1 00:00:00 --> 00:00:00,261 2 00:00:00,261 --> 00:00:07,579 We're going to talk about again some new concepts. 3 00:00:07,579 --> 00:00:14,748 And that's the concept of electrostatic potential 4 00:00:14,748 --> 00:00:23,111 electrostatic potential energy. For which we will use the 5 00:00:23,111 --> 00:00:29,832 symbol U and independently electric potential. 6 00:00:29,832 --> 00:00:37,276 Which is very different, for which we will use the 7 00:00:37,276 --> 00:00:41,201 symbol V. Imagine that I have a charge Q 8 00:00:41,201 --> 00:00:44,924 one here and that's plus, plus charge, 9 00:00:44,924 --> 00:00:50,861 and here I have a charge plus Q two and they have a distant, 10 00:00:50,861 --> 00:00:55,591 they're a distance R apart. And that is point P. 11 00:00:55,591 --> 00:01:01,528 It's very clear that in order to bring these charges at this 12 00:01:01,528 --> 00:01:07,164 distance from each other I had to do work to 13 00:01:07,164 --> 00:01:10,569 bring them there because they repel each other. 14 00:01:10,569 --> 00:01:14,938 It's like pushing in a spring. If you release the spring you 15 00:01:14,938 --> 00:01:18,27 get the energy back. If they were -- they were 16 00:01:18,27 --> 00:01:22,786 connected with a little string, the string would be stretched, 17 00:01:22,786 --> 00:01:25,97 take scissors, cut the string fweet they fly 18 00:01:25,97 --> 00:01:29,154 apart again. So I have put work in there and 19 00:01:29,154 --> 00:01:33,152 that's what we call the electrostatic potential energy. 20 00:01:33,152 --> 00:01:37,594 So let's work this out in some detail how much work I have to 21 00:01:37,594 --> 00:01:39,519 do. Well, 22 00:01:39,519 --> 00:01:42,607 we first put Q one here, if space is empty, 23 00:01:42,607 --> 00:01:45,988 this doesn't take any work to place Q one here. 24 00:01:45,988 --> 00:01:50,251 But now I come from very far away, we always think of it as 25 00:01:50,251 --> 00:01:53,853 infinitely far away, of course that's a little bit 26 00:01:53,853 --> 00:01:57,234 of exaggeration, and we bring this charge Q two 27 00:01:57,234 --> 00:02:00,91 from infinity to that point P. And I, Walter Lewin, 28 00:02:00,91 --> 00:02:04,07 have to do work, I have to push and push and 29 00:02:04,07 --> 00:02:08,04 push and the closer I get the harder I have to push and 30 00:02:08,04 --> 00:02:11,47 finally I reach that point P. 31 00:02:11,47 --> 00:02:15,824 Suppose I am here and this separation is little R. 32 00:02:15,824 --> 00:02:19,822 I've reached that point. Then the force on me, 33 00:02:19,822 --> 00:02:22,577 the electric force, is outwards. 34 00:02:22,577 --> 00:02:27,908 And so I have to overcome that force and so my force F Walter 35 00:02:27,908 --> 00:02:32,35 Lewin is in this direction. And so you can see I do 36 00:02:32,35 --> 00:02:36,26 positive work, the force and the direction in 37 00:02:36,26 --> 00:02:43,102 which I'm moving are in the same direction, I do positive work. 38 00:02:43,102 --> 00:02:47,616 Now, the work that I do could be calculated. 39 00:02:47,616 --> 00:02:53,915 The work that Walter Lewin is doing in going all the way from 40 00:02:53,915 --> 00:02:59,899 infinity to that location P is the integral going from in- 41 00:02:59,899 --> 00:03:05,778 infinity to radius R of the force of Walter Lewin dot DR. 42 00:03:05,778 --> 00:03:10,292 But of course that work is exactly the same, 43 00:03:10,292 --> 00:03:15,331 either one is fine, to take the electric force in 44 00:03:15,331 --> 00:03:18,695 going from R to infinity. 45 00:03:18,695 --> 00:03:20,636 Dot DR. Because the force, 46 00:03:20,636 --> 00:03:24,285 the electric force, and Walter Lewin's force are 47 00:03:24,285 --> 00:03:27,7 the same in magnitude but opposite direction, 48 00:03:27,7 --> 00:03:31,504 and so by flipping over, going from infinity to R, 49 00:03:31,504 --> 00:03:34,144 to R to infinity, this is the same. 50 00:03:34,144 --> 00:03:38,801 This is one and the same thing. Let's calculate this integral 51 00:03:38,801 --> 00:03:43,459 because that's a little easy. We know what the electric force 52 00:03:43,459 --> 00:03:46,021 is, Coulomb's law, it's repelling, 53 00:03:46,021 --> 00:03:51,3 so the force and DR are now in the same direction, 54 00:03:51,3 --> 00:03:56,677 so the angle theta between them is zero, so the cosine of theta 55 00:03:56,677 --> 00:04:00,667 is one, so we can forget about all the vectors, 56 00:04:00,667 --> 00:04:04,743 and so we would get then that this equals Q one, 57 00:04:04,743 --> 00:04:08,039 Q two, divided by four pi epsilon zero. 58 00:04:08,039 --> 00:04:11,768 And now I have downstairs here an R squared. 59 00:04:11,768 --> 00:04:16,452 And so I have the integral now DR divided by R squared. 60 00:04:16,452 --> 00:04:21,396 From capital R to infinity. And this integral is minus one 61 00:04:21,396 --> 00:04:23,222 over R. 62 00:04:23,222 --> 00:04:27,426 Which I have to evaluate between R and infinity. 63 00:04:27,426 --> 00:04:32,346 And when I do that that becomes plus one over capital R. 64 00:04:32,346 --> 00:04:37,712 Right, the integral of DR over R squared I'm sure you can all 65 00:04:37,712 --> 00:04:42,631 do that is minus one over R. I evaluate it between R and 66 00:04:42,631 --> 00:04:46,12 infinity and so you get plus one over R. 67 00:04:46,12 --> 00:04:51,486 And so U, which is the energy that -- the work that I have to 68 00:04:51,486 --> 00:04:56,406 do to bring this charge at that position, 69 00:04:56,406 --> 00:05:00,995 that U is now Q one. Times Q two divided by four pi 70 00:05:00,995 --> 00:05:04,575 epsilon zero. Divided by that capital R. 71 00:05:04,575 --> 00:05:08,889 And this of course this is scalar, that is work, 72 00:05:08,889 --> 00:05:13,57 it's a number of joules. If Q one and Q two are both 73 00:05:13,57 --> 00:05:18,068 positive or both ne- negative, I do positive work, 74 00:05:18,068 --> 00:05:22,015 you can see that, minus times minus is plus. 75 00:05:22,015 --> 00:05:25,136 Because then they repel each other. 76 00:05:25,136 --> 00:05:29,726 If one is positive and the other is 77 00:05:29,726 --> 00:05:33,642 negative, then I do negative work, and you see that that 78 00:05:33,642 --> 00:05:37,63 comes out as a sign sensitive, minus times plus is minus, 79 00:05:37,63 --> 00:05:41,617 so I can do negative work. If the two don't have the same 80 00:05:41,617 --> 00:05:44,466 polarity. I want you to convince yourself 81 00:05:44,466 --> 00:05:48,667 that if I didn't come along a straight line from all the way 82 00:05:48,667 --> 00:05:51,658 from infinity, but I came in a very crooked 83 00:05:51,658 --> 00:05:55,005 way, finally ended up at point P, at that point, 84 00:05:55,005 --> 00:06:00,416 that the amount of work that I had to do is exactly the same. 85 00:06:00,416 --> 00:06:04,257 You see the parallel with eight oh one where we dealt with 86 00:06:04,257 --> 00:06:06,885 gravity. Gravity is a conservative force 87 00:06:06,885 --> 00:06:09,715 and when you deal with conservative forces, 88 00:06:09,715 --> 00:06:13,623 the work that has to be done in going from one point to the 89 00:06:13,623 --> 00:06:15,779 other is independent of the path. 90 00:06:15,779 --> 00:06:18,744 That is the definition of conservative force. 