 14:34.280 If you now think in terms of lines of force you're going to 14:34.283 --> 14:36.423 get into a big mess because lines of force, 14:36.418 --> 14:38.528 due to two charges is a complicated thing, 14:38.529 --> 14:40.409 you remember drawing some pictures. 14:40.409 --> 14:42.009 But now here's the beauty. 14:42.009 --> 14:45.289 Forget the lines of force completely, not interested. 14:45.288 --> 14:48.628 I'm interested in the electric field. 14:48.629 --> 14:52.269 Focus on the electric field for which there's a very simple 14:52.269 --> 14:52.709 rule. 14:52.710 --> 14:58.250 The electric field due to two charges 1 and 2 is the electric 14:58.245 --> 15:02.855 field due to 1 plus the electric field due to 2. 15:02.860 --> 15:10.150 That's the great principle of superposition. 15:10.149 --> 15:14.369 Therefore, on the left hand side you might say if both 15:14.370 --> 15:17.640 charges are inside the surface integral of 15:17.636 --> 15:21.936 E_1,2_ meaning when both are 15:21.937 --> 15:26.317 present is equal to surface integral of E_1 15:26.318 --> 15:30.698 ⋅dA surface integral of E_2 15:30.700 --> 15:35.370 ⋅dA, because E_1,2 15:35.371 --> 15:38.751 is E_1 E_2. 15:38.750 --> 15:40.850 But the first guy is q_1 15:40.851 --> 15:43.071 /ε _0, 15:43.070 --> 15:47.830 second is q_2 /ε 15:47.833 --> 15:49.943 _0. 15:49.940 --> 15:54.040 Once you can do two charges you can do any number of charges. 15:54.038 --> 15:57.948 And now we write Gauss's Law almost in its final form. 15:57.950 --> 16:02.280 The final form is that if you took the total electric field, 16:02.278 --> 16:05.808 I'm not going to put total, E will stand for the 16:05.811 --> 16:09.491 total, on a surface S that is 16:09.490 --> 16:13.540 equal to the sum of all the charges, 16:13.538 --> 16:16.578 i from 1 to whatever many charges you have, 16:16.580 --> 16:18.390 divided by ε_0. 16:18.389 --> 16:22.519 The condition is these are charges inside the volume 16:22.523 --> 16:24.553 bounded by the surface. 16:24.548 --> 16:27.538 You do not count the charge here. 16:27.538 --> 16:31.738 You only count charges which are inside not charges which are 16:31.740 --> 16:32.440 outside. 16:32.440 --> 16:37.790 So let's make a notation that tells you that i is 16:37.794 --> 16:40.624 inside the volume V. 16:40.620 --> 16:44.470 And this surface is the boundary of the volume V. 16:44.470 --> 16:49.340 This partial derivative symbol is the universal convention for 16:49.336 --> 16:51.646 the boundary of any region. 16:51.649 --> 16:56.149 Okay, if you draw a real circle, this is the circle. 16:56.149 --> 17:00.579 The boundary of the circle is this. 17:00.580 --> 17:02.840 If you a have a cube, or if you have an orange the 17:02.836 --> 17:05.506 boundary of the volume orange is the skin of the orange. 17:05.509 --> 17:07.049 That's the definition of boundary. 17:07.048 --> 17:09.738 You can see this is like the skin of the solid region, 17:09.740 --> 17:14.650 and the surface integral on the skin of the electric field is a 17:14.650 --> 17:18.770 charge inside the volume whatever be the shape of the 17:18.768 --> 17:19.638 volume. 17:19.640 --> 17:21.440 And it works for all kinds of surfaces. 17:21.440 --> 17:23.690 I mean, here's one example. 17:23.690 --> 17:25.510 Suppose you have a charge here. 17:25.509 --> 17:29.009 You take a surface that kind of intersects itself and so on. 17:29.009 --> 17:33.649 Then what can happen is if I put a charge here the lines may 17:33.650 --> 17:34.830 go like this. 17:34.828 --> 17:38.768 It doesn't matter because over this when it switches back on 17:38.769 --> 17:41.509 itself some lines leave, some lines enter, 17:41.509 --> 17:43.579 and some lines leave again. 17:43.578 --> 17:48.848 So it might as well be that surface because double entering 17:48.846 --> 17:51.476 and leaving doesn't matter. 17:51.480 --> 17:55.760 So the rule is if q is inside it contributes to the 17:55.757 --> 17:58.907 surface integral, if q is outside it 17:58.909 --> 17:59.809 doesn't. 17:59.808 --> 18:03.608 The last variation is what if q is not made up of a 18:03.608 --> 18:07.038 discrete set of charges, which you can say one is here 18:07.037 --> 18:09.867 one is there, but there's some continuous 18:09.866 --> 18:13.766 blob, some lump of charge where the charge density-- 18:13.769 --> 18:20.599 so then we say suppose q is continuous. 18:20.598 --> 18:22.868 You've got to realize in real life it's never continuous. 18:22.868 --> 18:25.618 Real charges are made up of electrons and protons. 18:25.618 --> 18:28.648 They are discrete, but if your eye is not looking 18:28.654 --> 18:32.644 at it in great detail the charge can appear to be continuous. 18:32.640 --> 18:34.760 For example, if you take a glass of water 18:34.763 --> 18:37.423 the water is really made up of point molecules, 18:37.420 --> 18:40.130 water molecules, but to your eye it looks like a 18:40.134 --> 18:41.294 continuous medium. 18:41.288 --> 18:45.488 So for certain purposes we can replace a discrete distribution 18:45.492 --> 18:47.012 by a continuous one. 18:47.009 --> 18:50.549 For a continuous distribution what you have to tell me is the 18:50.550 --> 18:51.260 following. 18:51.259 --> 18:53.559 This is space x, y, and z. 18:53.559 --> 18:55.479 Here's the point. 18:55.480 --> 19:01.670 I take a tiny cube whose sides are dx, 19:01.670 --> 19:07.520 dy and dz, and ρ of x, 19:07.519 --> 19:16.379 y, and z times dxdydz is equal to charge 19:16.380 --> 19:19.630 in that little cube. 19:19.630 --> 19:22.910 In other words, like the usual density with 19:22.909 --> 19:27.359 this mass per unit volume the ρ is the charge per unit 19:27.361 --> 19:28.221 volume. 19:28.220 --> 19:30.860 So you multiply by volume you get the charge, 19:30.858 --> 19:32.598 but it need not be uniform. 19:32.598 --> 19:41.448 So if you have a solid region then the charge density could be 19:41.453 --> 19:46.393 varying from point to point, so. 19:46.390 --> 19:48.550 Oh, let's keep it here. 19:48.548 --> 19:54.188 In that case you will write the surface integral of 19:54.186 --> 20:00.606 E⋅dA over a surface that binds a volume 20:00.614 --> 20:05.804 V would be 1/ε_0 times 20:05.799 --> 20:10.309 the volume integral of ρ of (x, y, 20:10.309 --> 20:13.579 z) dxdydz. 20:13.578 --> 20:24.668 This is the final form of Gauss's Law. 20:24.670 --> 20:31.980 Now, how many people have ever done integrals in more than one 20:31.980 --> 20:33.420 dimension? 20:33.420 --> 20:38.190 Great, very good, made my day because I didn't 20:38.185 --> 20:39.665 have Plan B. 20:39.670 --> 20:44.090 But I did--no actually I had a Plan Tiny B which is to give you 20:44.093 --> 20:45.523 a quick refresher. 20:45.519 --> 20:49.289 If there is time at the end of the class I will remind you of 20:49.290 --> 20:51.680 how the multiple integrals are done. 20:51.680 --> 20:54.760 But right now I'm going to show you the efficacy of this law and 20:54.758 --> 20:57.248 that doesn't require doing complicated integrals. 20:57.250 --> 21:01.770 All the integrals will be so trivial you can just read them 21:01.771 --> 21:02.241 off. 21:02.240 --> 21:06.040 So when we have time we'll do questions of how to do these 21:06.038 --> 21:10.368 integrals, but I'm assuming this symbol means something to you. 21:10.368 --> 21:12.558 In words it means all the charge inside. 21:12.558 --> 21:15.458 So, again, by taking the volume, dividing into tiny 21:15.461 --> 21:17.431 regions, multiplying each volume, 21:17.425 --> 21:19.385 tiny volume, by the density to get the 21:19.393 --> 21:21.603 charge there, and summing over all the 21:21.602 --> 21:24.092 volumes inside the big volume V. 21:24.089 --> 21:25.069 That's a charge. 21:25.068 --> 21:28.668 So Gauss's Law still says the surface integral of E is 21:28.670 --> 21:30.770 the charge inside over epsilon 0. 21:30.769 --> 21:33.989 All right, so this is the great law. 21:33.990 --> 21:37.780 Now we're going to get some mileage out of it to solve 21:37.776 --> 21:41.916 certain problems which normally would be very difficult. 21:41.920 --> 21:45.480 The first problem, I think I probably did that 21:45.479 --> 21:50.149 last time but it's worth going over, is what is the electric 21:50.145 --> 21:53.385 field of a uniform density of charge. 21:53.390 --> 21:59.600 Suppose there's a charge Q and this is sphere of 21:59.602 --> 22:04.432 radius r, and it is a ball of radius 22:04.433 --> 22:05.933 r. 22:05.930 --> 22:07.890 You take all these charges and you pile them on top of each 22:07.888 --> 22:08.158 other. 22:08.160 --> 22:10.040 They don't like it, but you glue them or do 22:10.042 --> 22:11.212 whatever you have to do. 22:11.210 --> 22:13.720 You've got a ball of charge Q. 22:13.720 --> 22:16.160 You want the field due to that. 22:16.160 --> 22:18.660 It's the same problem Newton had with gravity. 22:18.660 --> 22:20.530 The earth is a solid ball. 22:20.528 --> 22:24.238 It's got mass everywhere and you would like to find the force 22:24.242 --> 22:25.112 on an apple. 22:25.108 --> 22:29.188 And it's not so evident that the force of the earth on the 22:29.189 --> 22:33.769 apple is as if the entire mass of the earth but at the center. 