1 00:00:00 --> 00:00:03,138 2 00:00:03,138 --> 00:00:08,508 You see here the topics the way I see them, you will get three 3 00:00:08,508 --> 00:00:13,086 problems to -- on the exam, and not all subjects can, 4 00:00:13,086 --> 00:00:16,344 of course, be represented on the exam. 5 00:00:16,344 --> 00:00:20,218 Nor can I cover all of them in fifty minutes. 6 00:00:20,218 --> 00:00:25,324 I will test some very basic ideas, the math will be utterly 7 00:00:25,324 --> 00:00:30,343 trivial, and if it becomes complicated, then you just know 8 00:00:30,343 --> 00:00:35,802 that you're on the wrong track. If you get stuck, 9 00:00:35,802 --> 00:00:39,249 somehow, on a problem, my advice is, 10 00:00:39,249 --> 00:00:44,764 move on, don't stay with the problem, but move on and try 11 00:00:44,764 --> 00:00:49,294 some others first. There is a reason why Gauss' 12 00:00:49,294 --> 00:00:53,431 Law there is in red, because Gauss' Law is, 13 00:00:53,431 --> 00:00:59,537 of course, extremely important in the early part of the course, 14 00:00:59,537 --> 00:01:05,151 the closed surface integral of E dot D A is the sum of the 15 00:01:05,151 --> 00:01:10,113 enclosed charged, divided by epsilon zero. 16 00:01:10,113 --> 00:01:15,133 And that is so important that you can be sure that there will 17 00:01:15,133 --> 00:01:18,313 be one problem dealing with Gauss' Law. 18 00:01:18,313 --> 00:01:23,333 Now, when you have Gauss' Law problems, there's always one of 19 00:01:23,333 --> 00:01:25,759 three. You must have symmetry, 20 00:01:25,759 --> 00:01:29,691 you must have a special distribution of charges, 21 00:01:29,691 --> 00:01:33,373 because otherwise, Gauss' Law doesn't get you 22 00:01:33,373 --> 00:01:36,636 anywhere. So we have spherical symmetry, 23 00:01:36,636 --> 00:01:40,087 we have cylindrical symmetry, 24 00:01:40,087 --> 00:01:43,902 and we have plain symmetry, and that's all there is. 25 00:01:43,902 --> 00:01:46,969 So you're going to get one of those three. 26 00:01:46,969 --> 00:01:49,438 I will do one now, you may choose. 27 00:01:49,438 --> 00:01:53,029 We're going to have a vote. One is a possibility, 28 00:01:53,029 --> 00:01:57,667 I do one on spherical symmetry, another one I do on cylindrical 29 00:01:57,667 --> 00:02:00,585 symmetry, or I do one on slab symmetry. 30 00:02:00,585 --> 00:02:02,979 Who wants the spherical symmetry? 31 00:02:02,979 --> 00:02:05,298 Hands. Who wants the cylindrical 32 00:02:05,298 --> 00:02:07,094 symmetry? Way more hands. 33 00:02:07,094 --> 00:02:11,209 Who wants plain? I think the 34 00:02:11,209 --> 00:02:14,55 cylinders have it. But if you're clever, 35 00:02:14,55 --> 00:02:19,435 you can stay for the next lecture, and then you can try to 36 00:02:19,435 --> 00:02:23,291 get the other one. We need a little bit of fun 37 00:02:23,291 --> 00:02:25,004 today. And therefore, 38 00:02:25,004 --> 00:02:29,717 I want to introduce you first to something very special, 39 00:02:29,717 --> 00:02:33,659 which is close to my heart, it is a secret top, 40 00:02:33,659 --> 00:02:37,858 you're going to see it there, and that secret top, 41 00:02:37,858 --> 00:02:41,714 I'm going to spin, and if you're a believer in 42 00:02:41,714 --> 00:02:46,306 eight oh one, which you should by now, 43 00:02:46,306 --> 00:02:51,527 then we will -- should be able to predict that that stop cannot 44 00:02:51,527 --> 00:02:55,653 spin for very long, there is friction with the air 45 00:02:55,653 --> 00:03:00,79 and friction with the surface, and so chances are it will soon 46 00:03:00,79 --> 00:03:04,159 fall over. We'll take a look at it later, 47 00:03:04,159 --> 00:03:07,022 again. So let's now start our first 48 00:03:07,022 --> 00:03:10,643 problem, and that is a cylindrical symmetry. 49 00:03:10,643 --> 00:03:16,454 Well, we have a cylinder -- and here is the cylinder -- 50 00:03:16,454 --> 00:03:19,536 it's very long, has radius R, 51 00:03:19,536 --> 00:03:25,261 and I have uniform charge distribution throughout the 52 00:03:25,261 --> 00:03:30,325 whole cylinder, and the density is rho Coulombs 53 00:03:30,325 --> 00:03:35,279 per cubic meter. Uniformly distributed through 54 00:03:35,279 --> 00:03:39,242 the cylinder. I want to know what the 55 00:03:39,242 --> 00:03:44,967 electric field inside the cylinder is and outside the 56 00:03:44,967 --> 00:03:48,82 cylinder. Let's first do outside the 57 00:03:48,82 --> 00:03:53,272 cylinder. The gauss surface, 58 00:03:53,272 --> 00:03:59,503 clearly, is going to be itself a cylinder, there it goes -- you 59 00:03:59,503 --> 00:04:05,333 can give it any random length, L, cannot have any effect on 60 00:04:05,333 --> 00:04:11,564 the answer -- and so the end if flat, perpendicular to the axis 61 00:04:11,564 --> 00:04:17,595 of symmetry, and this front part is flat, and this is curved. 