91 00:06:18,744 --> 00:06:21,237 Electric forces are also conservative. 92 00:06:21,237 --> 00:06:25,078 And so it doesn't make any difference whether I come along 93 00:06:25,078 --> 00:06:28,784 a straight line to this point or whether I do that in an 94 00:06:28,784 --> 00:06:33,163 extremely crooked way and finally end up here. 95 00:06:33,163 --> 00:06:37,566 That's the same amount of work. Now if we do have a collection 96 00:06:37,566 --> 00:06:40,958 of charges, so we have pluses and minus charges, 97 00:06:40,958 --> 00:06:43,556 some pluses, some minus, some pluses, 98 00:06:43,556 --> 00:06:46,299 minus, pluses, pluses, then you now can 99 00:06:46,299 --> 00:06:49,835 calculate the amount of work that I, Walter Lewin, 100 00:06:49,835 --> 00:06:54,165 have to do in assembling that. You bring one from infinity to 101 00:06:54,165 --> 00:06:56,331 here, another one, another one, 102 00:06:56,331 --> 00:07:00,3 and you add up all that work, some work may be positive, 103 00:07:00,3 --> 00:07:04,197 some work may be negative. Finally you h- arrive at the 104 00:07:04,197 --> 00:07:09,847 total amount of work that you have to 105 00:07:09,847 --> 00:07:16,796 do to assemble these charges. And that is the meaning of 106 00:07:16,796 --> 00:07:20,84 capital U. Now I turn to electric 107 00:07:20,84 --> 00:07:25,768 potential. And for that I start off here 108 00:07:25,768 --> 00:07:31,454 with a charge which I now call plus capital Q. 109 00:07:31,454 --> 00:07:36,761 It's located here. And at a position P at a 110 00:07:36,761 --> 00:07:42,982 distance R away I place a test charge plus Q. 111 00:07:42,982 --> 00:07:47,924 Make it positive for now, you can change it later to 112 00:07:47,924 --> 00:07:51,993 become a negative. And so the electrostatic 113 00:07:51,993 --> 00:07:57,903 potential energy we -- we know already, we just calculated it, 114 00:07:57,903 --> 00:08:03,426 that would be Q times Q divided by four pi epsilon zero R. 115 00:08:03,426 --> 00:08:06,914 That's exactly the same that we have. 116 00:08:06,914 --> 00:08:14,943 So the electric potential, electrostatic potential energy, 117 00:08:14,943 --> 00:08:22,658 is the work that I have to do to bring this charge here. 118 00:08:22,658 --> 00:08:28,969 Now I'm going to introduce electric potential. 119 00:08:28,969 --> 00:08:35,701 Electric potential. And that is the work per unit 120 00:08:35,701 --> 00:08:43,025 charge that I have to do to go from infinity to that 121 00:08:43,025 --> 00:08:46,701 position. So Q doesn't enter into it 122 00:08:46,701 --> 00:08:50,692 anymore. It is the work per unit charge 123 00:08:50,692 --> 00:08:54,683 to go from infinity to that location P. 124 00:08:54,683 --> 00:09:00,984 And so if it is the work per unit charge, that means little Q 125 00:09:00,984 --> 00:09:06,025 fweet disappears. And so now we write down that V 126 00:09:06,025 --> 00:09:09,491 at that location P, the potential, 127 00:09:09,491 --> 00:09:14,844 electric potential at that location P, 128 00:09:14,844 --> 00:09:19,459 is now only Q divided four pi epsilon zero R. 129 00:09:19,459 --> 00:09:24,178 Little Q has disappeared. It is also a scalar. 130 00:09:24,178 --> 00:09:29,317 This has unit joules. The units here is joules per 131 00:09:29,317 --> 00:09:33,407 coulombs. I have divided out one charge. 132 00:09:33,407 --> 00:09:38,965 It's work per unit charge. No one would ever call this 133 00:09:38,965 --> 00:09:44,418 joules per coulombs, we call this volts, 134 00:09:44,418 --> 00:09:48,876 called after the great Volta, who did a lot of research on 135 00:09:48,876 --> 00:09:50,987 this. So we call this volts. 136 00:09:50,987 --> 00:09:54,115 But it's the same as joules per coulombs. 137 00:09:54,115 --> 00:09:58,181 If we have a very simple situation like we have here, 138 00:09:58,181 --> 00:10:02,482 that we only have one charge, then this is the potential 139 00:10:02,482 --> 00:10:06,548 anywhere, at any distance you want, from this charge. 140 00:10:06,548 --> 00:10:09,363 If R goes up, if you're further away, 141 00:10:09,363 --> 00:10:11,787 the potential will become lower. 142 00:10:11,787 --> 00:10:15,853 If this Q is positive, the potential is everywhere in 143 00:10:15,853 --> 00:10:20,069 space positive for a single charge. 144 00:10:20,069 --> 00:10:23,7 If this Q is negative, everywhere in space the 145 00:10:23,7 --> 00:10:27,412 potential is negative. Electro- electric static 146 00:10:27,412 --> 00:10:31,689 potential can be negative. The work that I do per unit 147 00:10:31,689 --> 00:10:35,32 charge coming from infinity would be negative, 148 00:10:35,32 --> 00:10:39,677 if that's a negative charge. And the potential when I'm 149 00:10:39,677 --> 00:10:43,711 infinitely far away, when this R becomes infinitely 150 00:10:43,711 --> 00:10:47,745 large, is zero. So that's the way we 151 00:10:47,745 --> 00:10:52,035 define our zero. So you can have positive 152 00:10:52,035 --> 00:10:57,718 potentials, near positive charge, negative potentials, 153 00:10:57,718 --> 00:11:02,864 near negative charge, and if you're very very far 154 00:11:02,864 --> 00:11:08,226 away, then potential is zero. Let's now turn to our 155 00:11:08,226 --> 00:11:11,764 Vandegraaff. It's a hollow sphere, 156 00:11:11,764 --> 00:11:16,053 has a radius R. About thirty centimeters. 157 00:11:16,053 --> 00:11:22,04 And I'm going to put on here plus ten microcoulombs. 158 00:11:22,04 --> 00:11:24,461 It will distribute itself uniformly. 159 00:11:24,461 --> 00:11:27,227 We will discuss that next time in detail. 160 00:11:27,227 --> 00:11:30,685 Because it's a conductor. We already discussed last 161 00:11:30,685 --> 00:11:34,627 lecture that the electric field inside the sphere is zero. 162 00:11:34,627 --> 00:11:38,639 And that the electric field outside is not zero but that we 163 00:11:38,639 --> 00:11:42,235 can think of all the charge being at this point here, 164 00:11:42,235 --> 00:11:46,385 the plus ten microcoulombs is all here, as long as we want to 165 00:11:46,385 --> 00:11:50,258 know what the electric field outside is. 166 00:11:50,258 --> 00:11:54,892 So you can forget the fact that it is a -- a sphere. 167 00:11:54,892 --> 00:12:00,526 And so now I want to know what the electric potential is at any 168 00:12:00,526 --> 00:12:04,616 point in space. I want to know what it is here 169 00:12:04,616 --> 00:12:09,978 and I want to know what it is here at point P which is now a 170 00:12:09,978 --> 00:12:15,067 distance R from the center. And I want to know what it is 171 00:12:15,067 --> 00:12:18,338 here. At a distance little R from the 172 00:12:18,338 --> 00:12:21,791 center. So let's first do the potential 173 00:12:21,791 --> 00:12:26,153 here. The potential at point P is 174 00:12:26,153 --> 00:12:33,724 an integral going from R to infinity if I take the electric 175 00:12:33,724 --> 00:12:41,034 force divided by my test charge Q dot DR, but this is the 176 00:12:41,034 --> 00:12:46,908 electric field, see, this force times distance 177 00:12:46,908 --> 00:12:55,262 is work, but it is work per unit charge, so I take my test charge 178 00:12:55,262 --> 00:12:59,82 out. And so this is the integral in 179 00:12:59,82 --> 00:13:03,444 R to infinity of E dot DL -- DR, sorry. 