22:33.769 --> 22:35.209 You have to prove that. 22:35.210 --> 22:38.180 You have to prove that by dividing the earth into little 22:38.180 --> 22:40.880 pieces and finding the gravitational force of every 22:40.881 --> 22:44.231 little piece on the apple and adding all those vectors up. 22:44.230 --> 22:45.940 And when you do all the hard work you will find, 22:45.940 --> 22:48.800 in fact, that it's true, but it took Newton several 22:48.798 --> 22:52.288 years to prove that even though he was pretty certain that was 22:52.285 --> 22:52.795 true. 22:52.798 --> 22:55.558 That's very important because when you do the force law for 22:55.555 --> 22:57.735 apple versus the earth the distance you use for 22:57.740 --> 23:00.070 m_1m _2/r^(2) is 23:00.068 --> 23:02.968 a center-to-center distance as if the entire earth were at the 23:02.967 --> 23:03.677 center. 23:03.680 --> 23:06.490 A similar law is true for electric forces. 23:06.490 --> 23:08.810 In other words, if you've got a spherical 23:08.806 --> 23:12.506 distribution of charge the field due to that outside that ball is 23:12.511 --> 23:15.061 as if all the charge were at the center. 23:15.058 --> 23:18.498 That's what we want to prove, but we want to prove that 23:18.497 --> 23:22.187 without doing any difficult integrals, and the trick is the 23:22.188 --> 23:23.078 following. 23:23.078 --> 23:26.968 You are going to use Gauss's Law, and you can use Gauss's Law 23:26.972 --> 23:28.662 on any surface you like. 23:28.660 --> 23:32.030 For example, you can take that surface and 23:32.029 --> 23:36.719 you can say surface integral of E on the surface is 23:36.715 --> 23:40.695 Q/ε_0, but that's not very helpful 23:40.695 --> 23:44.065 because what you're trying to find out is not the electric 23:44.066 --> 23:46.486 field integrated over a whole region, 23:46.490 --> 23:48.500 but the electric field at each point. 23:48.500 --> 23:52.690 You want to know what's going on right there. 23:52.690 --> 23:55.820 So even though it's a true statement that whatever E 23:55.823 --> 23:57.883 is doing its integral on any surface is 23:57.875 --> 24:00.675 Q/ε_0 that doesn't tell you what is 24:00.684 --> 24:04.454 E at every point, and we're going to find that by 24:04.451 --> 24:05.871 the following trick. 24:05.869 --> 24:08.979 The trick is to first decide. 24:08.980 --> 24:12.040 This surface S, by the way, it's called a 24:12.041 --> 24:13.281 Gaussian surface. 24:13.279 --> 24:16.419 It's not a real surface. 24:16.420 --> 24:18.670 In real life there are no Gaussian surfaces. 24:18.670 --> 24:21.450 You draw them for a calculation, and you can move it 24:21.453 --> 24:24.673 around in your mind anyway you like, for each surface you'll 24:24.674 --> 24:26.154 get a different answer. 24:26.150 --> 24:34.170 We'll pick a Gaussian surface equal to a sphere of radius 24:34.173 --> 24:35.753 r. 24:35.750 --> 24:49.420 So I'm going to pick a Gaussian surface of radius r. 24:49.420 --> 24:52.400 I'm going to argue two things and you have to follow the 24:52.395 --> 24:55.205 reasoning because all the work is in the argument. 24:55.210 --> 24:58.530 You don't do any integrals, but you do some serious 24:58.526 --> 24:59.186 arguing. 24:59.190 --> 25:03.140 The arguing says it's a ball, positive charge. 25:03.140 --> 25:04.630 It's going to repel a test charge. 25:04.630 --> 25:06.950 Which way is the repulsion going to be at this point? 25:06.950 --> 25:09.700 It has to be radially away from the center. 25:09.700 --> 25:11.330 That's your argument. 25:11.328 --> 25:15.578 At each point the repulsion has to be away from the center. 25:15.578 --> 25:18.138 In other words the electric field has to have this 25:18.144 --> 25:18.934 distribution. 25:18.930 --> 25:21.690 We're not saying how it varies with distance, 25:21.694 --> 25:23.334 but it has to be radial. 25:23.328 --> 25:27.058 Again, the argument is the radially outgoing thing like a 25:27.057 --> 25:27.787 hedgehog. 25:27.788 --> 25:30.998 It's the only distribution of lines which will look the same 25:31.001 --> 25:34.161 if you rotate them because then one line will go to another 25:34.160 --> 25:34.650 line. 25:34.650 --> 25:37.570 The hedgehog will look exactly the same. 25:37.568 --> 25:40.758 And that is a necessary requirement because if you took 25:40.757 --> 25:44.057 this ball of charge you all understand that if you rotate 25:44.063 --> 25:47.663 the ball the field lines have to rotate due to any calculation 25:47.663 --> 25:50.273 you did, but if you rotate the ball it 25:50.269 --> 25:51.219 looks the same. 25:51.220 --> 25:52.660 So the field lines have to look the same. 25:52.660 --> 25:55.970 So the distribution you draw must be invariant under that 25:55.969 --> 25:56.619 rotation. 25:56.618 --> 25:59.588 This is a very powerful principle of symmetry and 25:59.586 --> 26:00.386 invariance. 26:00.390 --> 26:03.960 The principle says if something is the cause of something, 26:03.960 --> 26:07.390 and if I do something to the cause that leaves it alone, 26:07.390 --> 26:09.760 namely it looks the same, the effect has to look the 26:09.755 --> 26:10.075 same. 26:10.079 --> 26:12.009 It's a very reasonable argument. 26:12.009 --> 26:15.219 And the cause here is a ball of charge which is completely 26:15.217 --> 26:16.677 isotropic and spherical. 26:16.680 --> 26:18.160 You turn it, it looks the same, 26:18.163 --> 26:20.293 so the field distribution looks the same. 26:20.288 --> 26:23.618 That's all you need to proceed because that distribution then 26:23.624 --> 26:26.574 is going to have radial lines of unknown strength, 26:26.568 --> 26:29.058 but the strength can vary only with r, 26:29.059 --> 26:30.369 but not at a given r. 26:30.368 --> 26:32.608 At a given r the direction varies, 26:32.612 --> 26:35.142 but the strength is some E of r, 26:35.135 --> 26:37.935 and we're asking what is E of r? 26:37.940 --> 26:41.710 So we know that E looks like r, 26:41.711 --> 26:46.691 which is the vector of length in the radial direction times 26:46.685 --> 26:48.825 E of r. 26:48.828 --> 26:52.438 And sometimes I also write this as e_r times 26:52.438 --> 26:55.568 E of r, and we just want to know what E 26:55.565 --> 26:57.365 is a function of r. 26:57.369 --> 27:00.979 But now we can use Gauss's Law. 27:00.980 --> 27:04.780 So the left hand side is a surface integral of this radial 27:04.778 --> 27:07.578 E on a sphere of radius r. 27:07.578 --> 27:12.228 That's going to be E times 4Πr^(2), 27:12.230 --> 27:16.610 because E is a constant on the surface. 27:16.608 --> 27:18.878 The integral of a constant on the surface is the constant 27:18.881 --> 27:20.181 times the area of the surface. 27:20.180 --> 27:23.790 It's like saying if you want to integrate a function which is 27:23.794 --> 27:27.474 constant from here to here it's the constant height multiplied 27:27.468 --> 27:30.478 by the interval over which you're integrating. 27:30.480 --> 27:31.750 There's no integral to do. 27:31.750 --> 27:33.120 It's very trivial. 27:33.119 --> 27:34.389 It's a rectangle. 27:34.390 --> 27:37.400 Similarly a constant function integrated on a sphere is the 27:37.396 --> 27:39.676 area times this, but E is the function 27:39.678 --> 27:41.128 of, of course, r. 27:41.130 --> 27:43.020 That's going to be charge enclosed. 27:43.019 --> 27:45.429 The charge enclosed is Q and you've got to put the 27:45.432 --> 27:48.232 ε_0, and then you find 27:48.229 --> 27:52.979 E(r) is Q/4Πε 27:52.980 --> 27:55.510 _0r^(2). 27:55.509 --> 28:02.199 And if you want to put all the vectors back you can write here 28:02.199 --> 28:05.379 times e_r. 28:05.380 --> 28:10.840 So by this trick one can show that the field of a sphere is as 28:10.837 --> 28:14.147 if the charges were at the center, 28:14.150 --> 28:21.650 provided you are talking about a point outside the sphere. 28:21.650 --> 28:25.020 As long as this field is outside the real ball no matter 28:25.020 --> 28:27.960 what its radius is, the charge enclosed is always 28:27.962 --> 28:28.762 Q. 28:28.759 --> 28:30.869 The right hand side doesn't vary with Q. 28:30.868 --> 28:33.828 The left hand side looks like r squared times electric field so 28:33.829 --> 28:35.549 E goes like 1/r^(2). 28:35.548 --> 28:39.178 But now I'm going to do some variations on this. 28:39.180 --> 28:46.710 The first variation is what if I want the electric field at 28:46.712 --> 28:50.222 some point here, in here? 28:50.220 --> 28:51.210 You're allowed to ask that. 28:51.210 --> 28:53.950 You can take an instrument and put it there and ask what field 28:53.953 --> 28:54.453 you have. 28:54.450 --> 28:57.570 You have to calculate that too, so let's calculate it for a 28:57.566 --> 28:58.046 sphere. 28:58.048 --> 29:02.538 The formula is the same, same formula. 29:02.538 --> 29:06.988 Everything is the same, but Q is now the charge 29:06.994 --> 29:07.924 enclosed. 29:07.920 --> 29:11.960 So I'm going to write, again, 4Πε 29:11.961 --> 29:16.071 _0, I'm sorry, 4Πr^(2) 29:16.073 --> 29:20.363 times E(r) is the charge enclosed over 29:20.358 --> 29:22.458 ε_0. 29:22.460 --> 29:26.320 What is the charge enclosed in this sphere, now, 29:26.321 --> 29:31.341 which is a mathematical sphere with radius as little r, 29:31.335 --> 29:34.535 but it's smaller than big R? 29:34.539 --> 29:41.819 Can you make a guess? 29:41.818 --> 29:45.938 How much charge do you think that is in a smaller sphere? 29:45.940 --> 29:46.950 Yeah? 29:46.950 --> 29:49.370 Student: The charge density times volume. 29:49.369 --> 29:50.399 Prof: Right. 29:50.400 --> 29:54.360 So he said it's the charge density times the volume. 29:54.358 --> 29:57.288 I'm going to write down the answer without going through the 29:57.294 --> 29:59.734 intermediate step, but it'll coincide with what he 29:59.731 --> 30:00.181 said. 30:00.180 --> 30:03.860 The answer I'm going to write down is the total charge is 30:03.855 --> 30:05.755 Q on the whole ball. 30:05.759 --> 30:08.039 The charge is proportional to the volume. 