62 00:04:17,595 --> 00:04:23,684 I give this a radius little R, and so I know that everywhere 63 00:04:23,684 --> 00:04:28,176 on the surface of that cylinder outside, that the electric field 64 00:04:28,176 --> 00:04:32,027 must be the same everywhere because the distance is the 65 00:04:32,027 --> 00:04:34,452 same, that's the symmetry argument. 66 00:04:34,452 --> 00:04:38,66 Electric field cannot be any stronger here than it is there, 67 00:04:38,66 --> 00:04:42,368 if I'm on that surface. Symmetry argument number one. 68 00:04:42,368 --> 00:04:46,576 Symmetry argument number two is, given the fact that this is 69 00:04:46,576 --> 00:04:50,997 a cylinder, the electric field must everywhere be perpendicular 70 00:04:50,997 --> 00:04:54,562 to this axis, coming out -- I call it 71 00:04:54,562 --> 00:04:57,482 radially, but, of course, it is not radially, 72 00:04:57,482 --> 00:05:01,264 like a sphere -- it's radially coming out of this surface, 73 00:05:01,264 --> 00:05:04,25 always perpendicular to this axis of symmetry. 74 00:05:04,25 --> 00:05:06,705 Nature could not decide any other way. 75 00:05:06,705 --> 00:05:09,028 That's the second symmetry argument. 76 00:05:09,028 --> 00:05:12,279 One you recognize that argument, the electric flux 77 00:05:12,279 --> 00:05:16,195 through this flat surface and through that flat surface must 78 00:05:16,195 --> 00:05:17,588 be zero. Because then, 79 00:05:17,588 --> 00:05:20,442 the electric field and the local D A vector, 80 00:05:20,442 --> 00:05:23,959 which is the perpendicular to the 81 00:05:23,959 --> 00:05:27,579 surface, make angles of ninety degrees with each other, 82 00:05:27,579 --> 00:05:30,999 because E would be like this here, but D A is in the 83 00:05:30,999 --> 00:05:33,212 direction of the axis of symmetry. 84 00:05:33,212 --> 00:05:36,296 So no flux can, therefore, get out here and get 85 00:05:36,296 --> 00:05:38,777 out here. But only through this curved 86 00:05:38,777 --> 00:05:41,124 surface. But on this curved surface, 87 00:05:41,124 --> 00:05:44,879 if it is a positive charge, then the E vector and the D A 88 00:05:44,879 --> 00:05:48,433 are in the same direction, if it is a negative charge, 89 00:05:48,433 --> 00:05:50,512 they are in opposite directions. 90 00:05:50,512 --> 00:05:53,999 Later, you can change the sign of 91 00:05:53,999 --> 00:05:56,738 rho, let's just make it positive for now. 92 00:05:56,738 --> 00:06:00,848 So if, now, I apply Gauss' Law, then I only have to take this 93 00:06:00,848 --> 00:06:04,204 surface into account and not these two end pieces. 94 00:06:04,204 --> 00:06:07,491 And so I need to know, now, what this surface is, 95 00:06:07,491 --> 00:06:11,669 because E and D A are always in the same direction everywhere, 96 00:06:11,669 --> 00:06:15,299 thus the cosine of the angle between them is plus one. 97 00:06:15,299 --> 00:06:17,423 And so what is the surface area? 98 00:06:17,423 --> 00:06:21,258 That is going to be L times two pi little R, and then the 99 00:06:21,258 --> 00:06:25,854 electric vector is everywhere, the same there, 100 00:06:25,854 --> 00:06:30,494 this was our symmetry argument, and that is now the charge 101 00:06:30,494 --> 00:06:34,158 inside this cylinder, divided by epsilon zero. 102 00:06:34,158 --> 00:06:37,903 But, of course, the charge inside the cylinder, 103 00:06:37,903 --> 00:06:42,3 that's only the portion that is in this inner cylinder, 104 00:06:42,3 --> 00:06:45,23 and so that has also, then, length L. 105 00:06:45,23 --> 00:06:50,115 The cross-section here is pi r squared, so this is the volume 106 00:06:50,115 --> 00:06:55,245 of the volume of the charge that I have inside by 107 00:06:55,245 --> 00:06:58,803 Gaussian surface, I must multiply by rho, 108 00:06:58,803 --> 00:07:03,606 that gives in the charge, and I divide by epsilon zero. 109 00:07:03,606 --> 00:07:06,097 And of course, the L cancels, 110 00:07:06,097 --> 00:07:09,833 as it always does, and the pi cancels here, 111 00:07:09,833 --> 00:07:14,992 too, and so I find that the electric field equals r squared 112 00:07:14,992 --> 00:07:20,508 times rho divided by two epsilon zero r, and if you want to see 113 00:07:20,508 --> 00:07:24,333 it vectorially, you can put an r roof there, 114 00:07:24,333 --> 00:07:28,425 r roof, then, would be a vector 115 00:07:28,425 --> 00:07:33,742 which is perpendicular to the axis -- I mentioned earlier, 116 00:07:33,742 --> 00:07:36,633 I called that radially outwards. 117 00:07:36,633 --> 00:07:41,297 So this is the electric field outside the cylinder. 