180 00:13:03,444 --> 00:13:06,4 And that's a very easy integral. 181 00:13:06,4 --> 00:13:11,836 Because we know what E is. The electric field we have done 182 00:13:11,836 --> 00:13:15,46 several times. Follows immediately from 183 00:13:15,46 --> 00:13:21,183 Coulomb's law and so when you calculate this integral you get 184 00:13:21,183 --> 00:13:26,428 Q divided by four pi epsilon zero R which is no surprise 185 00:13:26,428 --> 00:13:31,482 because we already had that for a point 186 00:13:31,482 --> 00:13:35,209 charge. So this is the situation if R, 187 00:13:35,209 --> 00:13:38,634 little R, is larger than capital R. 188 00:13:38,634 --> 00:13:44,274 Precisely what we had before. We can put in some numbers. 189 00:13:44,274 --> 00:13:49,411 If you put in R equals R, which is uh oh point three 190 00:13:49,411 --> 00:13:54,347 meters, and you put in here the ten microcoulombs, 191 00:13:54,347 --> 00:13:59,887 and here the -- the thirty centimeters, then you'll find 192 00:13:59,887 --> 00:14:03,653 three hundred thousand volts. 193 00:14:03,653 --> 00:14:07,096 So you get three times ten to the fifth volts. 194 00:14:07,096 --> 00:14:11,38 If you um take R equals sixty centimeters, you double it, 195 00:14:11,38 --> 00:14:15,587 if you double the distance, the potential goes down by a 196 00:14:15,587 --> 00:14:17,882 factor of two, it's one over R, 197 00:14:17,882 --> 00:14:21,249 so it would be a hundred and fifty kilovolts. 198 00:14:21,249 --> 00:14:25,762 And if you go to three meters, then it is ten times smaller, 199 00:14:25,762 --> 00:14:30,276 then it is thirty kilovolts. And if you go to infinity which 200 00:14:30,276 --> 00:14:34,178 for all practical purposes would be 201 00:14:34,178 --> 00:14:38,373 Lobby seven, if you go to Lobby seven, 202 00:14:38,373 --> 00:14:44,382 then the potential for all practical purposes is about 203 00:14:44,382 --> 00:14:47,897 zero. Because R is so large that 204 00:14:47,897 --> 00:14:52,545 there is no potential left. So if I, if I, 205 00:14:52,545 --> 00:14:57,081 Walter Lewin, march from infinity to this 206 00:14:57,081 --> 00:15:03,09 surface of the Vandegraaff, and I put a charge Q in my 207 00:15:03,09 --> 00:15:07,771 pocket, and I march to the Vandegraaff, 208 00:15:07,771 --> 00:15:11,507 by the time I reach that point, I have done work, 209 00:15:11,507 --> 00:15:15,165 I multiply the charge now back to the potential, 210 00:15:15,165 --> 00:15:19,836 that gives you the work again, because potential was work per 211 00:15:19,836 --> 00:15:24,195 unit charge, and so the work that I have done then is the 212 00:15:24,195 --> 00:15:28,165 charge that I have in my pocket times the potential, 213 00:15:28,165 --> 00:15:31,667 in this case the potential of the Vandegraaff. 214 00:15:31,667 --> 00:15:35,559 If I go all the way to this surface, 215 00:15:35,559 --> 00:15:38,164 which is three hundred thousand volts. 216 00:15:38,164 --> 00:15:42,106 If I were a strong man then I would put one coulomb in my 217 00:15:42,106 --> 00:15:44,218 pocket. That's a lot of charge. 218 00:15:44,218 --> 00:15:48,442 Then I would have done three hundred thousand joules of work. 219 00:15:48,442 --> 00:15:52,313 By just carrying the one coulomb from Lobby seven to the 220 00:15:52,313 --> 00:15:55,129 Vandegraaff. That's about the same work I 221 00:15:55,129 --> 00:15:58,508 have to do to climb up the Empire State Building. 222 00:15:58,508 --> 00:16:01,324 The famous MGH, my mass times G times the 223 00:16:01,324 --> 00:16:05,759 height that I have to climb. So I know how the 224 00:16:05,759 --> 00:16:09,11 electric potential goes with distance. 225 00:16:09,11 --> 00:16:14,183 It's a one over R relationship. Now I have arrived at the 226 00:16:14,183 --> 00:16:18,984 Vandegraaff, I am at the surface, with my test charge, 227 00:16:18,984 --> 00:16:23,15 and now I go inside. And I slosh around inside, 228 00:16:23,15 --> 00:16:27,68 I feel no force anymore. There is no electric field 229 00:16:27,68 --> 00:16:30,759 inside. So as I move around inside, 230 00:16:30,759 --> 00:16:34,926 I experience no force. That means I do no work. 231 00:16:34,926 --> 00:16:39,183 So that means that the potential 232 00:16:39,183 --> 00:16:43,59 must remain constant. So the absence of an electric 233 00:16:43,59 --> 00:16:48,791 field here implies that the electric potential everywhere is 234 00:16:48,791 --> 00:16:53,374 exactly the same inside is the same as on the sphere. 235 00:16:53,374 --> 00:16:58,486 Because no further work is needed in marching around with a 236 00:16:58,486 --> 00:17:02,188 test charge. And so for this special case I 237 00:17:02,188 --> 00:17:07,123 could make a graph of the electric potential versus R and 238 00:17:07,123 --> 00:17:12,324 this is then the radius of the Vandegraaff and 239 00:17:12,324 --> 00:17:18,513 that would be a constant all the way up to this point and 240 00:17:18,513 --> 00:17:23,045 then it would fall off as one over R here. 241 00:17:23,045 --> 00:17:29,566 And in for the numbers that we have chosen, the potential at 242 00:17:29,566 --> 00:17:35,535 the maximum here would be three hundred thousand volts. 243 00:17:35,535 --> 00:17:42,167 Just as when you look at maps where you see contours of equal 244 00:17:42,167 --> 00:17:47,154 height of mountains, which we call equal 245 00:17:47,154 --> 00:17:51,689 altitudes, here we have surfaces of equipotential. 246 00:17:51,689 --> 00:17:57,149 And if you had a point charge or if you had the Vandegraaff, 247 00:17:57,149 --> 00:18:01,036 these surfaces would be concentric spheres. 248 00:18:01,036 --> 00:18:05,571 The further out you go, if the charge is positive, 249 00:18:05,571 --> 00:18:08,533 the lower the potential would be. 250 00:18:08,533 --> 00:18:12,142 They would be nicely spherical surfaces. 251 00:18:12,142 --> 00:18:17,147 Suppose now we had more than one charge, 252 00:18:17,147 --> 00:18:22,185 we had a plus Q one charge, and we had a minus Q two 253 00:18:22,185 --> 00:18:27,321 charge, for instance. And you're being asked now what 254 00:18:27,321 --> 00:18:32,26 is the potential at point P. Well, now the electric 255 00:18:32,26 --> 00:18:37,199 potential at point P, VP, is the potential that you 256 00:18:37,199 --> 00:18:42,039 would have measured if Q one had been there alone. 257 00:18:42,039 --> 00:18:47,175 And you have to add the potential that you would have 258 00:18:47,175 --> 00:18:51,517 seen if Q two had been there alone. 259 00:18:51,517 --> 00:18:55,742 Just adding work per unit charge for one with work per 260 00:18:55,742 --> 00:18:59,648 unit charge of the other. And if this is negative, 261 00:18:59,648 --> 00:19:03,793 then this quantity is negative, and this is positive. 