30:08.038 --> 30:11.168 Then the volume of this little guy is r^(3) and the 30:11.169 --> 30:13.419 volume of the big guy big R^(3). 30:13.420 --> 30:15.140 That's how much charge will be enclosed. 30:15.140 --> 30:17.470 Can you see that? 30:17.470 --> 30:19.950 If you take a square, and you take a part of the 30:19.950 --> 30:22.220 square that's inside that's half as big, 30:23.710 --> 30:26.220 because areas go like 1 over the distance squared. 30:26.220 --> 30:28.320 Volumes go like 1 over distance cubed. 30:28.318 --> 30:30.838 But we can also do what he just said. 30:30.838 --> 30:34.828 You can take the Q divide it by the volume 30:34.829 --> 30:36.659 4/3Πr^(3). 30:36.660 --> 30:39.530 That's the density. 30:39.529 --> 30:41.989 Then you can multiply the density by the volume of the 30:41.989 --> 30:44.169 little sphere 4/3Π little r cubed, 30:44.170 --> 30:47.070 and all I'm saying is forget these 4 thirds pis. 30:47.068 --> 30:53.028 It's just little r^(3) over big R^(3). 30:53.029 --> 30:56.949 Therefore, the electric field now, E(r), 30:56.950 --> 31:03.330 looks like Q over 4Πε 31:03.330 --> 31:10.180 _0r divided by r^(3). 31:10.180 --> 31:14.680 This is for r < R, 31:14.680 --> 31:18.670 is equal to Q/4Π ε_0 31:18.665 --> 31:22.555 (1/r^(2)) for r > 31:22.561 --> 31:23.741 R. 31:23.740 --> 31:27.990 Student: Isn't it supposed to be r cubed ________? 31:27.990 --> 31:29.430 Prof: Sorry, here? 31:29.430 --> 31:32.830 Student: Yeah, down on the second one. 31:32.829 --> 31:33.409 Prof: Here you mean? 31:33.410 --> 31:34.790 This line? 31:34.788 --> 31:38.258 Student: No, I mean like on the line above. 31:38.259 --> 31:40.329 You have like r over r cubed. 31:40.328 --> 31:42.308 Prof: I have little r cubed over big R 31:42.307 --> 31:42.577 cubed. 31:42.578 --> 31:43.558 Student: Right, but no. 31:43.558 --> 31:44.308 I mean the next line. 31:44.309 --> 31:44.879 Prof: Here? 31:44.880 --> 31:46.140 Student: Yeah. 31:46.140 --> 31:48.140 Prof: This is r less than R, 31:48.136 --> 31:49.466 r bigger than R. 31:49.470 --> 31:50.960 Student: > 31:50.960 --> 31:51.420 Oh, gotcha. 31:51.420 --> 31:53.240 Prof: Oh, you mean this 4Πr 31:53.244 --> 31:54.354 squared came downstairs? 31:54.349 --> 31:55.739 Yeah, that's right. 31:55.740 --> 31:57.690 Student: Gotcha. 31:57.690 --> 32:00.470 Prof: Okay, eternally vigilant, 32:00.474 --> 32:01.534 that's good. 32:01.529 --> 32:02.589 Yes? 32:02.588 --> 32:05.418 Student: What if little r equals big R? 32:05.420 --> 32:06.310 Prof: Let's ask. 32:06.308 --> 32:09.598 What if little r grows up and become big R, 32:09.598 --> 32:10.068 right? 32:10.069 --> 32:11.289 We'll find out. 32:11.289 --> 32:13.769 What do you expect? 32:13.769 --> 32:16.499 You've got to get the same answer at the field at a certain 32:16.501 --> 32:18.201 point no matter how we approached it, 32:18.198 --> 32:18.668 right? 32:18.670 --> 32:20.890 You'll find that's true in this formula, 32:20.890 --> 32:25.370 because if you put little r equals big R and 32:25.373 --> 32:29.383 big R over big R^(3) you'll get 1 over 32:29.384 --> 32:33.634 big R^(2) and this will also give you 1 over big 32:33.632 --> 32:38.042 R^(2). These formulas will match on the surface of 32:38.036 --> 32:39.606 this sphere. 32:39.609 --> 32:42.039 Yes? 32:42.038 --> 32:46.068 Student: So can you say again why we can assume that we 32:46.071 --> 32:49.781 don't have to do the integral ________________________? 32:49.779 --> 32:51.869 Prof: Why did I not do the integral? 32:51.868 --> 32:56.958 Student: Why we ________ 4 pi squared times E of 32:56.959 --> 32:57.329 r. 32:57.328 --> 32:58.418 Prof: Why do I do that you mean? 32:58.420 --> 32:59.000 Student: Yeah. 32:59.000 --> 33:00.850 Prof: Right. 33:00.846 --> 33:05.216 So the correct way to do it is to really do 33:05.220 --> 33:07.160 E⋅dA, right? 33:07.160 --> 33:11.350 If I did that it'll be fine with you, correct? 33:11.348 --> 33:17.488 This is what you're saying I should do, right? 33:17.490 --> 33:19.660 You asked me why I didn't do this integral, 33:19.663 --> 33:21.423 why I just wrote 4Πr^(2), 33:21.423 --> 33:22.773 is that your question? 33:22.769 --> 33:24.429 Right. 33:24.430 --> 33:29.100 The assumed form for E is the function that depends on 33:29.096 --> 33:33.216 r times unit vector in the radial direction. 33:33.220 --> 33:38.150 And since I surrounded it with this sphere the tiny area vector 33:38.154 --> 33:42.534 is also e_r times the value of the tiny 33:42.532 --> 33:43.252 area. 33:43.250 --> 33:48.060 In other words, dA and E are both 33:48.061 --> 33:52.771 parallel if the surface is a sphere, right? 33:52.769 --> 33:55.739 So when you take the dot product this will just become 33:55.740 --> 33:58.600 E(r) times dA times cosine of 0 33:58.601 --> 34:00.341 which you don't care about. 34:00.339 --> 34:01.879 That's the first thing. 34:01.880 --> 34:04.330 You still have to integrate over this sphere, 34:04.328 --> 34:07.668 but E of r is a constant on this sphere because 34:07.667 --> 34:09.447 it depends only on r. 34:09.449 --> 34:12.019 So throughout the whole sphere if it's a constant, 34:12.018 --> 34:16.438 it's like the number 19, you just pull it and then you 34:16.440 --> 34:21.440 get E of r times integral dA and that is 34:21.443 --> 34:23.533 my 4Πr^(2). 34:23.530 --> 34:26.870 So in general you have to do an integral, but if you're lucky 34:26.865 --> 34:30.365 and the integrand is a constant then the integral is trivial. 34:30.369 --> 34:34.239 So for a charge at the center of the spherical ball, 34:34.244 --> 34:38.044 and if you went to a sphere then the integrant is a 34:38.041 --> 34:39.031 constant. 34:39.030 --> 34:40.010 Okay, so this is what you get. 34:40.010 --> 34:43.000 If you draw a picture of this it looks like this. 34:43.000 --> 34:49.470 Electric field as the function of r it will grow with 34:49.467 --> 34:54.947 r then it will decline as 1/r^(2). 34:54.949 --> 34:59.029 And here is little r equals big R. 34:59.030 --> 34:59.160 Yep? 34:59.155 --> 35:00.985 Student: What if it were hollow and all the charges 35:00.994 --> 35:02.014 were on the outside ________? 35:02.010 --> 35:04.230 Prof: We'll come to that. 35:04.230 --> 35:05.610 Yes? 35:05.610 --> 35:08.660 Student: Why can we ignore the charge that's outside 35:08.657 --> 35:09.637 the small sphere? 35:09.639 --> 35:11.099 Prof: Why don't we count it? 35:11.099 --> 35:12.219 Student: Uh-huh, did you explain that already? 35:12.219 --> 35:14.109 Prof: Yeah, well it's a very interesting 35:14.105 --> 35:14.715 point, right? 35:14.719 --> 35:17.739 Suppose you make a hole in the earth, of if you go deep into 35:17.735 --> 35:18.345 the earth. 35:18.349 --> 35:21.779 We said if you're outside the earth the entire ball is pulling 35:21.777 --> 35:23.707 you in, but the claim I'm making is if 35:23.710 --> 35:25.730 you're inside the earth, if you're here, 35:25.726 --> 35:28.586 only the ball underneath your feet is pulling you, 35:28.590 --> 35:30.110 but not these guys, right? 35:30.110 --> 35:33.080 That's the question, why don't they exert a force, 35:33.083 --> 35:33.573 right? 35:33.570 --> 35:40.900 Well, Gauss's Law says, no, but we can try to verify 35:40.902 --> 35:43.492 that explicitly. 35:43.489 --> 35:46.549 So we are going to do the hollow sphere, 35:46.545 --> 35:49.675 but you followed the solid sphere now? 35:49.679 --> 35:52.349 So why is the field getting stronger as you leave the 35:52.349 --> 35:55.379 center, why is it falling like 1/r^(2) once you leave 35:55.378 --> 35:56.148 the sphere? 35:56.150 --> 35:59.850 You've got to think about that. 35:59.849 --> 36:05.239 Why is it growing when you leave the origin? 36:05.239 --> 36:06.369 Yep? 36:06.369 --> 36:08.549 Student: Because there is more charge _________. 36:08.550 --> 36:09.480 Prof: Right. 36:09.483 --> 36:11.793 The charge you are enclosing grows like r cubed, 36:11.793 --> 36:14.303 and the distance from the center of that charge is r 36:14.300 --> 36:16.070 squared which comes downstairs. 36:16.070 --> 36:19.240 So as you go further out inside the ball you are gaining, 36:19.239 --> 36:22.059 but once you reach the surface of the ball and you go further 36:22.061 --> 36:24.321 out you're getting more area for Gauss's Law, 36:24.320 --> 36:27.610 but not more charge which is stuck at capital Q. 36:27.610 --> 36:31.490 That's why once you leave the sphere it starts declining, 36:31.490 --> 36:34.750 but if you're inside the sphere it increases. 36:34.750 --> 36:36.510 This is also true for the gravitational force. 36:36.510 --> 36:40.140 The gravitational force if you go to the center of the earth 36:40.143 --> 36:43.593 and you move away by 1 inch you'll be pulled back with an 36:43.594 --> 36:46.984 extremely weak force that's proportional to the distance 36:46.983 --> 36:49.143 from the center of the earth. 36:49.139 --> 36:54.339 Okay, then we come to another question which came up, 36:54.340 --> 36:58.140 which is let's take a hollow sphere. 36:58.139 --> 37:10.989 I take a sphere and I scoop a hole in it and I want the field 37:10.989 --> 37:12.489 here. 37:12.489 --> 37:16.439 Okay, so if you want the field here you take a Gauss's Law for 37:16.436 --> 37:19.616 a circle like that, or a sphere like that and 37:19.619 --> 37:23.559 you'll get E times 4Πr^(2) is equal to 1 37:23.561 --> 37:26.661 over ε_0 at times what? 37:26.659 --> 37:28.129 Student: Nothing. 37:28.130 --> 37:29.490 Prof: Nothing. 37:29.489 --> 37:34.239 So the electric field inside a hollow sphere is 0. 