118 00:07:41,297 --> 00:07:44,282 R squared rho two epsilon zero r. 119 00:07:44,282 --> 00:07:49,786 So it falls off as one over r. So now I want to know what it 120 00:07:49,786 --> 00:07:53,517 is inside the cylinder. So now I go to r, 121 00:07:53,517 --> 00:07:58,088 less than equal to r. So it's clear that what I do 122 00:07:58,088 --> 00:08:02,291 now, I'm going to have a Gaussian 123 00:08:02,291 --> 00:08:05,328 surface which, again, is a cylinder, 124 00:08:05,328 --> 00:08:08,018 has length L, and it has, again, 125 00:08:08,018 --> 00:08:12,618 two flat pieces at the end, so no flux will go through 126 00:08:12,618 --> 00:08:16,697 those two pieces, so my first term of Gauss' Law 127 00:08:16,697 --> 00:08:21,47 is going to be the same, I have L times two pi little R, 128 00:08:21,47 --> 00:08:25,723 because the radius of this inner circle is also r, 129 00:08:25,723 --> 00:08:30,322 L two pi r times the electric field, the arguments are 130 00:08:30,322 --> 00:08:35,401 identical -- but now, there is less charge 131 00:08:35,401 --> 00:08:39,727 inside by Gaussian surface. Uh, the volume is L, 132 00:08:39,727 --> 00:08:45,158 now times pi little R squared, and then I get rho to convert 133 00:08:45,158 --> 00:08:48,563 it to charge, divided by epsilon zero. 134 00:08:48,563 --> 00:08:51,049 I lose my L, as I always do, 135 00:08:51,049 --> 00:08:56,48 my pi goes, and so now I get E equals -- there is an r here, 136 00:08:56,48 --> 00:09:02,187 and there is an R squared here, an so I only end up with one R, 137 00:09:02,187 --> 00:09:08,201 divide by two epsilon zero, and if you like that vector 138 00:09:08,201 --> 00:09:10,857 notation, you can always do this. 139 00:09:10,857 --> 00:09:13,763 And of course, if rho were negative, 140 00:09:13,763 --> 00:09:17,83 then automatically, you see, if you put a negative 141 00:09:17,83 --> 00:09:22,062 charge density in here, then the E field flips over, 142 00:09:22,062 --> 00:09:26,628 so that's automatically taken into account both here and 143 00:09:26,628 --> 00:09:29,366 there. So let's take a look at it, 144 00:09:29,366 --> 00:09:33,682 I'm quite happy with that. If you substitute little R 145 00:09:33,682 --> 00:09:38,566 equals capital R, you are right at the 146 00:09:38,566 --> 00:09:44,089 surface of your cylinder, then you get the same answer in 147 00:09:44,089 --> 00:09:46,456 both cases. Substitute R, 148 00:09:46,456 --> 00:09:50,796 capital R in here, then the magnitude of E -- 149 00:09:50,796 --> 00:09:56,911 don't worry about the direction now -- is rho capital R divided 150 00:09:56,911 --> 00:10:01,941 by two epsilon zero, and if you put in here for this 151 00:10:01,941 --> 00:10:06,38 little R, capital R, you find exactly the same 152 00:10:06,38 --> 00:10:09,832 answer. So we can now make a plot of 153 00:10:09,832 --> 00:10:15,159 the electric field as a function of 154 00:10:15,159 --> 00:10:21,845 distance R, here being capital R, and here being the electric 155 00:10:21,845 --> 00:10:25,633 field strength. It's a linear line, 156 00:10:25,633 --> 00:10:30,312 zero here, it goes up to a certain maximum, 157 00:10:30,312 --> 00:10:34,212 and then it falls off as one over R. 158 00:10:34,212 --> 00:10:37,889 And this value here is this value. 159 00:10:37,889 --> 00:10:43,125 That's where little R is capital R. 160 00:10:43,125 --> 00:10:46,444 It is obvious and pleasing that the electric field, 161 00:10:46,444 --> 00:10:49,231 on the axis itself, where little r is zero, 162 00:10:49,231 --> 00:10:53,147 that that electric field is zero, that is something that you 163 00:10:53,147 --> 00:10:56,399 could have predicted almost without any knowledge, 164 00:10:56,399 --> 00:11:00,248 because you have symmetry all around it, there is charge on 165 00:11:00,248 --> 00:11:03,5 the left, there is charge on the -- one the right, 166 00:11:03,5 --> 00:11:07,283 there's charge on north and south, and the electric fields 167 00:11:07,283 --> 00:11:10,602 right at the center, of course, all pair each other 168 00:11:10,602 --> 00:11:13,588 [unintelligible], so you get no electric field 169 00:11:13,588 --> 00:11:17,37 right at the center. If the charge, 170 00:11:17,37 --> 00:11:20,865 for some reason, would all be at the outer 171 00:11:20,865 --> 00:11:25,468 surface, if it were a solid conductor that would be the 172 00:11:25,468 --> 00:11:30,668 case, then the electric field would be zero everywhere inside, 173 00:11:30,668 --> 00:11:34,249 and this would be unchanged, assuming then, 174 00:11:34,249 --> 00:11:39,108 that you have the same amount of charge on the outside per 175 00:11:39,108 --> 00:11:42,603 unit length as you now have on the inside. 