262 00:19:03,793 --> 00:19:08,815 So when you have configurations of positive and negative charges 263 00:19:08,815 --> 00:19:12,96 then of course depending upon where you are in space, 264 00:19:12,96 --> 00:19:17,584 if you're close to the plus charge, the potential is almost 265 00:19:17,584 --> 00:19:22,367 certainly positive, because the one over R is huge. 266 00:19:22,367 --> 00:19:26,255 If you're very close to the negative charge again the one 267 00:19:26,255 --> 00:19:30,282 over R of this little charge will dominate and so you get a 268 00:19:30,282 --> 00:19:33,476 negative potential. And so you have surfaces of 269 00:19:33,476 --> 00:19:37,364 positive potential and you have equipotential surfaces of 270 00:19:37,364 --> 00:19:41,53 negative potentials and so there are surfaces which have zero 271 00:19:41,53 --> 00:19:44,098 potential. And they're not always very 272 00:19:44,098 --> 00:19:47,362 easy to envision. But what I want to show you is 273 00:19:47,362 --> 00:19:51,041 some work that Maxwell himself did in 274 00:19:51,041 --> 00:19:53,631 figuring out these equipotentials. 275 00:19:53,631 --> 00:19:57,556 And so I have here a transparency of publication by 276 00:19:57,556 --> 00:19:59,518 Maxwell. You see a charge, 277 00:19:59,518 --> 00:20:02,735 let's assume it is plus four and plus one, 278 00:20:02,735 --> 00:20:06,895 it could be minus four and minus one, but let's assume 279 00:20:06,895 --> 00:20:10,113 they're plus. And you see the green lines, 280 00:20:10,113 --> 00:20:14,194 which we have seen before, which are the field lines. 281 00:20:14,194 --> 00:20:18,275 Don't pay any attention to the green field lines now. 282 00:20:18,275 --> 00:20:20,786 The red lines are equipotentials. 283 00:20:20,786 --> 00:20:24,867 And you have to rotate them about 284 00:20:24,867 --> 00:20:27,42 the vertical, because they're of course 285 00:20:27,42 --> 00:20:29,772 surfaces, this is three-dimensional. 286 00:20:29,772 --> 00:20:33,332 I have not drawn all the equipotential surfaces in red 287 00:20:33,332 --> 00:20:35,885 because they become too cluttered here. 288 00:20:35,885 --> 00:20:38,639 But I've tried to put most of them in red. 289 00:20:38,639 --> 00:20:42,469 Since this charge is positive and that charge is positive, 290 00:20:42,469 --> 00:20:45,425 everywhere in space, no matter where you are, 291 00:20:45,425 --> 00:20:47,574 the potential has to be positive. 292 00:20:47,574 --> 00:20:51,202 There is not a single point where it could be negative. 293 00:20:51,202 --> 00:20:56,577 If you are very far away from the plus four and the plus one, 294 00:20:56,577 --> 00:20:59,946 then you expect that the equipotential surfaces are 295 00:20:59,946 --> 00:21:03,989 spheres, because it's almost as if you were looking at a plus 296 00:21:03,989 --> 00:21:06,887 five charge. So it doesn't surprise you that 297 00:21:06,887 --> 00:21:10,93 when you go far out that you ultimately get spherical shapes. 298 00:21:10,93 --> 00:21:14,636 When you're very close to the plus four they are perfect 299 00:21:14,636 --> 00:21:17,803 spheres, when you're very close to the plus one, 300 00:21:17,803 --> 00:21:21,577 they are perfect spheres. But then when you're sort of in 301 00:21:21,577 --> 00:21:26,631 between, neither close to the plus four nor to the plus one, 302 00:21:26,631 --> 00:21:29,235 they have this very funny shape. 303 00:21:29,235 --> 00:21:33,605 It reminds me the shape of this balloon a little bit. 304 00:21:33,605 --> 00:21:35,789 Sort of like this. You see. 305 00:21:35,789 --> 00:21:40,747 And there is one surface which is most unusual equipotential 306 00:21:40,747 --> 00:21:45,537 surface which here has a point where the electric field is 307 00:21:45,537 --> 00:21:48,478 zero. It's sort of like twisting the 308 00:21:48,478 --> 00:21:52,175 neck of a goose, you get something like this, 309 00:21:52,175 --> 00:21:57,384 and so you have here a surface which has a point here and it is 310 00:21:57,384 --> 00:22:02,215 exactly at that point where the electric field is 311 00:22:02,215 --> 00:22:05,309 zero, that does not mean that the potential is zero, 312 00:22:05,309 --> 00:22:08,038 of course not, the potential is positive here. 313 00:22:08,038 --> 00:22:11,556 If you come with a positive charge from the Lobby seven and 314 00:22:11,556 --> 00:22:15,074 you have to march up to that point, you have to do positive 315 00:22:15,074 --> 00:22:17,137 work. You have to overcome both the 316 00:22:17,137 --> 00:22:20,594 repelling force from the plus four and the repelling force 317 00:22:20,594 --> 00:22:23,566 from the plus one. But finally when you reach that 318 00:22:23,566 --> 00:22:27,084 point you can rest because there is no force on you at that 319 00:22:27,084 --> 00:22:29,207 point. That's what it means that the 320 00:22:29,207 --> 00:22:33,552 electric field is zero. It does not mean that you 321 00:22:33,552 --> 00:22:36,818 haven't done any work. So never confuse electric 322 00:22:36,818 --> 00:22:40,431 fields with potentials. I want to draw your attention 323 00:22:40,431 --> 00:22:43,835 to the fact that the green lines, the field lines, 324 00:22:43,835 --> 00:22:47,309 are everywhere perpendicular to the equipotentials. 325 00:22:47,309 --> 00:22:50,505 I will get back to that during my next lecture. 326 00:22:50,505 --> 00:22:53,84 That is not an accident. That is always the case. 327 00:22:53,84 --> 00:22:57,453 Now, Maxwell shows you something that is a little bit 328 00:22:57,453 --> 00:23:00,719 more complicated. Here, he calculated for us the 329 00:23:00,719 --> 00:23:05,736 equipotential surfaces, the red ones are the surfaces, 330 00:23:05,736 --> 00:23:09,929 again you have to rotate them about the vertical to make it 331 00:23:09,929 --> 00:23:13,183 three-dimensional, and now we have a minus one 332 00:23:13,183 --> 00:23:16,725 charge and a plus four. And so whenever it is red, 333 00:23:16,725 --> 00:23:19,472 the surface, the potential is positive, 334 00:23:19,472 --> 00:23:23,81 and whenever I have drawn it blue, the potential is negative. 335 00:23:23,81 --> 00:23:28,292 First, if we were very far away from both the plus four and the 336 00:23:28,292 --> 00:23:32,269 minus one, you expect to be looking at a charge which is 337 00:23:32,269 --> 00:23:36,1 effectively plus three. And so if you go very far away 338 00:23:36,1 --> 00:23:39,278 for sure the potential is 339 00:23:39,278 --> 00:23:43,544 everywhere positive and you expect them to be spherical 340 00:23:43,544 --> 00:23:46,23 again. If you look here you're very 341 00:23:46,23 --> 00:23:49,785 far away from the plus four and the minus one, 342 00:23:49,785 --> 00:23:53,34 indeed this has already the shape of a sphere. 343 00:23:53,34 --> 00:23:58,08 So that's clear that the plus four and the minus one far away 344 00:23:58,08 --> 00:24:02,188 behave like a plus three. If you're very close to the 345 00:24:02,188 --> 00:24:06,296 plus four, you get nice spheres around the plus four, 346 00:24:06,296 --> 00:24:10,96 positive potential, if you're very close to the 347 00:24:10,96 --> 00:24:14,827 minus one, notice that the blue surfaces are almost nice 348 00:24:14,827 --> 00:24:18,765 spheres, but now they're all negative because you're very 349 00:24:18,765 --> 00:24:22,07 close to the minus one. So a negative potential. 