37:34.239 --> 37:36.619 In other words, if you took the earth and you 37:36.619 --> 37:39.269 scooped a big hole in the middle you can float. 37:39.269 --> 37:42.789 It won't pull you at all. 37:42.789 --> 37:45.349 So inside a hollow sphere there is no electric field. 37:45.349 --> 37:47.939 Even though there's charge everywhere there's no electric 37:47.940 --> 37:48.310 field. 37:48.309 --> 37:51.759 We can understand that in some cases, and other cases require a 37:51.764 --> 37:52.994 little more thought. 37:52.989 --> 37:57.169 Let me first show you that for a very, very thin shell why that 37:57.166 --> 37:57.836 happens. 37:57.840 --> 38:00.390 Once it's true for a very, very thin shell and it doesn't 38:00.387 --> 38:03.067 do anything you can put one shell inside another and make it 38:03.072 --> 38:05.852 as thick as you like because 0 is going to be the contribution 38:05.849 --> 38:06.759 from each guy. 38:06.760 --> 38:08.890 So let me take a very thin shell. 38:08.889 --> 38:11.539 Here's a very thin shell. 38:11.539 --> 38:14.029 So remember, this is not a ball. 38:14.030 --> 38:18.230 This is a shell whose thickness is the thickness of the chalk 38:18.230 --> 38:18.720 here. 38:18.719 --> 38:22.309 I think you will agree that at the center of this sphere the 38:22.311 --> 38:23.531 force has to be 0. 38:23.530 --> 38:25.950 You agree with that? 38:25.949 --> 38:27.759 Because which way should it go? 38:27.760 --> 38:29.970 For every guy pushing, say from here in that 38:29.974 --> 38:32.864 direction, there's another one from there pushing in this 38:32.858 --> 38:33.578 direction. 38:33.579 --> 38:35.749 They all cancel. 38:35.750 --> 38:38.100 It's very clear that at the center you don't feel a force. 38:38.099 --> 38:43.869 But what is truly amazing is that even when you're off center 38:43.873 --> 38:46.283 you don't feel a force. 38:46.280 --> 38:50.250 So I'm going to give you the argument for that one, 38:50.253 --> 38:54.703 argument for that namely I'm once again going to pair off 38:54.704 --> 38:56.934 canceling charges, okay? 38:56.929 --> 39:00.479 I'm going to cancel one thing against another thing. 39:00.480 --> 39:04.060 When you were at the center you canceled a little charge here 39:04.061 --> 39:06.151 with a little charge here, right? 39:06.150 --> 39:08.580 When you are here here's what you do. 39:08.579 --> 39:16.469 Take a cone like that. 39:16.469 --> 39:17.519 It is not two lengths. 39:17.518 --> 39:20.918 It is a three dimensional cone, but I'm able to show you only a 39:20.923 --> 39:23.453 slice right down the middle, you understand? 39:23.449 --> 39:26.389 This is really a cone that'll cut the sphere there; 39:26.389 --> 39:28.889 it'll cut the sphere there. 39:28.889 --> 39:30.899 Take all these guys. 39:30.900 --> 39:33.360 They have a certain charge, which is some charge density 39:33.364 --> 39:37.514 which is uniform, times the area of this thing 39:37.507 --> 39:44.467 divided by the square of the distance r_1. 39:44.469 --> 39:47.729 Here, these charges here will push you to the right with a 39:47.733 --> 39:49.913 force which is the area of density, 39:49.909 --> 39:54.639 the charge density per unit area times the area of the 39:54.635 --> 39:59.715 smaller circle divided by r_2 squared. 39:59.719 --> 40:15.019 But now I claim that I can equate these two. 40:15.018 --> 40:19.018 I can equate these two if this area of vector and this area of 40:19.018 --> 40:22.818 vector were radiating outwards, and this is actually a true 40:22.822 --> 40:23.742 statement. 40:23.739 --> 40:25.519 This area is proportional to r_1 squared. 40:25.518 --> 40:32.758 This area is proportional to r_1 squared, 40:32.764 --> 40:36.454 therefore they will cancel. 40:36.449 --> 40:38.449 But this is a very subtle argument. 40:38.449 --> 40:41.169 Sometimes even textbooks get it wrong. 40:41.170 --> 40:46.410 And generally if you take two cones like this you can see that 40:46.409 --> 40:51.649 the area vector points that way, but the electric field vector 40:51.650 --> 40:53.370 points this way. 40:53.369 --> 40:54.439 They are not the same. 40:54.440 --> 40:56.740 The area vector, area on the sphere, 40:56.742 --> 40:59.772 right, will point in one direction, but the 40:59.769 --> 41:02.349 r vector is different. 41:02.349 --> 41:06.789 So you have to take the cosine θ on both sides, 41:06.789 --> 41:10.079 but it turns out cosine θ is also the same 41:10.083 --> 41:13.743 because if you took a line and you cut it this way that angle 41:13.744 --> 41:15.334 will equal that angle. 41:15.329 --> 41:18.479 So that's a subtly we don't need, but that's how you really 41:18.476 --> 41:19.776 prove for every shell. 41:19.780 --> 41:20.910 There's a cancellation. 41:20.909 --> 41:23.189 If you know what a solid angle is it's very easy to say. 41:23.190 --> 41:26.050 This is called, with the cosine thetas in it, 41:26.050 --> 41:29.510 it's called a solid angle enclosed by this point, 41:29.510 --> 41:32.480 and the two things have the same solid angle and they 41:32.476 --> 41:32.986 cancel. 41:32.989 --> 41:37.699 But if you don't want the most complicated proof you can take 41:37.704 --> 41:42.264 points like this where it's clear that this area vector and 41:42.262 --> 41:45.802 this area vector are both perpendicular, 41:45.800 --> 41:49.380 and to the radial direction, and therefore the electric 41:49.378 --> 41:51.828 field and this vector are parallel. 41:51.829 --> 41:53.809 Then you don't need the cosine thetas. 41:53.809 --> 41:56.439 They are clearly equal at possibly even 1. 41:56.440 --> 42:00.570 Then A/r^(2) being the same is the reason it works. 42:00.570 --> 42:03.660 In other words, if the force of gravity or the 42:03.659 --> 42:07.639 electric force did not fall exactly like 1/r^(2) the 42:07.641 --> 42:10.871 field inside a hollow sphere will not be 0. 42:10.869 --> 42:13.989 So this is how people tested the 1/r^(2) force law in 42:13.987 --> 42:14.777 the old days. 42:14.780 --> 42:17.360 They went inside a hollow sphere and tried to see if there 42:17.356 --> 42:20.026 is any field inside by putting test charges and finding they 42:20.025 --> 42:20.835 don't respond. 42:20.840 --> 42:23.880 That's the most reliable way to prove 0 field. 42:23.880 --> 42:24.740 Yes? 42:24.739 --> 42:27.939 Student: Can you explain again why the A over r 42:27.936 --> 42:29.176 squareds are equal? 42:29.179 --> 42:32.489 Prof: Well, you can see that if you took a 42:32.487 --> 42:37.877 cone and if you just did similar triangles you can see that this 42:37.878 --> 42:41.558 area is growing like that radius square. 42:41.559 --> 42:44.949 It's a matter of how areas grow when you scale anything. 42:44.949 --> 42:48.819 Take a cone, okay, and you slice it at some 42:48.818 --> 42:51.948 point, and slice it even higher. 42:51.949 --> 42:54.209 Here's a cone. 42:54.210 --> 42:56.920 You slice it there and you slice it there, 42:56.918 --> 43:01.008 and my claim is this area is to that area that distance squared 43:01.014 --> 43:03.134 is to that distance squared. 43:03.130 --> 43:03.900 Why is that, right? 43:03.900 --> 43:04.960 That's your question. 43:04.960 --> 43:10.380 The area of this cone is something, something radius 43:10.380 --> 43:13.890 times something, something height, 43:13.887 --> 43:14.947 right? 43:14.949 --> 43:18.269 So you can calculate, if you like. 43:18.268 --> 43:22.858 The radius of this circle and the radius of that circle are 43:22.864 --> 43:25.484 growing linearly as you go out. 43:25.480 --> 43:28.510 If you go distance r here, and you get a radius here, 43:28.510 --> 43:31.540 if you go distance r there you get another radius. 43:31.539 --> 43:35.089 Is it clear to you that the radii are proportional to the 43:35.085 --> 43:37.615 distance from the center in this cone? 43:37.619 --> 43:38.569 Let me draw it again. 43:38.570 --> 43:39.310 This is a bad picture. 43:39.309 --> 43:46.729 Here's my cone. 43:46.730 --> 43:47.950 Take two circles. 43:47.949 --> 43:48.979 Well again, I screwed up. 43:48.980 --> 43:50.320 This is supposed to be circles. 43:50.320 --> 43:51.910 This is the center. 43:51.909 --> 43:57.029 That radius is to that height what this radius is to this 43:57.027 --> 43:57.847 height. 43:57.849 --> 44:00.089 They're just similar triangles. 44:00.090 --> 44:03.690 Therefore, this height is what I'm calling r_1 44:03.692 --> 44:07.122 and this height is what I'm calling r_2. 44:07.119 --> 44:09.469 Therefore, if this radius is proportional to r_2 44:09.474 --> 44:11.914 the area of that circle is Πr_2^(2). 44:11.909 --> 44:14.439 This is Πr_1^(2). 44:14.440 --> 44:19.400 So that'll go like that distance squared is to that 44:19.402 --> 44:21.292 distance squared. 44:21.289 --> 44:24.979 Okay, whenever you scale things for an area by factor of 2 it'll 44:24.980 --> 44:27.090 go like the square of that factor. 44:27.090 --> 44:30.140 You can already see that if you draw these cones this'll 44:30.139 --> 44:33.189 intersect the circle, the sphere on the small circle. 44:33.190 --> 44:34.840 This will intercept it on a bigger circle. 44:34.840 --> 44:37.710 That much is clear, but the extra is not linearly 44:37.711 --> 44:38.611 proportional. 44:38.610 --> 44:41.630 it is quadratic because the circle has two dimensions both 44:41.626 --> 44:44.796 of which are growing linearly with the distance from the apex 44:44.802 --> 44:45.652 of the cone. 44:45.650 --> 44:51.940 That's why it's r squared. 44:51.940 --> 45:06.760 All right, another thing one can calculate is 45:06.760 --> 45:15.380 the electric field due to an infinitely long wire. 45:15.380 --> 45:23.000 Here's an infinitely long wire with lambda coulombs per meter. 