176 00:11:42,603 --> 00:11:45,246 So that is cylindrical symmetry. 177 00:11:45,246 --> 00:11:49,678 I am dying to take a look at my top. 178 00:11:49,678 --> 00:11:55,283 I am really -- and much to my shock, do I see that it stopped? 179 00:11:55,283 --> 00:11:59,802 It's still rotating. So maybe I have to come to the 180 00:11:59,802 --> 00:12:04,593 conclusion that there is something wrong with eight oh 181 00:12:04,593 --> 00:12:07,485 one. There must be a layer deeper 182 00:12:07,485 --> 00:12:11,733 than eight oh one -- there is friction, and yet, 183 00:12:11,733 --> 00:12:14,535 this top doesn't come to a halt. 184 00:12:14,535 --> 00:12:19,868 And so that layer deeper -- maybe that layer deeper is eight 185 00:12:19,868 --> 00:12:24,798 oh two. Give that some thought, 186 00:12:24,798 --> 00:12:28,311 it may add to your sleepless nights. 187 00:12:28,311 --> 00:12:33,53 We'll visit it later, because maybe it will come to a 188 00:12:33,53 --> 00:12:35,035 stop. Very well. 189 00:12:35,035 --> 00:12:38,749 Let's now do something very different. 190 00:12:38,749 --> 00:12:43,968 I have two conducting flat plates for the plane plate 191 00:12:43,968 --> 00:12:49,89 capacitor, and it has a certain thickness, the material is a 192 00:12:49,89 --> 00:12:55,812 conducting material, has a certain thickness, 193 00:12:55,812 --> 00:13:00,762 it's very large in size. Way bigger is the plane 194 00:13:00,762 --> 00:13:06,555 linearly than the separation. And let this separation be 195 00:13:06,555 --> 00:13:13,191 little d, and I charge the upper plate with a positive charge so 196 00:13:13,191 --> 00:13:17,826 I get here surface charge density plus sigma, 197 00:13:17,826 --> 00:13:22,776 and here minus sigma, area is A, of both plates. 198 00:13:22,776 --> 00:13:29,938 I know that in the conductor itself, there is no current, 199 00:13:29,938 --> 00:13:33,153 and therefore, in the conductor itself, 200 00:13:33,153 --> 00:13:35,775 the electric field must be zero. 201 00:13:35,775 --> 00:13:39,751 I have an electric field between the two plates, 202 00:13:39,751 --> 00:13:44,572 which can be derived using Gauss' Law, but we have already 203 00:13:44,572 --> 00:13:49,309 used up our time on Gauss' Law, which is sigma divided by 204 00:13:49,309 --> 00:13:52,947 epsilon zero. And the electric outside these 205 00:13:52,947 --> 00:13:58,022 two plates is very close to zero here, and also very close to 206 00:13:58,022 --> 00:14:01,538 zero here. We find that from the 207 00:14:01,538 --> 00:14:04,54 superposition principle -- because this plate, 208 00:14:04,54 --> 00:14:08,41 which is negatively charged, will add to the electric field 209 00:14:08,41 --> 00:14:11,213 pointing down, and as you perhaps remember, 210 00:14:11,213 --> 00:14:13,615 that that is independent of distance. 211 00:14:13,615 --> 00:14:17,818 Well, provided that you are not so far away from the plates that 212 00:14:17,818 --> 00:14:21,088 the dimension of the plates is going to interfere. 213 00:14:21,088 --> 00:14:24,958 The -- if the plate is ten by ten meters, then it's fine as 214 00:14:24,958 --> 00:14:27,693 long as you are, say, within a few meters. 215 00:14:27,693 --> 00:14:31,296 But it you go a hundred meters away, then it's not true 216 00:14:31,296 --> 00:14:35,164 anymore. So if we assume that we don't 217 00:14:35,164 --> 00:14:38,267 go too far out, then the electric field due to 218 00:14:38,267 --> 00:14:41,991 this one pointing down, due to this one is pointing up, 219 00:14:41,991 --> 00:14:45,576 and they have equal strength, they are independent of 220 00:14:45,576 --> 00:14:49,783 distance, so they cancel each other here, and they cancel each 221 00:14:49,783 --> 00:14:52,196 other there, superposition argument. 222 00:14:52,196 --> 00:14:56,402 Let this point be P inside the conductor, and this point be S, 223 00:14:56,402 --> 00:14:59,781 and the first thing I would ask you, for instance, 224 00:14:59,781 --> 00:15:03,504 is what is V P minus V S? It's the potential difference 225 00:15:03,504 --> 00:15:06,411 over this capacitor, 226 00:15:06,411 --> 00:15:09,56 if you want to call it a capacitor. 