350 00:24:22,07 --> 00:24:25,867 There is here one surface which now has zero potential. 351 00:24:25,867 --> 00:24:30,156 It has to be because if you're negative potential close to the 352 00:24:30,156 --> 00:24:33,954 minus one and you have positive potential very far out, 353 00:24:33,954 --> 00:24:37,258 you got to go through a surface where it's zero. 354 00:24:37,258 --> 00:24:42,603 And so there is here a surface, I still have put it in blue, 355 00:24:42,603 --> 00:24:47,118 which is actually everywhere on this surface the potential is 356 00:24:47,118 --> 00:24:49,45 zero. Is the electric field zero 357 00:24:49,45 --> 00:24:51,031 there? Absolutely not. 358 00:24:51,031 --> 00:24:54,944 Electric field should not be confused with potential. 359 00:24:54,944 --> 00:24:59,007 What it means is that if you take a test charge in your 360 00:24:59,007 --> 00:25:02,996 pocket and you come from infinity and you walk to that 361 00:25:02,996 --> 00:25:07,134 surface, that by the time you have reached that surface, 362 00:25:07,134 --> 00:25:10,37 you've done zero work. That's what it means. 363 00:25:10,37 --> 00:25:13,51 That the potential is zero. 364 00:25:13,51 --> 00:25:17,357 There is here one point which we discussed earlier in my 365 00:25:17,357 --> 00:25:20,225 lectures where the electric field is zero. 366 00:25:20,225 --> 00:25:22,393 The potential is not zero there. 367 00:25:22,393 --> 00:25:25,26 The potential is definitely positive here. 368 00:25:25,26 --> 00:25:27,568 Because here was the zero surface. 369 00:25:27,568 --> 00:25:31,415 Here is already positive surface, and this is a positive 370 00:25:31,415 --> 00:25:34,003 surface. So the potential is positive. 371 00:25:34,003 --> 00:25:37,919 However, if you reach that point there's no force on your 372 00:25:37,919 --> 00:25:40,577 charge. So that means electric field is 373 00:25:40,577 --> 00:25:44,779 zero. And it's not so easy of course 374 00:25:44,779 --> 00:25:49,359 to calculate these surfaces. Maxwell was capable of doing 375 00:25:49,359 --> 00:25:53,939 that a hundred ten years ago. And nowadays we can do that 376 00:25:53,939 --> 00:25:58,437 very easily with computers. Equipotential surfaces which 377 00:25:58,437 --> 00:26:01,79 have different values can never intersect. 378 00:26:01,79 --> 00:26:06,125 Plus five volt surface can never intersect with a plus 379 00:26:06,125 --> 00:26:10,133 three or a minus one. And you think about why that 380 00:26:10,133 --> 00:26:11,359 is. Why that is, 381 00:26:11,359 --> 00:26:17,09 that would be a total violation of the conservation 382 00:26:17,09 --> 00:26:20,448 of energy. So equipotential surfaces, 383 00:26:20,448 --> 00:26:23,9 different values, can never intersect. 384 00:26:23,9 --> 00:26:27,351 All right. So you've seen that for the 385 00:26:27,351 --> 00:26:33,041 various charge configurations, the equipotential surfaces have 386 00:26:33,041 --> 00:26:38,544 very complicated shapes and cannot always be calculated in a 387 00:26:38,544 --> 00:26:42,555 very easy way. Now comes the question why do 388 00:26:42,555 --> 00:26:46,939 we introduce electric potentials, 389 00:26:46,939 --> 00:26:49,653 who needs them? And who needs equipotential 390 00:26:49,653 --> 00:26:52,109 surfaces? Isn't it true that if we know 391 00:26:52,109 --> 00:26:55,664 the electric field vectors everywhere in space that that 392 00:26:55,664 --> 00:26:59,477 determines uniquely how charges will move, what acceleration 393 00:26:59,477 --> 00:27:02,385 they will obtain, that means how their kinetic 394 00:27:02,385 --> 00:27:05,1 energy will change, and the answer is yeah, 395 00:27:05,1 --> 00:27:08,655 if you know the electric field everywhere in space sure. 396 00:27:08,655 --> 00:27:12,533 Then you can predict everything that happens with a charge in 397 00:27:12,533 --> 00:27:16,217 that field. But there are examples where 398 00:27:16,217 --> 00:27:20,528 the electric fields are so incredibly complicated that it 399 00:27:20,528 --> 00:27:24,223 is easier to work with equipotentials because the 400 00:27:24,223 --> 00:27:28,842 change in kinetic energy as I will discuss now really depends 401 00:27:28,842 --> 00:27:33,077 only on the change in the potential when you go from one 402 00:27:33,077 --> 00:27:36,541 point to another. So you will see very shortly 403 00:27:36,541 --> 00:27:41,16 that sometimes if you're only interested in change of kinetic 404 00:27:41,16 --> 00:27:45,625 energy and not necessarily interested in the details of the 405 00:27:45,625 --> 00:27:50,167 trajectory, then equipotentials come in very 406 00:27:50,167 --> 00:27:53,708 handy. Never confuse U which is 407 00:27:53,708 --> 00:28:00,08 electrostatic potential energy with V which is electric 408 00:28:00,08 --> 00:28:03,738 potential. This has unit joules. 409 00:28:03,738 --> 00:28:10,582 And this has unit joules per coulombs, which we call volts. 410 00:28:10,582 --> 00:28:16,836 If I have a collection of charges, pluses and minuses, 411 00:28:16,836 --> 00:28:22,465 U has only one value. It is the work that I have to 412 00:28:22,465 --> 00:28:26,043 do to put all these crazy charges exactly where they are. 413 00:28:26,043 --> 00:28:29,813 But the electric potential is different here from there from 414 00:28:29,813 --> 00:28:31,857 there to there to there to there. 415 00:28:31,857 --> 00:28:35,818 If you're very close to a plus charge, you can be sure that the 416 00:28:35,818 --> 00:28:39,14 potential is positive. If you're very close to a -- a 417 00:28:39,14 --> 00:28:41,696 negative charge, you can be sure that the 418 00:28:41,696 --> 00:28:44,763 potential is negative. But U has only one number. 419 00:28:44,763 --> 00:28:47,382 It's only one value. They're both scalars. 420 00:28:47,382 --> 00:28:49,426 Don't confuse one with the other. 421 00:28:49,426 --> 00:28:55 In a gravitational field, matter, like a piece of chalk, 422 00:28:55 --> 00:28:59,317 wants to go from high potential to low potential. 423 00:28:59,317 --> 00:29:03,903 If I just release it with zero speed, there it goes, 424 00:29:03,903 --> 00:29:06,691 high potential to low potential. 425 00:29:06,691 --> 00:29:11,368 In analogy, positive charges will also go from a high 426 00:29:11,368 --> 00:29:15,505 electric potential to a low electric potential. 