45:23.000 --> 45:25.950 So you want to find the electric field. 45:25.949 --> 45:31.069 By the way, I should tell you that Gauss's Law--let me do this 45:31.074 --> 45:32.254 one example. 45:32.250 --> 45:34.360 I'll tell you what the restriction of Gauss's Law is. 45:34.360 --> 45:38.780 So we come to this problem and we can again argue by symmetry 45:38.782 --> 45:42.912 the field lines have to be radially away from the axis of 45:42.909 --> 45:46.309 their line, and if you look at it from the 45:46.311 --> 45:51.931 edge, from one edge it'll look like 45:51.934 --> 45:52.994 this. 45:52.989 --> 45:55.239 If you look at the wire from the edge the lines should be 45:55.244 --> 45:56.134 going out like that. 45:56.130 --> 45:57.980 It's like a hedgehog but it's cylindrical. 45:57.980 --> 45:59.620 It's not in all three dimensions. 45:59.619 --> 46:03.369 It's radial in this direction. 46:03.369 --> 46:06.709 And the question is and the field has to be constant along 46:06.713 --> 46:10.003 the length of the infinite wire at each point at the same 46:09.998 --> 46:13.638 distance because if you move the wire horizontally it looks the 46:13.635 --> 46:16.505 same so the field distribution cannot vary. 46:16.510 --> 46:19.750 If it is a finite wire you cannot make the argument. 46:19.750 --> 46:23.610 In a finite wire as you come near the edges lines will start 46:23.614 --> 46:24.274 tilting. 46:24.268 --> 46:27.318 But a finite wire doesn't look the same when you move it; 46:27.320 --> 46:29.100 an infinite wire does. 46:29.099 --> 46:32.489 Therefore, for an infinite wire if you don't stop here the lines 46:32.485 --> 46:35.495 will always look the same, so that if you shift them over 46:35.496 --> 46:37.106 they should look the same. 46:37.110 --> 46:41.860 So the only unknown question is - I know the field is in this 46:41.858 --> 46:46.608 direction radially away from the wire in all directions whose 46:46.608 --> 46:50.328 magnitude is fixed at a distance r, 46:50.329 --> 46:53.589 but I want to know what the magnitude is. 46:53.590 --> 46:57.600 Again, I'm going to take Gauss's Law and I'm going to 46:57.597 --> 47:00.137 apply to the following surface. 47:00.139 --> 47:08.559 Surface is a cylinder and the cylinder has some length 47:08.561 --> 47:11.731 l, and it's got some radius 47:11.726 --> 47:14.816 r and the flat faces of the cylinder are parallel, 47:14.820 --> 47:29.930 so if you really want to give the nice picture here they look 47:29.927 --> 47:32.947 like this. 47:32.949 --> 47:35.619 So Gauss's Law can be applied only to a closed surface. 47:35.619 --> 47:36.769 You understand that? 47:36.768 --> 47:40.248 You cannot do it for an open surface because only if you trap 47:40.248 --> 47:43.668 the charge completely in all directions will you count every 47:43.668 --> 47:44.188 line. 47:44.190 --> 47:46.900 If you've got holes in your surface then stuff can escape 47:46.902 --> 47:48.552 and you cannot promise anything. 47:48.550 --> 47:52.660 So I need a closed surface and my closed surface is the 47:52.661 --> 47:53.501 cylinder. 47:53.500 --> 47:56.120 So I'm going to write, once again, the surface 47:56.115 --> 47:59.595 integral of the electric field on that cylinder is the charge 47:59.603 --> 48:00.363 enclosed. 48:00.360 --> 48:03.150 The charge enclosed is the easiest part. 48:03.150 --> 48:06.260 I'm going to give you 10 seconds to think in your head. 48:06.260 --> 48:11.480 What is a charge enclosed by the cylinder? 48:11.480 --> 48:12.790 Okay? 48:12.789 --> 48:16.059 It's charge per unit length times the length of wire trapped 48:16.063 --> 48:20.553 inside the cylinder and then, of course, I have this 48:20.548 --> 48:24.488 1/ε_0. 48:24.489 --> 48:27.529 It still is a statement about the integral of E on the 48:27.530 --> 48:30.520 surface, and it could be any surface, but the beauty of this 48:30.519 --> 48:31.989 surface is the following. 48:31.989 --> 48:34.809 There are contributions to E⋅dA from 48:34.806 --> 48:38.276 the curvy side of the cylinder and from the flat side. 48:38.280 --> 48:42.370 In the flat side the surface vector, area vector dA 48:42.373 --> 48:46.183 for any small portion, or in fact for the entire face, 48:46.179 --> 48:47.399 is like that. 48:47.400 --> 48:52.780 The electric field runs along that face so the dot product is 48:52.780 --> 48:53.140 0. 48:53.139 --> 48:58.329 So I got 0 from the left side, 0 from the right side, 48:58.333 --> 49:02.933 then I got non-zero from the curvy cylinder. 49:02.929 --> 49:07.569 On the curved cylinder you can, I hope, see that every area 49:07.570 --> 49:11.490 vector is actually parallel to the field lines. 49:11.489 --> 49:16.629 So just like on a sphere this integral will be E(r) times the 49:16.628 --> 49:21.678 surface of the curvy part of the cylinder which is 2Πr 49:21.679 --> 49:23.649 times L. 49:23.650 --> 49:27.490 That's the area of the curved part of the cylinder. 49:27.489 --> 49:30.299 So you see L cancels out, and it better cancel out 49:30.300 --> 49:32.760 because L is an artificial construction. 49:32.760 --> 49:34.770 We made up L. 49:34.768 --> 49:37.378 The answer should not depend on the Gaussian surface you picked. 49:37.380 --> 49:40.050 I told you it's a figment of our imagination, 49:40.050 --> 49:44.750 so it doesn't depend on it, and we find E(r) 49:44.753 --> 49:48.883 = λ/2Πε _0r, 49:48.880 --> 49:52.890 which is the result we got earlier on by doing all the 49:52.885 --> 49:54.845 brute force integration. 49:54.849 --> 49:57.549 Remember we took a point here. 49:57.550 --> 49:58.530 We took some segment. 49:58.530 --> 50:01.640 We drew the arrows, did the integrals, 50:01.637 --> 50:05.917 sine thetas and whatnot, but you can get that in one 50:05.922 --> 50:06.682 shot. 50:06.679 --> 50:10.309 So it looks like Gauss's Law is the easy way to do stuff, 50:10.313 --> 50:13.623 and you may wonder why we bother to do any difficult 50:13.621 --> 50:14.531 integrals. 50:14.530 --> 50:19.150 The reason is that if I change the symmetry of the problem in 50:19.152 --> 50:22.932 the slightest way I cannot calculate the field. 50:22.929 --> 50:26.479 For example, if on the spherical charge 50:26.476 --> 50:31.886 distribution instead of a sphere I made a little blip here, 50:31.891 --> 50:36.841 maybe did a little surgery and put that guy here. 50:36.840 --> 50:39.260 We're dead. 50:39.260 --> 50:41.720 I mean, there's a formula for the field, but no one can 50:41.724 --> 50:43.234 calculate it in any simple way. 50:43.230 --> 50:46.230 You can, again, take a Gaussian surface and 50:46.233 --> 50:50.383 it'll still be true that the integral of the electric field 50:50.382 --> 50:54.172 on that surface will be the same Q/ε 50:54.172 --> 50:58.112 _0, but the problem is the E 50:58.108 --> 51:00.828 on the surface is no longer constant. 51:00.829 --> 51:01.609 You understand? 51:01.610 --> 51:04.260 Not every point is symmetric any more. 51:04.260 --> 51:06.420 E may be stronger where there's a bulge. 51:06.420 --> 51:08.670 E may be weaker where there is hole, 51:08.666 --> 51:11.606 so and also its direction is changing in a crazy way. 51:11.610 --> 51:13.970 So you can make one true statement about the integral of 51:13.971 --> 51:15.861 that crazy function over the whole region. 51:15.860 --> 51:20.380 That cannot be used to deduce the value of E at every 51:20.376 --> 51:20.986 point. 51:20.989 --> 51:22.609 So you still need the integral. 51:22.610 --> 51:25.060 You may have to do the integral maybe on a computer, 51:25.059 --> 51:26.779 but that's the answer to all problems, 51:26.780 --> 51:32.330 but for simple problems with a great deal of symmetry we can 51:32.331 --> 51:37.131 use Gauss's Law to get these things very easily, 51:37.130 --> 51:38.670 okay? 51:38.670 --> 51:42.570 That's the Gauss. 51:42.570 --> 51:46.220 Now, I'm going to introduce you to a second notion which is 51:46.221 --> 51:49.491 pretty important to study electricity, and that's the 51:49.494 --> 51:51.074 notion of conductors. 51:51.070 --> 51:54.960 So we're going to divide the world into two things, 51:54.960 --> 51:57.140 conductors and insulators. 51:57.139 --> 51:59.649 As you know, matter is made up of positive 51:59.648 --> 52:03.008 and negative charges and the negative charges circle the 52:03.012 --> 52:05.982 positive charges, and they pretty much stay near 52:05.983 --> 52:09.563 their parent atoms, near the parent nuclei, 52:09.557 --> 52:12.737 and you cannot separate them. 52:12.739 --> 52:17.439 But in a metal at least some of the electrons from each atom 52:17.443 --> 52:18.883 become communal. 52:18.880 --> 52:21.970 In other words, they can run around the whole 52:21.967 --> 52:22.527 solid. 52:22.530 --> 52:25.770 They don't belong to any one nucleus. 52:25.769 --> 52:26.719 So that's a conductor. 52:26.719 --> 52:30.849 In a conductor the negative charges, if you like, 52:30.853 --> 52:32.493 are free to move. 52:32.489 --> 52:41.149 So here's the first result. 52:41.150 --> 52:46.480 E is equal to 0 inside a conductor. 52:46.480 --> 52:51.130 That really follows from the meaning of the word conductor. 52:51.130 --> 52:54.110 If you took a chunk of perfect conductor, maybe copper is good 52:54.110 --> 52:57.140 enough; they'll be no electric field 52:57.141 --> 52:58.531 inside copper. 52:58.530 --> 52:59.670 Why? 52:59.670 --> 53:02.780 Because if this is the chunk of material, 53:02.780 --> 53:05.650 this electric field, then the charges that I said 53:05.652 --> 53:09.362 are free to move will respond to the electric field and they'll 53:09.362 --> 53:10.202 be moving. 