227 00:15:09,56 --> 00:15:13,451 That is the integral, in going from P to S, 228 00:15:13,451 --> 00:15:17,341 of E dot D L. For reasons that I have never 229 00:15:17,341 --> 00:15:21,602 understood, your book will switch these around, 230 00:15:21,602 --> 00:15:25,863 and put here a minus sign, which is, of course, 231 00:15:25,863 --> 00:15:30,587 exactly the same thing. And so now, we can calculate 232 00:15:30,587 --> 00:15:34,107 the potential difference. If I am here, 233 00:15:34,107 --> 00:15:39,863 and I'm going to walk down -- suppose I walk down 234 00:15:39,863 --> 00:15:44,374 [unintelligible] a straight line, do D L is in the same 235 00:15:44,374 --> 00:15:48,05 direction as E, then it's immediately obvious 236 00:15:48,05 --> 00:15:53,147 that this is simply E times that distance D, because E and D L 237 00:15:53,147 --> 00:15:57,491 are in the same direction. So the cosine of the angle 238 00:15:57,491 --> 00:16:02,002 between them is plus one. And so I find that this is E, 239 00:16:02,002 --> 00:16:05,595 which is, um, sigma divided by epsilon zero, 240 00:16:05,595 --> 00:16:11,435 times that distance D. If I had chosen another route, 241 00:16:11,435 --> 00:16:16,724 I would have found the same answer, because we're dealing 242 00:16:16,724 --> 00:16:22,201 here with conservative fields, so the path does not matter. 243 00:16:22,201 --> 00:16:26,922 So as long as you go from this plate to this plate, 244 00:16:26,922 --> 00:16:30,038 that integral is always E times D. 245 00:16:30,038 --> 00:16:34,854 What is the potential difference between point P and 246 00:16:34,854 --> 00:16:37,404 point T here? V P minus V T, 247 00:16:37,404 --> 00:16:43,468 it's clear that that is zero, because there is no electric 248 00:16:43,468 --> 00:16:47,547 field here, and there is no electric field anywhere there, 249 00:16:47,547 --> 00:16:50,052 so the integral, obviously, is zero. 250 00:16:50,052 --> 00:16:52,7 What is the capacitance of this plate? 251 00:16:52,7 --> 00:16:55,705 The capacitance is the charge on one plate, 252 00:16:55,705 --> 00:16:59,713 divided by the potential difference, which I just use the 253 00:16:59,713 --> 00:17:03,362 word V now, it means it is this value V P minus V S. 254 00:17:03,362 --> 00:17:06,368 So what is the charge on one of the plates? 255 00:17:06,368 --> 00:17:10,59 It doesn't matter which one you take, that is sigma times A. 256 00:17:10,59 --> 00:17:14,581 That's the definition of sigma, right? 257 00:17:14,581 --> 00:17:18,86 It's charge per unit area. So that's the charge on the 258 00:17:18,86 --> 00:17:23,219 plate, and the potential difference we just calculated, 259 00:17:23,219 --> 00:17:26,368 that is sigma D divided by epsilon zero, 260 00:17:26,368 --> 00:17:30,404 so the epsilon zero comes upstairs, and so we find, 261 00:17:30,404 --> 00:17:34,441 now, that it is A times epsilon zero, divided by D. 262 00:17:34,441 --> 00:17:37,912 Notice it's independent of sigma, of course. 263 00:17:37,912 --> 00:17:42,191 Capacitance is geometry, it has nothing to do with how 264 00:17:42,191 --> 00:17:46,55 much charge you have on the capacitor. 265 00:17:46,55 --> 00:17:50,889 I could ask you what is the electrostatic potential energy. 266 00:17:50,889 --> 00:17:54,928 The electrostatic potential energy is the work that you 267 00:17:54,928 --> 00:17:59,49 would have to do to assemble the positive charges here and the 268 00:17:59,49 --> 00:18:03,305 negative charges there. You could also look at it at 269 00:18:03,305 --> 00:18:07,344 the energy that it takes to create that electric field. 270 00:18:07,344 --> 00:18:10,411 Same question. So this work that has to be 271 00:18:10,411 --> 00:18:14,525 done to assemble it is the charge on one plate times the 272 00:18:14,525 --> 00:18:20,516 potential difference times one-half, or -- which is the 273 00:18:20,516 --> 00:18:26,157 same -- one-half C V squared. So, what is one-half Q V? 274 00:18:26,157 --> 00:18:30,753 Let's first take this one, there is one-half, 275 00:18:30,753 --> 00:18:35,14 Q is sigma times A, potential difference V, 276 00:18:35,14 --> 00:18:39,632 we have, is here, sigma D divided by epsilon 277 00:18:39,632 --> 00:18:44,333 zero, so this is the answer that we will find. 278 00:18:44,333 --> 00:18:48,615 From this one, and the answer that we find 279 00:18:48,615 --> 00:18:52,294 from the other one must, 280 00:18:52,294 --> 00:18:55,973 of course, by the same. Let's check that. 281 00:18:55,973 --> 00:19:01,309 This is C, [unintelligible] is A epsilon zero divided by D, 282 00:19:01,309 --> 00:19:06,738 and now I must multiply by the potential difference squared, 283 00:19:06,738 --> 00:19:10,693 so I get a sigma squared, I get a D squared, 284 00:19:10,693 --> 00:19:15,938 and I get an epsilon zero squared, and these two better be 285 00:19:15,938 --> 00:19:18,973 the same. I have sigma sigma here, 286 00:19:18,973 --> 00:19:23,958 sigma squared, I have D squared divided by D, 287 00:19:23,958 --> 00:19:27,65 so I have only one D, I have an epsilon z- zero 288 00:19:27,65 --> 00:19:30,38 squared here and here, and epsilon, 289 00:19:30,38 --> 00:19:34,313 so I have only one over epsilon zero, so they are, 290 00:19:34,313 --> 00:19:37,925 indeed, the same. Now I could ask you where is 291 00:19:37,925 --> 00:19:41,457 the charge located? Let's first go to the top 292 00:19:41,457 --> 00:19:45,712 plate, where is the charge located on the upper plate? 