427 00:29:15,505 --> 00:29:19,462 And of course this is unique for electricity, 428 00:29:19,462 --> 00:29:25,128 negative charges will go from a low potential to a high electric 429 00:29:25,128 --> 00:29:28,636 potential. Suppose I had a position A in 430 00:29:28,636 --> 00:29:35,527 space and I had another position B 431 00:29:35,527 --> 00:29:43,622 and I specify the potentials. So here we have A, 432 00:29:43,622 --> 00:29:51,545 potential is VA, and here we have point B where 433 00:29:51,545 --> 00:29:57,401 the potential is VB. By definition, 434 00:29:57,401 --> 00:30:04,943 the potential of VA as we discussed before is 435 00:30:04,943 --> 00:30:12,036 the integral -- by the way if these are separated by some 436 00:30:12,036 --> 00:30:16,595 random distance R, whatever you want. 437 00:30:16,595 --> 00:30:23,307 So the potential of A is defined as the integral going 438 00:30:23,307 --> 00:30:30,905 from A to infinity of E dot DR. That is the definition of the 439 00:30:30,905 --> 00:30:37,327 potential of A. There is an E here which is 440 00:30:37,327 --> 00:30:42,149 force per unit charge. So it is not work. 441 00:30:42,149 --> 00:30:49,744 If there were force DR it would be work but it is force per unit 442 00:30:49,744 --> 00:30:55,531 charge that makes it E. So the potential of B for 443 00:30:55,531 --> 00:31:02,403 definition is the integral from B to infinity of E dot DR. 444 00:31:02,403 --> 00:31:10,832 And so therefore the potential difference between point A and 445 00:31:10,832 --> 00:31:16,362 B, VA minus VB, equals the integral from A to B 446 00:31:16,362 --> 00:31:23,214 of E dot DR, and for reasons that I still don't understand 447 00:31:23,214 --> 00:31:29,225 after having been in this business for a long time, 448 00:31:29,225 --> 00:31:37,519 books will always tell you they reverse VA and B so they give 449 00:31:37,519 --> 00:31:42,399 you VB, VB minus VA. And then they say well we have 450 00:31:42,399 --> 00:31:46,986 to put a minus sign in front of the uh integral. 451 00:31:46,986 --> 00:31:51,866 It's the same thing. So books always give it to you 452 00:31:51,866 --> 00:31:55,77 in this form. But it is exactly the same. 453 00:31:55,77 --> 00:32:00,845 Hope you realize that. This is the two equations that 454 00:32:00,845 --> 00:32:05,92 I have here are the same. VA minus VB is the integral 455 00:32:05,92 --> 00:32:12,044 from A to B of E dot DR. If I flip this over then all I 456 00:32:12,044 --> 00:32:17,154 have to do is put a minus sign here and the two are identical. 457 00:32:17,154 --> 00:32:21,845 Notice that if there is no electric field between A and B 458 00:32:21,845 --> 00:32:25,112 they have the same potential, of course. 459 00:32:25,112 --> 00:32:29,719 Because when you march from A to B with a charge in your 460 00:32:29,719 --> 00:32:33,99 pocket no work is done. So the potential remains the 461 00:32:33,99 --> 00:32:36,587 same. I will change this DR to a 462 00:32:36,587 --> 00:32:39,351 different symbol, which I call DL. 463 00:32:39,351 --> 00:32:43,539 DR would mean that we go from A to 464 00:32:43,539 --> 00:32:49,628 infinity along this straight line and then we go from B to 465 00:32:49,628 --> 00:32:56,251 infinity along the straight line but it makes no difference how 466 00:32:56,251 --> 00:32:59,776 you go. If you go from A to B this 467 00:32:59,776 --> 00:33:06,399 potential difference and you go in this way then VA minus VB is 468 00:33:06,399 --> 00:33:11,74 not going to change. And so if now I introduce here 469 00:33:11,74 --> 00:33:16,7 a element DL, which is a small vector, 470 00:33:16,7 --> 00:33:22,79 and if the local E vector here is like so, at this point here, 471 00:33:22,79 --> 00:33:27,681 then VA minus VB is then the integral of E dot DL. 472 00:33:27,681 --> 00:33:33,672 In other words I can replace the R by an L and you may choose 473 00:33:33,672 --> 00:33:39,262 any path that you prefer. And that's the way that we will 474 00:33:39,262 --> 00:33:43,156 show you this equation most of the time. 475 00:33:43,156 --> 00:33:47,648 So it makes no difference how you 476 00:33:47,648 --> 00:33:53,354 march because we are dealing here with conservative fields. 477 00:33:53,354 --> 00:33:58,469 So let's now make the assumption that VA is a hundred 478 00:33:58,469 --> 00:34:02,305 fifty volts. And that VB for instance is 479 00:34:02,305 --> 00:34:05,748 fifty volts. So it's a very specific 480 00:34:05,748 --> 00:34:08,699 example. What does it mean now? 481 00:34:08,699 --> 00:34:14,502 It means that if I put plus Q charge in my pocket and I come 482 00:34:14,502 --> 00:34:21,092 all the way from Lobby seven and I walk up to point B. 483 00:34:21,092 --> 00:34:27,768 So Walter Lewin plus Q charge in his pocket goes from Lobby 484 00:34:27,768 --> 00:34:33,178 seven to point B, I have to do work and the work 485 00:34:33,178 --> 00:34:40,199 I have to do is the product of my charge Q with the potential. 486 00:34:40,199 --> 00:34:45,724 So that is Q the work I have to do is Q times VB. 487 00:34:45,724 --> 00:34:52,399 So in this case it's fifty times Q, whatever that charge is 488 00:34:52,399 --> 00:34:57,119 that I have in my pocket. This 489 00:34:57,119 --> 00:35:01,494 is in joules. Now, I go from Lobby seven to 490 00:35:01,494 --> 00:35:04,723 point A. I have to do more work. 491 00:35:04,723 --> 00:35:09,411 I have to do a hundred fifty Q joules of work. 492 00:35:09,411 --> 00:35:13,786 You can think of it I first come to A to B, 493 00:35:13,786 --> 00:35:19,099 I'm already exhausted, I have to put in another work 494 00:35:19,099 --> 00:35:25,141 to get all the way to point A. So you can imagine if I have 495 00:35:25,141 --> 00:35:31,927 this plus Q charge at point A, where there it's it's a higher 496 00:35:31,927 --> 00:35:35,693 potential, it wants to go back all by itself to B. 497 00:35:35,693 --> 00:35:40,229 It wants to go from a higher potential to a lower potential. 498 00:35:40,229 --> 00:35:43,226 Look, the E vector is in this direction. 499 00:35:43,226 --> 00:35:46,609 Positive charge will go to a lower potential. 500 00:35:46,609 --> 00:35:50,145 And as it moves from A to B energy is released. 501 00:35:50,145 --> 00:35:53,45 How much energy? Well, this is the amount of 502 00:35:53,45 --> 00:35:57,832 work I have done to get to A, this is the amount of work I 503 00:35:57,832 --> 00:36:03,787 did to get to B, and so if now the charge goes 504 00:36:03,787 --> 00:36:08,729 back from A to B, it's the difference that 505 00:36:08,729 --> 00:36:14,032 becomes available in terms of kinetic energy. 506 00:36:14,032 --> 00:36:18,01 It's a change in potential energy. 507 00:36:18,01 --> 00:36:24,277 And that change in potential energy, so the change in 508 00:36:24,277 --> 00:36:29,58 potential energy, when the plus Q charge goes 509 00:36:29,58 --> 00:36:35,245 from A to B, that change is Q times VA minus VB. 510 00:36:35,245 --> 00:36:40,341 QVB at point B and QVA at point A. 511 00:36:40,341 --> 00:36:44,538 So this is the potential energy that is in principle available 512 00:36:44,538 --> 00:36:48,804 if the charge moves from A to B. And you remember from eight oh 513 00:36:48,804 --> 00:36:52,656 one the work energy theorem. If we deal with conservative 514 00:36:52,656 --> 00:36:56,165 forces, then the sum of potential energy and kinetic 515 00:36:56,165 --> 00:36:59,743 energy of an object is the same. That's also true for 516 00:36:59,743 --> 00:37:02,22 gravitational forces. In other words, 517 00:37:02,22 --> 00:37:06,141 this difference in potential energy that becomes available 518 00:37:06,141 --> 00:37:10,2 like potential energy becomes available when I drop my chalk 519 00:37:10,2 --> 00:37:14,488 from a high potential to a low potential, 520 00:37:14,488 --> 00:37:17,176 that's converted to kinetic energy. 521 00:37:17,176 --> 00:37:22,078 So this difference now is also converted into kinetic energy of 522 00:37:22,078 --> 00:37:25,478 that moving charge. And so that would be the 523 00:37:25,478 --> 00:37:30,3 kinetic energy at point B minus the kinetic energy at point A. 524 00:37:30,3 --> 00:37:33,384 Which is really the work energy theorem. 