53:10.199 --> 53:15.369 So I should say E equal to 0 inside a conductor in a 53:15.367 --> 53:17.057 static situation. 53:17.059 --> 53:20.699 In other words, once the charges have stopped 53:20.702 --> 53:25.092 moving the electric field will be 0 in a conductor. 53:25.090 --> 53:28.820 Let me explain to you a little more what's going on so it's not 53:28.824 --> 53:29.794 a big mystery. 53:29.789 --> 53:36.719 Suppose there is a uniform electric field going from left 53:36.717 --> 53:38.077 to right? 53:38.079 --> 53:42.739 In that uniform field I take a chunk of copper like a nice 53:42.740 --> 53:46.340 rectangular chunk and I stick it in there. 53:46.340 --> 53:49.180 What will happen? 53:49.179 --> 53:52.779 In the beginning the electric field will penetrate the copper 53:52.784 --> 53:55.854 and the field lines say to the positive charges, 53:55.849 --> 53:57.689 "You go to the right," and to the negative 53:57.692 --> 53:59.972 charges, "You go to the left." 53:59.969 --> 54:03.429 Negative charges will race to the left until they cannot go 54:03.425 --> 54:05.625 anymore without leaving the solid, 54:05.630 --> 54:07.480 and that they're not allowed to do, 54:07.480 --> 54:13.520 leaving behind some deficit on the other side. 54:13.519 --> 54:15.249 But look what's happening now. 54:15.250 --> 54:20.880 These guys produce their own electric field which goes from 54:20.875 --> 54:22.325 here to here. 54:22.329 --> 54:25.219 Therefore, inside the conductor, the electric field by 54:25.219 --> 54:28.219 the superposition principle is the field due to whatever 54:28.217 --> 54:30.397 outside agency produced this field, 54:30.400 --> 54:34.950 plus the field due to these guys, and they will not stop 54:34.952 --> 54:39.672 until the field they produced exactly cancels the external 54:39.672 --> 54:40.502 field. 54:40.500 --> 54:44.610 Then the migration will stop. 54:44.610 --> 54:46.290 So how do they know when to stop? 54:46.289 --> 54:47.599 They are not that smart, right? 54:47.599 --> 54:51.239 But the point is once they've produced the potential, 54:51.239 --> 54:55.089 I mean, once they produce the field that cancels the external 54:55.088 --> 54:58.548 field there'll be no field inside the bulk to encourage 54:58.552 --> 55:02.532 charges to move any more and that's when they stop moving. 55:02.530 --> 55:05.030 Okay, so a conductor, at least in this case, 55:05.030 --> 55:07.880 not so hard to understand what they have to do. 55:07.880 --> 55:09.170 Negative charges go to one side. 55:09.170 --> 55:13.170 Positive go to the other side until the field they produce is 55:13.168 --> 55:17.098 an arrow going to the left of the same magnitude as this one 55:17.099 --> 55:19.099 then E is 0 inside. 55:19.099 --> 55:25.009 But what's amazing about these metals is that if you take a 55:25.007 --> 55:30.507 potato shaped metal it's not so easy to see what charge 55:30.509 --> 55:36.109 arrangement will exactly neutralize the field everywhere 55:36.110 --> 55:37.640 in sight. 55:37.639 --> 55:38.679 Here it's easy to see. 55:38.679 --> 55:41.829 I want a right moving one canceled with a left moving one. 55:41.829 --> 55:44.029 I draw a line, a plane of charges here and 55:44.030 --> 55:46.230 here, and you can see they will cancel. 55:46.230 --> 55:49.690 But even this oddball object, I claim, 55:49.690 --> 55:52.060 will eventually acquire some density of charges, 55:52.059 --> 55:55.989 they're not uniform or anything, in some complicated 55:55.987 --> 56:00.067 fashion until the field inside the solid becomes 0. 56:00.070 --> 56:05.410 So these particles will always figure out a way to make the 56:05.407 --> 56:09.637 field 0 inside, because it's the definition. 56:09.639 --> 56:12.489 If it's not 0 they've got more work to do, 56:12.489 --> 56:15.499 and the charges have to separate even more until there 56:15.503 --> 56:19.033 is no hunger for the separation because they managed to produce 56:19.030 --> 56:20.680 a field inside that is 0. 56:20.679 --> 56:23.579 Then nobody else will join this flow and it'll stop. 56:23.579 --> 56:26.949 So that's a very short period, 10 to the minus something, 56:26.947 --> 56:30.557 when charges rearrange when you put a conductor in the field, 56:30.556 --> 56:32.236 then quickly it'll stop. 56:32.239 --> 56:35.129 If you put an alternating field going back and forth then, 56:35.130 --> 56:37.530 of course, it depends on how rapidly it's oscillating, 56:37.530 --> 56:39.880 and charges may not be able to keep up with that. 56:39.880 --> 56:42.650 That's called a plasma frequency, and beyond that a 56:42.650 --> 56:45.920 field would start penetrating because charges cannot keep in 56:45.920 --> 56:46.420 step. 56:46.420 --> 56:48.270 But a DC field, where you've got all the time 56:48.268 --> 56:50.708 in the world to settle down, they will very quickly come to 56:50.706 --> 56:51.586 this arrangement. 56:51.590 --> 56:53.310 They will stop. 56:53.309 --> 56:59.389 So remember field in a conductor is 0 by definition. 56:59.389 --> 57:14.369 Okay, now the next thing I will show you is that charge inside a 57:14.369 --> 57:19.839 conductor equal to 0. 57:19.840 --> 57:22.270 By that I mean the following. 57:22.268 --> 57:28.468 If you took a conductor and you threw some charge on it where 57:28.474 --> 57:29.824 will it go? 57:29.820 --> 57:31.340 Okay, here's a chunk of copper. 57:31.340 --> 57:33.780 Throw 10 million electrons. 57:33.780 --> 57:35.820 Well, the 10 million electrons don't like each other. 57:35.820 --> 57:37.960 They will try to run away from each other. 57:37.960 --> 57:41.950 In the end they will all sit somewhere here, 57:41.954 --> 57:47.344 but they will sit in such a way that the field inside is 0, 57:47.344 --> 57:51.624 because you cannot have an electric field. 57:51.619 --> 57:54.329 If the electric field is 0 everywhere then the charge is 57:54.333 --> 57:56.333 also 0 everywhere, because you can take any 57:56.327 --> 57:58.237 surface, any volume you like with the integral of 57:58.242 --> 58:02.162 E⋅dA, you're going to get 0. 58:02.159 --> 58:04.909 If you can get E⋅dA is 0 for 58:04.909 --> 58:08.199 every possible surface then the charge enclosed by every 58:08.195 --> 58:11.445 possible surface is 0, therefore there is no charge in 58:11.449 --> 58:11.819 sight. 58:11.820 --> 58:13.330 So where does the charge go? 58:13.329 --> 58:15.899 It goes to the boundary of the metal. 58:15.900 --> 58:19.850 Once they're at the boundary of the metal the metal starts 58:19.849 --> 58:23.729 calling you back because there is something called a work 58:23.728 --> 58:24.628 function. 58:24.630 --> 58:27.290 It's like all the electrons in a swimming pool. 58:27.289 --> 58:29.159 If they don't like each other they can go to the edge of the 58:29.155 --> 58:30.505 pool, but to leave the pool they've 58:30.512 --> 58:32.112 got to climb up over the vertical walls, 58:32.110 --> 58:34.620 and they cannot scale the walls. 58:34.619 --> 58:36.339 If you rip them hard enough they will. 58:36.340 --> 58:39.270 If you put enough charges, charges will start flying from 58:39.273 --> 58:43.023 here and maybe land there; that we call lightening. 58:43.018 --> 58:47.348 For that they'll have to break the air and produce a conducting 58:47.351 --> 58:51.611 path, but if you don't put such strong fields the charges will 58:51.612 --> 58:53.432 remain on the surface. 58:53.429 --> 58:57.489 Now, it turns out we can actually relate the electric 58:57.487 --> 59:02.087 field at the surface to the charge density of the surface by 59:02.092 --> 59:03.812 a following trick. 59:03.809 --> 59:05.689 Let's go to the surface here and ask, "What's the 59:05.686 --> 59:06.286 electric field? 59:06.289 --> 59:07.909 Which way can it point?" 59:07.909 --> 59:14.089 I claim the electric field can only point perpendicular to the 59:14.085 --> 59:16.485 surface, because if you had parallel 59:16.494 --> 59:19.184 components then the charges can move along the surface. 59:19.179 --> 59:21.749 No one says you cannot run it on the edge of the swimming 59:21.746 --> 59:23.256 pool, you just cannot leave it. 59:23.260 --> 59:26.770 So the electric field lines must point radial, 59:26.771 --> 59:29.271 I mean, normal to the surface. 59:29.268 --> 59:32.738 And I claim that we can actually calculate the electric 59:32.739 --> 59:35.889 field here if you knew the charge density here. 59:35.889 --> 59:37.569 So let's do that. 59:37.570 --> 59:43.850 So here's some surface, and I have--take a tiny region 59:43.851 --> 59:50.961 here and I'm going to take a Gaussian surface that looks like 59:50.963 --> 59:52.033 this. 59:52.030 --> 59:54.980 It's a cylinder, very tiny cylinder, 59:54.983 --> 59:58.533 and the field lines are like that there. 59:58.530 --> 1:00:02.050 There is no field lines here because there's no field inside. 1:00:02.050 --> 1:00:05.500 And the field here all of these things are 0, 1:00:05.501 --> 1:00:10.131 and the field here is parallel to the cylinder so it doesn't 1:00:10.132 --> 1:00:11.312 contribute. 1:00:11.309 --> 1:00:11.879 You see that? 1:00:11.880 --> 1:00:12.700 I got a cylinder. 1:00:12.699 --> 1:00:14.149 I rammed it into this solid. 1:00:14.150 --> 1:00:17.110 It's a mathematical Gaussian surface. 1:00:17.110 --> 1:00:19.820 It's got following faces, flat face, 1:00:19.820 --> 1:00:22.410 no E because there's nothing inside the metal, 1:00:22.409 --> 1:00:24.399 this part of the curvy-face, no E, 1:00:24.400 --> 1:00:25.950 nothing inside the metal. 