293 00:19:45,712 --> 00:19:49,725 Some of you may say, "Oh, well, maybe the charge is 294 00:19:49,725 --> 00:19:53,337 in the plate, somewhere here. 295 00:19:53,337 --> 00:19:56,289 That cannot be. I make a Gaussian surface all 296 00:19:56,289 --> 00:19:59,174 around that charge. Gauss' Law will tell me, 297 00:19:59,174 --> 00:20:02,595 then, that the surface, closed surface integral of E 298 00:20:02,595 --> 00:20:06,017 dot D A is not zero, because there is charge inside, 299 00:20:06,017 --> 00:20:10,109 and if there is charge inside, that closed surface integral is 300 00:20:10,109 --> 00:20:14,067 not zero -- but we know that the electric field must be zero 301 00:20:14,067 --> 00:20:17,757 everywhere, zero in the conductor, everywhere it must be 302 00:20:17,757 --> 00:20:21,044 zero, so the closed surface integral must be zero. 303 00:20:21,044 --> 00:20:23,392 So there cannot be any charge there. 304 00:20:23,392 --> 00:20:27,832 Simple argument. Some of you may say, 305 00:20:27,832 --> 00:20:32,188 "OK, maybe some of the charge here is at the top surface. 306 00:20:32,188 --> 00:20:35,378 Not allowed either, in this configuration. 307 00:20:35,378 --> 00:20:39,734 I make myself a small pillbox, which is my Gauss surface, 308 00:20:39,734 --> 00:20:43,467 these are flat ends. Electric field is zero here, 309 00:20:43,467 --> 00:20:46,89 electric field is zero there, sot he surface, 310 00:20:46,89 --> 00:20:51,713 closed surface integral must be zero because the electric field 311 00:20:51,713 --> 00:20:55,524 is zero everywhere, but there is charge inside the 312 00:20:55,524 --> 00:20:59,803 pillbox, and so Gauss' Law says that it 313 00:20:59,803 --> 00:21:02,4 cannot be zero. Since it is zero, 314 00:21:02,4 --> 00:21:06,296 there's no charge inside. And so there's only one 315 00:21:06,296 --> 00:21:11,004 solution, nature puts all the positive charge right here at 316 00:21:11,004 --> 00:21:15,388 the bottom of this plate, and the negative charge right 317 00:21:15,388 --> 00:21:20,096 there at the top of that plate. That's the only solution in 318 00:21:20,096 --> 00:21:23,424 this case. Charge cannot be anywhere else. 319 00:21:23,424 --> 00:21:27,239 I can't believe it. That thing is still running. 320 00:21:27,239 --> 00:21:31,217 I want you to take a look at that. 321 00:21:31,217 --> 00:21:35,193 You see that top is still happily running, 322 00:21:35,193 --> 00:21:40,818 so either this has to be a violation of the conservation of 323 00:21:40,818 --> 00:21:45,182 energy somehow, or it is black magic -- it can 324 00:21:45,182 --> 00:21:49,934 never be excluded in twenty six one hundred -- or, 325 00:21:49,934 --> 00:21:54,492 perhaps, there is some simple physics behind it. 326 00:21:54,492 --> 00:21:59,827 And whatever that simple physics is, I would like you to 327 00:21:59,827 --> 00:22:02,833 start thinking about. All right. 328 00:22:02,833 --> 00:22:08,318 Next subject. Again, let me come with a -- 329 00:22:08,318 --> 00:22:12,967 with two plates, because I want to start talking 330 00:22:12,967 --> 00:22:17,418 about dielectrics, and I want to massage this 331 00:22:17,418 --> 00:22:20,979 idea of capacitance a little further. 332 00:22:20,979 --> 00:22:24,737 I have here a parallel plate capacitor. 333 00:22:24,737 --> 00:22:28,1 And I put on here, positive charge, 334 00:22:28,1 --> 00:22:32,947 plus sigma, I call it now, sigma free, there is no 335 00:22:32,947 --> 00:22:38,916 dielectric yet, but, later, there will be, 336 00:22:38,916 --> 00:22:43,825 so I call it sigma free, and here is minus sigma free, 337 00:22:43,825 --> 00:22:46,975 separation is D, surface area is A. 338 00:22:46,975 --> 00:22:51,05 So in the beginning, it's going to be boring, 339 00:22:51,05 --> 00:22:56,886 we know that the electric field here is going in this direction, 340 00:22:56,886 --> 00:23:01,98 and that electric field is sigma free divided by epsilon 341 00:23:01,98 --> 00:23:05,13 zero. The free goes with the sigma. 342 00:23:05,13 --> 00:23:11,052 I charge it up using a power supply, 343 00:23:11,052 --> 00:23:19,434 and now -- this is crucial -- I disconnect the power supply. 344 00:23:19,434 --> 00:23:25,97 I take the leads off. So I disconnect the power 345 00:23:25,97 --> 00:23:30,516 supply. That means -- and this is 346 00:23:30,516 --> 00:23:40,035 crucial -- that whatever comes that this charge is trapped, 347 00:23:40,035 --> 00:23:44,029 can never change. No matter what we're going to 348 00:23:44,029 --> 00:23:46,114 do. Power supply has been 349 00:23:46,114 --> 00:23:49,588 disconnected, that sigma free is trapped. 