525 00:37:33,384 --> 00:37:35,835 It's the conservation of energy. 526 00:37:35,835 --> 00:37:40,025 Now any piece of metal, no matter how crumby or dented 527 00:37:40,025 --> 00:37:45,223 it is, is an equipotential. As long as there is no charge 528 00:37:45,223 --> 00:37:48,635 moving inside the metal. And that's obvious that it's an 529 00:37:48,635 --> 00:37:51,24 equipotential. Because these charges inside 530 00:37:51,24 --> 00:37:54,341 the metal, these electrons, when they experience an 531 00:37:54,341 --> 00:37:57,132 electric field, they begin to move immediately 532 00:37:57,132 --> 00:38:00,357 in the electric field, and they will move until there 533 00:38:00,357 --> 00:38:03,583 is no force on them anymore, and that means they have 534 00:38:03,583 --> 00:38:06,064 effectively made the electric field zero. 535 00:38:06,064 --> 00:38:09,723 So charges inside the conductor always move automatically in 536 00:38:09,723 --> 00:38:13,197 such a way that they kill the electric 537 00:38:13,197 --> 00:38:16,33 field inside. If the electric field hadn't 538 00:38:16,33 --> 00:38:19,464 been zero yet, they would still be moving. 539 00:38:19,464 --> 00:38:23,821 And so each metal that you have, no matter where you bring 540 00:38:23,821 --> 00:38:27,795 it, as long as there are no electric currents inside, 541 00:38:27,795 --> 00:38:30,165 will always be an equipotential. 542 00:38:30,165 --> 00:38:34,751 So I can take a trash can and bring it into an external field 543 00:38:34,751 --> 00:38:39,49 and then very shortly after I've brought it in when things have 544 00:38:39,49 --> 00:38:43,465 calmed down, the trash can will be an 545 00:38:43,465 --> 00:38:48,403 equipotential and the electric field inside the metal will 546 00:38:48,403 --> 00:38:51,348 everywhere will everywhere be zero. 547 00:38:51,348 --> 00:38:55,939 So I could for instance attach point A to a trash can, 548 00:38:55,939 --> 00:39:00,011 metal trash can, so the whole trash can would be 549 00:39:00,011 --> 00:39:04,256 at a hundred fifty volts, and I could put point B, 550 00:39:04,256 --> 00:39:08,934 make it part of my -- of my soda, which is also made of 551 00:39:08,934 --> 00:39:12,053 metal. And so the whole soda would be 552 00:39:12,053 --> 00:39:16,65 at fifty volts and the entire trash can 553 00:39:16,65 --> 00:39:19,514 would be at a hundred fifty volts. 554 00:39:19,514 --> 00:39:24,115 I place the whole thing in vacuum and now I release an 555 00:39:24,115 --> 00:39:26,893 electron at point B. An electron. 556 00:39:26,893 --> 00:39:30,626 An electron wants to go to higher potential. 557 00:39:30,626 --> 00:39:35,835 A proton would go from A to B, electron wants to go from B to 558 00:39:35,835 --> 00:39:38,699 A. And so now energy is available. 559 00:39:38,699 --> 00:39:43,734 The electric potential energy is available and the electron 560 00:39:43,734 --> 00:39:47,35 will start to pick up speed and 561 00:39:47,35 --> 00:39:50,966 finally end up at A. Now how it will travel I don't 562 00:39:50,966 --> 00:39:52,629 know. The electric field 563 00:39:52,629 --> 00:39:55,449 configuration is enormously complicated. 564 00:39:55,449 --> 00:39:57,907 Between the can and this trash can. 565 00:39:57,907 --> 00:40:01,522 Amazingly complicated. If you were to see the field 566 00:40:01,522 --> 00:40:05,354 lines it would be weird. But if we all we want to know 567 00:40:05,354 --> 00:40:08,825 is what the kinetic energy is, what the speed is, 568 00:40:08,825 --> 00:40:11,717 with which this electron reaches the can, 569 00:40:11,717 --> 00:40:15,405 so what? Then we can use the work 570 00:40:15,405 --> 00:40:19,887 energy theorem and find out immediately what that kinetic 571 00:40:19,887 --> 00:40:23,17 energy is. Because the available potential 572 00:40:23,17 --> 00:40:27,573 energy is the charge of the electron times the potential 573 00:40:27,573 --> 00:40:30,454 difference between these two objects. 574 00:40:30,454 --> 00:40:35,258 Well the charge of the electron is one point six times ten to 575 00:40:35,258 --> 00:40:39,821 the minus nineteen coulombs. The potential difference is a 576 00:40:39,821 --> 00:40:43,263 hundred volts. And that is the difference in 577 00:40:43,263 --> 00:40:47,611 kinetic energy. If I assume that I release the 578 00:40:47,611 --> 00:40:50,855 electron at zero speed, then I have immediately the 579 00:40:50,855 --> 00:40:54,683 kinetic energy that it has at point A which is one-half M of 580 00:40:54,683 --> 00:40:57,344 the electron times the speed at A squared. 581 00:40:57,344 --> 00:41:00,847 So now you see that accepting the fact that we know the 582 00:41:00,847 --> 00:41:03,702 equipotentials, we can very quickly calculate 583 00:41:03,702 --> 00:41:07,466 the kinetic energy and therefore the speed of the electron, 584 00:41:07,466 --> 00:41:10,58 as they arrive at A, without any knowledge of the 585 00:41:10,58 --> 00:41:14,213 complicated electric field. If you put in the numbers for 586 00:41:14,213 --> 00:41:19,698 the mass of the electron, then, which is nine times ten 587 00:41:19,698 --> 00:41:24,808 to the minus thirty-one kilograms, then you'll find that 588 00:41:24,808 --> 00:41:29,732 this speed is about two percent of the speed of light. 589 00:41:29,732 --> 00:41:33,356 A substantial speed. All our potentials, 590 00:41:33,356 --> 00:41:37,351 electric potentials, are defined relative to 591 00:41:37,351 --> 00:41:41,067 infinity. That means at infinity they are 592 00:41:41,067 --> 00:41:44,412 zero. That is because of the one over 593 00:41:44,412 --> 00:41:49,214 R relationship. That's very nice and dandy and 594 00:41:49,214 --> 00:41:51,884 it works. However, there are situations 595 00:41:51,884 --> 00:41:56,169 whereby it really doesn't matter where you think of your zero. 596 00:41:56,169 --> 00:41:59,541 Remember with gravity we had a similar situation. 597 00:41:59,541 --> 00:42:03,686 With gravity we always worried about difference in potential 598 00:42:03,686 --> 00:42:07,34 energy but sometimes we call this zero and this plus. 599 00:42:07,34 --> 00:42:10,361 Sometimes you call this plus and this minus. 600 00:42:10,361 --> 00:42:14,084 It doesn't really matter because the change in kinetic 601 00:42:14,084 --> 00:42:18,088 energy is dictated only by the difference 602 00:42:18,088 --> 00:42:21,182 in potentials. So it is very nice and dandy to 603 00:42:21,182 --> 00:42:25,581 call that a hundred fifty and to call that fifty but you wouldn't 604 00:42:25,581 --> 00:42:29,775 have found any different answer for the electron if you called 605 00:42:29,775 --> 00:42:34,105 this potential one hundred volts and you called this one zero or 606 00:42:34,105 --> 00:42:38,024 you called this one zero and this one minus one hundred or 607 00:42:38,024 --> 00:42:41,461 you called this one fifty and this one minus fifty. 