1:00:25.949 --> 1:00:28.449 Here E is parallel to the sides of the cylinder, 1:00:28.447 --> 1:00:29.417 so there's no flux. 1:00:29.420 --> 1:00:32.510 The top end, if the area's A or 1:00:32.512 --> 1:00:35.532 dA, the top end will have a flux 1:00:35.534 --> 1:00:39.684 which will be the E that I'm trying to calculate times 1:00:39.681 --> 1:00:40.581 dA. 1:00:40.579 --> 1:00:44.269 That's going to be equal to the charge enclosed is 1:00:44.271 --> 1:00:48.041 σdA/ ε_0, 1:00:48.039 --> 1:00:52.469 because σ is the charge density which I 1:00:52.474 --> 1:00:54.394 assume I'm given. 1:00:54.389 --> 1:00:57.239 Then you can see the dA cancels. 1:00:57.239 --> 1:01:11.019 The electric field is σ/ε_0. 1:01:11.018 --> 1:01:15.928 Do you remember ever seeing a formula like this sigma or 1:01:15.932 --> 1:01:17.812 anything like this? 1:01:17.809 --> 1:01:24.859 Yes? 1:01:24.860 --> 1:01:25.480 Okay. 1:01:25.480 --> 1:01:33.680 Let me remind you where you saw it. 1:01:33.679 --> 1:01:38.869 I showed you that if you took an infinite plane of charge 1:01:38.867 --> 1:01:42.297 density σ the electric field was 1:01:42.295 --> 1:01:47.215 σ/2ε_0, and I did that with a long 1:01:47.215 --> 1:01:48.005 calculation. 1:01:48.010 --> 1:01:48.920 I took a point there. 1:01:48.920 --> 1:01:51.050 I took rings and so on. 1:01:51.050 --> 1:01:53.270 Now, you don't have to do that. 1:01:53.268 --> 1:01:56.568 Let's use Gauss's Law again to calculate it. 1:01:56.570 --> 1:01:59.900 Once again, you argue that if you're in front of an infinite 1:01:59.902 --> 1:02:03.352 plane the only distribution of field lines that makes sense is 1:02:03.349 --> 1:02:06.739 if the field line's always perpendicular to the surface. 1:02:06.739 --> 1:02:08.929 They can get weaker as you go away, 1:02:08.929 --> 1:02:11.239 but at a given distance from the plane they should have the 1:02:11.240 --> 1:02:13.680 same value, because if you slide the plane 1:02:13.684 --> 1:02:15.764 up and down it shouldn't matter. 1:02:15.760 --> 1:02:17.810 So let's take the side view of the plane. 1:02:17.809 --> 1:02:19.869 There are your charges. 1:02:19.869 --> 1:02:23.089 And I'm going to take a Gaussian cylinder that looks 1:02:23.090 --> 1:02:23.850 like this. 1:02:23.849 --> 1:02:26.409 The field lines are going like that. 1:02:26.409 --> 1:02:27.519 They're going like that. 1:02:27.518 --> 1:02:30.408 There is some charge trapped here. 1:02:30.409 --> 1:02:37.669 And the area of that guy is some dA. 1:02:37.670 --> 1:02:40.270 So again, I apply Gauss's Law here. 1:02:40.268 --> 1:02:43.608 Of the cylinder I got the curvy sides with no contribution, 1:02:43.610 --> 1:02:46.490 because the field is parallel to the curvy side. 1:02:46.489 --> 1:02:48.429 I got the flat faces. 1:02:48.429 --> 1:02:50.959 So I get E times dA for one of them, 1:02:50.961 --> 1:02:53.341 and E times dA for the other one, 1:02:53.338 --> 1:02:54.888 because both are outgoing. 1:02:54.889 --> 1:02:57.579 That is equal to charge enclosed with the σ 1:02:57.579 --> 1:03:00.439 times dA divided by ε_0. 1:03:00.440 --> 1:03:05.040 If you cancel the dA you find E is 1:03:05.039 --> 1:03:08.269 σ/2ε_0. 1:03:08.268 --> 1:03:10.858 So the electric field on either side of the plane is 1:03:10.856 --> 1:03:12.526 σ/2ε_0. 1:03:12.530 --> 1:03:14.370 You just get it from Gauss's Law. 1:03:14.369 --> 1:03:18.379 Again, there's no reason to do the complicated integral. 1:03:18.380 --> 1:03:20.900 That's because we've reduced everything to one unknown, 1:03:20.900 --> 1:03:23.470 namely what was the magnitude of the electric field at a 1:03:23.469 --> 1:03:25.149 certain distance from the plane? 1:03:25.150 --> 1:03:28.490 Direction is known to be perpendicular and to be constant 1:03:28.485 --> 1:03:29.375 on this line. 1:03:29.380 --> 1:03:33.370 Then the one number came out to be, in fact, independent of how 1:03:33.369 --> 1:03:34.269 far you are. 1:03:34.268 --> 1:03:36.928 But now go back to the conductor. 1:03:36.929 --> 1:03:41.539 If you go back to the conductor you find the field at the 1:03:41.541 --> 1:03:45.991 surface of a conductor is σ/ε_0 1:03:45.989 --> 1:03:49.859 where σ is the charge density there, 1:03:49.860 --> 1:03:51.530 whereas for an infinite plane it's 1:03:51.534 --> 1:03:53.974 σ/2ε_0 on either side. 1:03:53.969 --> 1:03:56.019 And you can ask, what's going on? 1:03:56.018 --> 1:04:00.048 Why is it σ/ε_0? 1:04:00.050 --> 1:04:04.980 Can anybody guess what may be happening? 1:04:04.980 --> 1:04:07.140 In other words, if you take a tiny area on a 1:04:07.135 --> 1:04:10.515 conductor and you go very, very, very close it should be 1:04:10.523 --> 1:04:13.263 as if you're next to an infinite plane, 1:04:13.260 --> 1:04:15.590 because if you're very, very close to the surface you 1:04:15.594 --> 1:04:17.394 don't know if it's finite or infinite. 1:04:17.389 --> 1:04:19.989 So you should get the same answer as for the infinite 1:04:19.989 --> 1:04:20.389 plane. 1:04:20.389 --> 1:04:24.819 Therefore, if you took only the charge sitting there it should 1:04:24.815 --> 1:04:27.565 give you σ/2ε_0 1:04:27.572 --> 1:04:30.692 going out and σ/2ε_0 1:04:30.690 --> 1:04:31.780 going in. 1:04:31.780 --> 1:04:33.010 In other words, let me draw a picture here. 1:04:33.010 --> 1:04:34.380 That's my tiny area. 1:04:34.380 --> 1:04:38.890 I expect the field due to just the charge there to be like 1:04:38.887 --> 1:04:41.937 this, σ/2ε_0, 1:04:41.940 --> 1:04:46.710 and σ/2ε_0. 1:04:46.710 --> 1:04:51.160 But that is not the whole story because I've got the rest of the 1:04:51.157 --> 1:04:53.837 conductor that's got its own charge. 1:04:53.840 --> 1:04:56.790 It is going to produce a field. 1:04:56.789 --> 1:05:00.829 That field will have some value, and I claim I know the 1:05:00.831 --> 1:05:02.481 value of that field. 1:05:02.480 --> 1:05:07.220 It'll be precisely enough to cancel it on the inside, 1:05:07.219 --> 1:05:10.819 but if you cancel it on the inside you'll double it on the 1:05:10.818 --> 1:05:14.228 outside because it's got σ/2ε_0 1:05:14.228 --> 1:05:16.878 pointing outwards to cancel this guy, 1:05:16.880 --> 1:05:20.850 but then it'll aid the one due to the little area by doubling 1:05:20.849 --> 1:05:21.179 it. 1:05:21.179 --> 1:05:23.469 That's why it's σ/ε_0 1:05:23.471 --> 1:05:24.671 outside and 0 inside. 1:05:24.670 --> 1:05:26.540 In other words, here is the full story. 1:05:26.539 --> 1:05:29.979 If you go to a conductor, if you pick a region here, 1:05:29.981 --> 1:05:33.761 the field at this point is due to what's here and what is 1:05:33.760 --> 1:05:35.110 everywhere else. 1:05:35.110 --> 1:05:39.030 What is here does that as if it was an infinite plane. 1:05:39.030 --> 1:05:45.850 The rest of the guys do that and that cancels this here, 1:05:45.849 --> 1:05:50.059 but adds there, so you can see more 1:05:50.063 --> 1:05:54.283 ε_0 outside. 1:05:54.280 --> 1:05:57.050 Okay, then there are other variations to this theme. 1:05:57.050 --> 1:06:01.050 So here's a conductor, and I make a hole in it, 1:06:01.045 --> 1:06:03.125 and I put some charge. 1:06:03.130 --> 1:06:07.940 You can ask if I throw some charge on it where will it sit. 1:06:07.940 --> 1:06:12.970 Well, some charges will sit here if you put some coulombs, 1:06:12.971 --> 1:06:16.591 but what will happen on the inner wall? 1:06:16.590 --> 1:06:21.190 This is a hole. 1:06:21.190 --> 1:06:23.930 The claim is that there'll be no charges here. 1:06:23.929 --> 1:06:26.889 They'll all be outside even though you've got a hole in the 1:06:26.889 --> 1:06:27.349 middle. 1:06:27.349 --> 1:06:29.719 Again, Gauss's Law tells you why. 1:06:29.719 --> 1:06:33.179 If I take a surface like this and do Gauss's Law, 1:06:33.175 --> 1:06:36.485 since the electric field is 0 inside the metal, 1:06:36.485 --> 1:06:38.425 Q enclosed is 0. 1:06:38.429 --> 1:06:40.959 So the Q that's enclosed, which is 0, 1:06:40.960 --> 1:06:45.120 means that maybe some positive guys and some negative guys, 1:06:45.119 --> 1:06:48.619 but there's nothing from stopping the positive guys from 1:06:48.619 --> 1:06:51.799 rushing to meet the negative guys and canceling. 1:06:51.800 --> 1:06:53.670 They will, for there is no reason they just stay on 1:06:53.670 --> 1:06:54.830 opposite sides of the island. 1:06:54.829 --> 1:06:58.279 They'll just come together. 1:06:58.280 --> 1:07:01.660 Therefore, there is no stable equilibrium in which the charges 1:07:01.657 --> 1:07:03.537 will be ever inside a conductor. 1:07:03.539 --> 1:07:07.509 So if you have a conductor and you put charge on it, 1:07:07.510 --> 1:07:11.870 it goes to the surface even if there are holes inside the 1:07:11.869 --> 1:07:12.959 conductor. 1:07:12.960 --> 1:07:15.540 Now, that's a very interesting paradox. 1:07:15.539 --> 1:07:19.019 I showed you that the field inside a conductor even if 1:07:19.023 --> 1:07:22.113 there's a hole is 0, but let's ask the following 1:07:22.114 --> 1:07:22.974 question. 1:07:22.969 --> 1:07:26.669 I take a conductor with the hole in it and I put a charge 1:07:26.672 --> 1:07:27.732 q here. 1:07:27.730 --> 1:07:32.240 Will you know about it from the outside? 1:07:32.239 --> 1:07:33.509 You're not allowed to enter it. 1:07:33.510 --> 1:07:38.970 Will you know from the outside there's a charge inside or will 1:07:38.969 --> 1:07:41.029 it get also shielded? 1:07:41.030 --> 1:07:47.460 Will there be any flux lines coming out here? 1:07:47.460 --> 1:07:47.780 Yep? 1:07:47.780 --> 1:07:48.950 Student: Yes. 1:07:48.949 --> 1:07:49.879 Prof: Because? 