350 00:23:49,588 --> 00:23:52,54 I'm going to do various things now. 351 00:23:52,54 --> 00:23:56,97 I'm going to change the distance between the plates, 352 00:23:56,97 --> 00:24:00,965 and then independently, I'm going to shove in a 353 00:24:00,965 --> 00:24:05,567 dielectric, we will do that separately, one at a time. 354 00:24:05,567 --> 00:24:10,778 And so the equations that I can now trust, and that I will be 355 00:24:10,778 --> 00:24:16,25 looking at, are the following. The -- the free 356 00:24:16,25 --> 00:24:21,342 charge that I have, Q free, is obviously sigma free 357 00:24:21,342 --> 00:24:25,823 times the area. And that is the definition of 358 00:24:25,823 --> 00:24:30,61 surface charge density. So I can trust that one. 359 00:24:30,61 --> 00:24:36,823 The electric field between the plates is sigma free divided by 360 00:24:36,823 --> 00:24:40,999 epsilon zero, and now I get a kappa there, 361 00:24:40,999 --> 00:24:45,887 if we have a dielectric. The potential difference 362 00:24:45,887 --> 00:24:52,065 between the plates is E D. Provided I know the E, 363 00:24:52,065 --> 00:24:55,074 this is the E, it's always E D. 364 00:24:55,074 --> 00:24:59,084 We just had that in our previous problem. 365 00:24:59,084 --> 00:25:04,198 The capacitance itself, C, is the free charge on one 366 00:25:04,198 --> 00:25:10,215 plate divided by the potential difference between the plates, 367 00:25:10,215 --> 00:25:14,627 V is my potential difference, that is this V, 368 00:25:14,627 --> 00:25:21,244 and the electrostatic potential energy equals one-half 369 00:25:21,244 --> 00:25:25,722 Q free times V, but it is also one-half C V 370 00:25:25,722 --> 00:25:29,56 squared. And so keep these in mind in 371 00:25:29,56 --> 00:25:33,079 what follows. Take a look at them, 372 00:25:33,079 --> 00:25:38,089 the first one is correct, second one is correct, 373 00:25:38,089 --> 00:25:42,461 third one is correct, that one is correct, 374 00:25:42,461 --> 00:25:48,218 that one I can also live with. Or you could write down, 375 00:25:48,218 --> 00:25:53,229 for the capacitance, if you wanted 376 00:25:53,229 --> 00:25:58,402 that, you can write down A times epsilon zero divided by D, 377 00:25:58,402 --> 00:26:02,237 times kappa. The first thing I'm going to do 378 00:26:02,237 --> 00:26:05,269 with the power supply disconnected, 379 00:26:05,269 --> 00:26:10,175 I'm going to increase the distance D between the plates. 380 00:26:10,175 --> 00:26:15,437 And I'm going to increase them by a distance -- I'm going to 381 00:26:15,437 --> 00:26:19,629 double the distance. So D goes up by a factor of 382 00:26:19,629 --> 00:26:23,196 two. But kappa remains one. 383 00:26:23,196 --> 00:26:25,861 Just air. No dielectric yet. 384 00:26:25,861 --> 00:26:29,512 What happened with the electric field? 385 00:26:29,512 --> 00:26:31,387 E. E can not change, 386 00:26:31,387 --> 00:26:36,913 because sigma free cannot change, kappa is one -- there's 387 00:26:36,913 --> 00:26:42,538 no kappa -- and if this cannot change, this cannot change. 388 00:26:42,538 --> 00:26:48,065 So, as I move the plates apart, there is no change in the 389 00:26:48,065 --> 00:26:53,69 electric field. Non-intuitive as that may be 390 00:26:53,69 --> 00:26:56,974 for you, the electric field remains a constant. 391 00:26:56,974 --> 00:27:01,258 So what happens now with the potential difference between the 392 00:27:01,258 --> 00:27:03,828 plates? That, now, must increase by a 393 00:27:03,828 --> 00:27:06,827 factor of two, because if I increase D by a 394 00:27:06,827 --> 00:27:09,683 factor of two, and if E is doing nothing, 395 00:27:09,683 --> 00:27:12,253 then V must go up by a factor of two. 396 00:27:12,253 --> 00:27:14,681 So V must go up by a factor of two. 397 00:27:14,681 --> 00:27:18,751 And I did a demonstration here, during one of my lectures, 398 00:27:18,751 --> 00:27:22,892 whereby I changed D from one millimeter to ten millimeters, 399 00:27:22,892 --> 00:27:28,445 and I changed the potential difference from thousand volts 400 00:27:28,445 --> 00:27:32,454 to ten thousand volts, you've seen it in front of your 401 00:27:32,454 --> 00:27:36,689 own eyes, if you were here. So, indeed, when you separate 402 00:27:36,689 --> 00:27:40,092 the plates with the power supply disconnected, 403 00:27:40,092 --> 00:27:42,512 the potential difference goes up. 404 00:27:42,512 --> 00:27:45,008 What happens with the capacitance? 