608 00:42:41,461 --> 00:42:45,035 So the behavior of the electrons of the charges would 609 00:42:45,035 --> 00:42:49,228 of course not change. And of course electrical 610 00:42:49,228 --> 00:42:53,986 engineers would always per definition call the potential of 611 00:42:53,986 --> 00:42:57,677 the earth zero when they built their circuits. 612 00:42:57,677 --> 00:43:01,696 So now I would like to demonstrate to you with the 613 00:43:01,696 --> 00:43:06,535 Vandegraaff that if you get a strong electric field from the 614 00:43:06,535 --> 00:43:11,211 radially outwards from the Vandegraaff that you get a huge 615 00:43:11,211 --> 00:43:15,968 potential difference between this point here and this point 616 00:43:15,968 --> 00:43:18,839 there. Uh if I have my numbers still 617 00:43:18,839 --> 00:43:21,399 there, I hope I do, 618 00:43:21,399 --> 00:43:23,912 there they are, at the surface of the 619 00:43:23,912 --> 00:43:27,193 Vandegraaff which takes about ten microcoulombs, 620 00:43:27,193 --> 00:43:30,683 it will be three hundred thousand volts right here, 621 00:43:30,683 --> 00:43:33,964 here it would be a hundred fifty thousand volts, 622 00:43:33,964 --> 00:43:37,803 and here three meters from the center, it's about thirty 623 00:43:37,803 --> 00:43:40,526 kilovolts. So that means that if I place 624 00:43:40,526 --> 00:43:44,854 this fluorescent tube into that electric field that there would 625 00:43:44,854 --> 00:43:48,344 be a gigantic potential difference between here and 626 00:43:48,344 --> 00:43:50,997 there provided that I hold it radially. 627 00:43:50,997 --> 00:43:56,101 If I hold it like this then the potential difference 628 00:43:56,101 --> 00:43:59,548 between here and there would be zero of course, 629 00:43:59,548 --> 00:44:03,743 if I hold it tangentially, they would be both at the same 630 00:44:03,743 --> 00:44:07,34 electric potential. But when I hold them radially 631 00:44:07,34 --> 00:44:11,685 you will see perhaps that this fluorescent tube will show a 632 00:44:11,685 --> 00:44:15,206 little bit of light. Once you see light it means 633 00:44:15,206 --> 00:44:18,353 that electrons are moving through that gas. 634 00:44:18,353 --> 00:44:22,399 It means charge is moving. We haven't discussed current 635 00:44:22,399 --> 00:44:25,798 yet, but that's what it means. 636 00:44:25,798 --> 00:44:29,371 A current is flowing. And this current has to be 637 00:44:29,371 --> 00:44:34,161 delivered by the Vandegraaff and the Vandegraaff is only capable 638 00:44:34,161 --> 00:44:36,669 of providing very modest currents. 639 00:44:36,669 --> 00:44:39,786 So you're not going to see a lot of light. 640 00:44:39,786 --> 00:44:43,663 But I want to show you that you will see some light. 641 00:44:43,663 --> 00:44:45,792 No wires attached. Just here. 642 00:44:45,792 --> 00:44:50,125 And then I will rotate it tangentially and you will see no 643 00:44:50,125 --> 00:44:53,318 light at all. So if we can make it a little 644 00:44:53,318 --> 00:44:57,423 darker as a start and I'll start the 645 00:44:57,423 --> 00:45:02,565 Vandegraaff and then if Marcos comes to make it completely dark 646 00:45:02,565 --> 00:45:06,296 when necessary, because the light is so little 647 00:45:06,296 --> 00:45:10,111 that we really have to make it completely dark. 648 00:45:10,111 --> 00:45:14,755 I will put on a glove for safety reasons although I don't 649 00:45:14,755 --> 00:45:19,647 think it will do me much good. Notice I have here a piece of 650 00:45:19,647 --> 00:45:23,213 glass to well, to be well-insulated from the 651 00:45:23,213 --> 00:45:27,027 glass so that I don't mess up the 652 00:45:27,027 --> 00:45:32,08 demonstration by if I hold my fingers here it will be very 653 00:45:32,08 --> 00:45:35,27 different than holding my hands here. 654 00:45:35,27 --> 00:45:40,766 So let's go first close without -- with the lights still on and 655 00:45:40,766 --> 00:45:46,172 then OK why don't you turn the lights off now all the way off. 656 00:45:46,172 --> 00:45:49,186 OK I -- I think you can see a glow. 657 00:45:49,186 --> 00:45:53,794 It's radially outwards now. And Marcos can you give a 658 00:45:53,794 --> 00:45:57,428 little light? OK I will now go tangential, 659 00:45:57,428 --> 00:46:03,761 can you turn uh the lights off? And now you see nothing, 660 00:46:03,761 --> 00:46:07,333 very little. And now I go radial again. 661 00:46:07,333 --> 00:46:11,281 And there you go. Now if I -- if I'm crazy, 662 00:46:11,281 --> 00:46:15,511 if I were crazy, then I would touch the end of 663 00:46:15,511 --> 00:46:21,057 this tube with my finger thereby allowing this current to go 664 00:46:21,057 --> 00:46:26,603 straight through my body to the earth which may increase the 665 00:46:26,603 --> 00:46:28,671 light. Let me try that. 666 00:46:28,671 --> 00:46:33,865 So -- so I'm going to touch the -- the -- 667 00:46:33,865 --> 00:46:39,268 the -- this -- this fluorescent tube on your right side. 668 00:46:39,268 --> 00:46:39,858 Ah. Ah. 669 00:46:39,858 --> 00:46:43,1 Ah. Every time I -- I touch it ah. 670 00:46:43,1 --> 00:46:47,815 But that's not ah. But you see every time I touch 671 00:46:47,815 --> 00:46:53,611 it I make it easier for the current to flow and you see very 672 00:46:53,611 --> 00:46:58,621 clearly that it lights up. Now I want to do the same 673 00:46:58,621 --> 00:47:05,4 demonstration with a neon flash tube and the neon flash 674 00:47:05,4 --> 00:47:08,524 tube I will place at the end of a fishing rod. 675 00:47:08,524 --> 00:47:12,62 This neon flash tube we used during the first lecture when I 676 00:47:12,62 --> 00:47:16,994 was beating up students but I've learned not to do that anymore. 677 00:47:16,994 --> 00:47:20,883 Um this takes um several kilovolts to get a little bit of 678 00:47:20,883 --> 00:47:24,076 light out of it from one side to the other, oh, 679 00:47:24,076 --> 00:47:27,687 that's duck soup for the Vandegraaff, you know you're 680 00:47:27,687 --> 00:47:30,811 talking about hundreds and thousands of volts, 681 00:47:30,811 --> 00:47:34,838 and so here I will actually start spinning it and then when 682 00:47:34,838 --> 00:47:42,348 it is radially inwards maybe you will see light and when it 683 00:47:42,348 --> 00:47:51,264 is tangential you won't see much light and then if I feel very 684 00:47:51,264 --> 00:47:59,45 good I will do that again. OK uh so Marcos if you make it 685 00:47:59,45 --> 00:48:06,466 uh dark I'll give it a twist. OK, radial, radial, 686 00:48:06,466 --> 00:48:10,61 radial, radial, radial, 687 00:48:10,61 --> 00:48:14,512 radial, radial, radial, radial, 688 00:48:14,512 --> 00:48:18,805 OK. Now I ah OK I touched it now I 689 00:48:18,805 --> 00:48:23,488 touch it again. And I touch it again. 690 00:48:23,488 --> 00:48:26,09 And again. And again. 691 00:48:26,09 --> 00:48:30,253 Ah. You see every time I touch it 692 00:48:30,253 --> 00:48:35,586 it lights me. And it gives a nice flash of 693 00:48:35,586 --> 00:48:39,879 light. So you see here in front of 694 00:48:39,879 --> 00:48:46,475 your eyes without any wires attached 695 00:48:46,475 --> 00:48:53,947 that the potential difference created by the electric field 696 00:48:53,947 --> 00:49:01,033 that those potential differences make these lights work. 697 00:49:01,033 --> 49:06 All right, see you Friday.