1:07:49.880 --> 1:07:52.230 Student: Because there's a charge. 1:07:52.230 --> 1:07:52.730 Prof: Right. 1:07:52.730 --> 1:07:57.260 If there are no field lines here, and I do Gauss's Law on 1:07:57.255 --> 1:07:59.765 this surface, if the surface integral of 1:07:59.769 --> 1:08:01.789 E is 0 q_enclosed is 1:08:01.788 --> 1:08:05.108 0, but I know there is some 1:08:05.112 --> 1:08:07.272 q inside. 1:08:07.269 --> 1:08:09.209 So what's going to happen? 1:08:09.210 --> 1:08:13.820 I want the field lines to be present here and absent here. 1:08:13.820 --> 1:08:17.990 So what'll happen is if you put a q here the material 1:08:17.993 --> 1:08:22.023 conductor will separate into -q's where these lines 1:08:22.024 --> 1:08:27.314 can terminate, and then out here will be some 1:08:27.305 --> 1:08:33.855 compensating plus charges that'll produce the field 1:08:33.863 --> 1:08:38.983 that'll produce the lines going out. 1:08:38.979 --> 1:08:43.119 In other words, this is really like a chunk of 1:08:43.123 --> 1:08:48.563 copper I showed you where it screens the field by polarizing 1:08:48.557 --> 1:08:53.897 into a negative part in this wall and positive contribution 1:08:53.899 --> 1:08:59.149 in the outer wall so that inside there is no field. 1:08:59.149 --> 1:09:03.679 Okay, so these are different variations of this theme and you 1:09:03.680 --> 1:09:08.210 should be able to do a whole bunch of problems connected with 1:09:08.210 --> 1:09:08.890 that. 1:09:08.890 --> 1:09:13.850 Okay, I've used up my board, so I think I'll give you guys a 1:09:13.850 --> 1:09:18.900 five minute tour on how to do these integrals in case it ever 1:09:18.895 --> 1:09:19.985 comes up. 1:09:19.988 --> 1:09:22.658 I'm just going to do two very trivial integrals and stop, 1:09:22.659 --> 1:09:25.469 and I'm going to do them in two dimensions and you can worry 1:09:25.474 --> 1:09:26.814 about higher dimensions. 1:09:26.810 --> 1:09:28.640 It's just generalization. 1:09:28.640 --> 1:09:34.780 Suppose you're in the xy plane and there's a function in 1:09:34.778 --> 1:09:39.828 the xy plane, f (x, y), and someone says find 1:09:39.827 --> 1:09:41.607 f (x,y) dx dy. 1:09:41.609 --> 1:09:45.279 What's the procedure and what are you supposed to do is the 1:09:45.283 --> 1:09:45.983 question. 1:09:45.979 --> 1:09:49.229 What you do in Cartesian coordinates is you take a line 1:09:49.225 --> 1:09:52.475 y equal to something, y equal to something 1:09:52.481 --> 1:09:54.621 delta y, x equal to something, 1:09:54.621 --> 1:09:56.651 x equal to something delta x, 1:09:56.649 --> 1:09:59.059 and that region (bounded) by these contours of constant 1:09:59.063 --> 1:10:01.833 x at different values and constant y at different 1:10:01.832 --> 1:10:05.382 values, has an area dxdy. 1:10:05.380 --> 1:10:08.390 You multiply that area by this function. 1:10:08.390 --> 1:10:10.660 This could be the number of people living per square mile 1:10:10.655 --> 1:10:12.595 and this could be the number of square miles, 1:10:12.600 --> 1:10:16.650 and you add up all the little squares over the area that was 1:10:16.649 --> 1:10:17.679 given to you. 1:10:17.680 --> 1:10:21.570 So whoever tells you to do integrals better tell you over 1:10:21.567 --> 1:10:24.757 what range you're going to do that integral. 1:10:24.760 --> 1:10:26.600 Okay? 1:10:26.600 --> 1:10:33.980 So let's find out the area for triangle by this process. 1:10:33.979 --> 1:10:35.789 Here's a triangle. 1:10:35.788 --> 1:10:40.288 Let's say there's one here and one here, 0,1; 1:10:40.289 --> 1:10:41.279 1,0. 1:10:41.279 --> 1:10:42.379 This is y. 1:10:42.380 --> 1:10:42.940 This is x. 1:10:42.939 --> 1:10:44.759 I want to find the area of the triangle. 1:10:44.760 --> 1:10:45.520 Yep? 1:10:45.520 --> 1:10:47.690 Student: Isn't that point 1,1? 1:10:47.689 --> 1:10:50.159 Prof: Yes, thank you. 1:10:50.159 --> 1:10:52.559 That point is 1,1. 1:10:52.560 --> 1:10:56.640 You're supposed to take dxdy you write it like 1:10:56.640 --> 1:10:57.190 this. 1:10:57.189 --> 1:10:58.279 There's no function or integral. 1:10:58.279 --> 1:11:01.279 I just want 1 times dxdy, so the 1 is 1:11:01.282 --> 1:11:02.052 implicit. 1:11:02.050 --> 1:11:08.080 dy, you pick a certain x and y should go 1:11:08.078 --> 1:11:10.088 from here to here. 1:11:10.090 --> 1:11:13.300 The range of y is clearly 1 - x because 1:11:13.300 --> 1:11:15.770 this is a curve x y = 1. 1:11:15.770 --> 1:11:20.970 So y goes from 0 to 1 - x, and for every choice 1:11:20.967 --> 1:11:24.517 of x, x goes from 0 to 1. 1:11:24.520 --> 1:11:25.230 Yep? 1:11:25.229 --> 1:11:28.789 Student: Isn't the line defined as y = x ________. 1:11:28.789 --> 1:11:29.589 Prof: Pardon me? 1:11:29.590 --> 1:11:30.020 Student: Is the line ________? 1:11:30.020 --> 1:11:30.810 Prof: Oh, let me see. 1:11:30.810 --> 1:11:31.210 No, no. 1:11:31.210 --> 1:11:33.700 This is not the equation of the line. 1:11:33.698 --> 1:11:41.018 This height here is equal to 1 - x. 1:11:41.020 --> 1:11:42.720 Oh, you think that's wrong? 1:11:42.720 --> 1:11:46.960 Student: > 1:11:46.960 --> 1:11:49.300 Prof: The height of y, isn't there 1 - 1:11:49.296 --> 1:11:49.796 x? 1:11:49.800 --> 1:11:50.590 Student: > 1:11:50.590 --> 1:11:51.650 Prof: Or it's just x. 1:11:51.649 --> 1:11:52.669 I'm sorry. 1:11:52.670 --> 1:11:54.930 Okay, caught on tape. 1:11:54.930 --> 1:11:58.030 Very good. 1:11:58.029 --> 1:12:04.899 All right, so here is x, goes up to x and the 1:12:04.899 --> 1:12:10.349 y integral you do it as lower limit, 1:12:10.350 --> 1:12:13.110 upper limit gives you an x and x goes from 1:12:13.108 --> 1:12:13.558 0 to 1. 1:12:13.560 --> 1:12:16.650 That gives you x^(2)/2 from 0 to 1. 1:12:16.649 --> 1:12:22.819 That gives you 1 over 2. 1:12:22.819 --> 1:12:26.129 If you had a function of x and y you wanted 1:12:26.125 --> 1:12:27.925 to integrate, in other words instead of just 1:12:27.925 --> 1:12:29.475 the number one, if you had a function of 1:12:29.484 --> 1:12:31.684 x and y the rule is the following. 1:12:31.680 --> 1:12:35.370 You put the function here, and put x equal to a 1:12:35.365 --> 1:12:38.055 fixed value, don't take it as a variable, 1:12:38.055 --> 1:12:41.165 and treat y as a variable and do dy from 1:12:41.172 --> 1:12:42.822 this limit to this limit. 1:12:42.819 --> 1:12:47.259 When you are done you will get some mass that depends only on 1:12:47.255 --> 1:12:50.355 x that you integrate from 0 to 1. 1:12:50.359 --> 1:12:55.369 But the final thing I'm going to do is to do the same integral 1:12:55.373 --> 1:12:59.323 in polar coordinates, area of a triangle in polar 1:12:59.319 --> 1:13:01.949 coordinates looks like this. 1:13:01.948 --> 1:13:04.668 In polar coordinates, as you know, 1:13:04.670 --> 1:13:08.740 you draw circles of r and r dr, 1:13:08.738 --> 1:13:12.918 and lines at θ and θ dθ, 1:13:12.920 --> 1:13:17.310 and this shaded region here has got one side equal to 1:13:17.307 --> 1:13:22.117 rdθ and the other side equal to dr. 1:13:22.118 --> 1:13:26.788 So area integral will look like rdrdθ times some 1:13:26.787 --> 1:13:31.067 function of r and θ over the allowed 1:13:31.065 --> 1:13:35.175 region, but I want to do now a triangle 1:13:35.177 --> 1:13:37.637 of height 1 and base 1. 1:13:37.640 --> 1:13:39.730 Here's what I do. 1:13:39.729 --> 1:13:43.089 I'm going to pick theta first and hold it fixed, 1:13:43.091 --> 1:13:45.741 and theta will go from 0 to Π/4. 1:13:45.739 --> 1:13:46.819 Can you see that? 1:13:46.819 --> 1:13:48.969 This is supposed to be a square. 1:13:48.970 --> 1:13:52.280 Theta goes from 0 to Π/4, 1:13:52.282 --> 1:13:54.622 and I have rdr. 1:13:54.619 --> 1:13:58.689 What is the range of r? 1:13:58.689 --> 1:14:00.499 How far is this thing? 1:14:00.500 --> 1:14:03.940 You can see that if this angle is theta then 1:14:03.935 --> 1:14:07.925 rcosθ is 1, and for r is 1:14:07.931 --> 1:14:09.851 1/cosθ. 1:14:09.850 --> 1:14:14.640 So r dr is really d of r squared. 1:14:14.640 --> 1:14:21.230 d of r squared is equal to d of 1:14:21.226 --> 1:14:25.486 1/cos^(2)θ, right? 1:14:25.489 --> 1:14:26.439 I mean, let me do it this way. 1:14:26.439 --> 1:14:34.739 d of r squared is r^(2)/2 from 1/cosθ 1:14:34.742 --> 1:14:36.292 to 0. 1:14:36.288 --> 1:14:43.888 So that gives me 1/2 integral dθ 0 to Π/4 of 1:14:43.890 --> 1:14:47.240 1/cos^(2)θ. 1:14:47.238 --> 1:14:50.228 Well, you may not know this, but sec^(2)θ is 1:14:50.229 --> 1:14:52.259 the derivative of tanθ. 1:14:52.260 --> 1:14:56.660 So derivative of integral of tanθ will turn out 1:14:56.661 --> 1:15:00.761 to be simply tanθ calculated between Π/4 1:15:00.764 --> 1:15:01.514 and 0. 1:15:01.510 --> 1:15:04.940 That'll be just 1. 1:15:04.939 --> 1:15:07.999 So you're not supposed to follow at lightening speed this 1:15:08.000 --> 1:15:08.930 integral, okay? 1:15:08.930 --> 1:15:11.740 Either you've seen it before or if you need some help you can 1:15:11.740 --> 1:15:13.710 look at my math book I mentioned to you. 1:15:13.710 --> 1:15:17.180 All I'm trying to tell you here is if when you have multiple 1:15:17.181 --> 1:15:20.421 integrals you pick one coordinate and you freeze it, 1:15:20.420 --> 1:15:23.080 for example, x when you integrate to 1:15:23.083 --> 1:15:26.953 the other coordinate over the allowed region that keeps you in 1:15:26.953 --> 1:15:30.573 the boundary then integrate over the second variable. 1:15:30.569 --> 1:15:32.499 You may need that, but for most of the problems 1:15:32.496 --> 1:15:34.126 I've given you won't have to do that. 1:15:34.130 --> 1:15:35.130 They're pretty simple. 1:15:35.130 --> 1:15:40.000