405 00:27:45,008 --> 00:27:48,336 Well, the capacitance is Q free divided by V. 406 00:27:48,336 --> 00:27:52,117 There's an R missing here. This one doesn't change. 407 00:27:52,117 --> 00:27:56,504 This one goes up by a factor of two, so C must go down by a 408 00:27:56,504 --> 00:28:00,587 factor of two. What happens with the 409 00:28:00,587 --> 00:28:06,074 electrostatic potential energy? Well, it is one-half Q free 410 00:28:06,074 --> 00:28:11,655 times V, Q free cannot change, V went up by a factor of two, 411 00:28:11,655 --> 00:28:14,871 so U must go up by a factor of two. 412 00:28:14,871 --> 00:28:20,546 Remember that when I separated these plates and increased the 413 00:28:20,546 --> 00:28:25,181 potential difference, I told you I was doing work. 414 00:28:25,181 --> 00:28:29,249 U is increasing. If I move the plates apart, 415 00:28:29,249 --> 00:28:33,694 I have to do that work. All right. 416 00:28:33,694 --> 00:28:39,552 So this is the first part, whereby we change D. 417 00:28:39,552 --> 00:28:44,518 Now, I go back to D, leave it as it was, 418 00:28:44,518 --> 00:28:51,267 and now I want to change kappa. I'm going to move in a 419 00:28:51,267 --> 00:28:56,615 dielectric. I'd like to stay working on the 420 00:28:56,615 --> 00:29:04,256 center board I have to change, have to -- can't see this any 421 00:29:04,256 --> 00:29:09,898 more. So now D is as it was before, 422 00:29:09,898 --> 00:29:14,951 but now kappa becomes three. So I take dielectric and I 423 00:29:14,951 --> 00:29:19,442 shove it in, and sigma free is fixed, and so now, 424 00:29:19,442 --> 00:29:23,746 what happens with E? Well, sigma free is fixed. 425 00:29:23,746 --> 00:29:27,489 If kappa, all of a sudden, becomes three, 426 00:29:27,489 --> 00:29:31,606 [krrk] E field goes down. Is that surprising? 427 00:29:31,606 --> 00:29:37,781 No, that is not surprising, because as you move in the 428 00:29:37,781 --> 00:29:42,643 dielectric, the -- this surface charge density is not going to 429 00:29:42,643 --> 00:29:46,788 change, but you are inducing now, on your dielectric, 430 00:29:46,788 --> 00:29:51,251 negative charge here and positive charge here as a result 431 00:29:51,251 --> 00:29:56,192 of that external electric field, and so that creates an induced 432 00:29:56,192 --> 00:29:59,38 electric field in this direction, and so, 433 00:29:59,38 --> 00:30:03,126 as a result of that, the net electric field goes 434 00:30:03,126 --> 00:30:05,836 down. And that's what you see here, 435 00:30:05,836 --> 00:30:09,039 it goes down, in this case, 436 00:30:09,039 --> 00:30:12,927 by a factor of three. What happens with the potential 437 00:30:12,927 --> 00:30:16,739 difference over the plates? Well, D wasn't changing, 438 00:30:16,739 --> 00:30:19,132 remember? We kept D constant now. 439 00:30:19,132 --> 00:30:23,467 So if E goes down by a factor of three, V must go down by a 440 00:30:23,467 --> 00:30:26,233 factor of three. What happens with the 441 00:30:26,233 --> 00:30:29,373 capacitance, C? Well, the capacitor is free 442 00:30:29,373 --> 00:30:32,513 charge divided by the potential difference. 443 00:30:32,513 --> 00:30:35,877 The free charge is not changing, it's trapped. 444 00:30:35,877 --> 00:30:40,181 The potential difference goes down by a 445 00:30:40,181 --> 00:30:43,915 factor of three, capacitance goes up by a factor 446 00:30:43,915 --> 00:30:46,299 of three. What happens with the 447 00:30:46,299 --> 00:30:50,59 electrostatic potential energy? Well, the electrostatic 448 00:30:50,59 --> 00:30:53,132 potential energy is one-half Q V. 449 00:30:53,132 --> 00:30:57,423 But Q free could not change. V went down by a factor of 450 00:30:57,423 --> 00:31:00,204 three. So U must go down by a factor 451 00:31:00,204 --> 00:31:03,382 of three. That means if the electrostatic 452 00:31:03,382 --> 00:31:08,626 potential energy goes down, that as I move in this 453 00:31:08,626 --> 00:31:11,939 dielectric, that I do negative work. 454 00:31:11,939 --> 00:31:16,292 If I had to push it in, you would have gone up. 455 00:31:16,292 --> 00:31:20,172 So, in a way, as I move the dielectric in, 456 00:31:20,172 --> 00:31:24,999 it's being sucked in. There is a force that pulls it 457 00:31:24,999 --> 00:31:27,743 in. Interesting all by itself. 458 00:31:27,743 --> 00:31:32,096 I would like you, at home, to go to ex- through 459 00:31:32,096 --> 00:31:37,491 exactly the same questions, verbatim, with one difference. 460 00:31:37,491 --> 00:31:40,897 And that is, you keep 461 00:31:40,897 --> 00:31:45,505 the power supply connected. Now your answers are going to 462 00:31:45,505 --> 00:31:48,138 be very different. For one thing, 463 00:31:48,138 --> 00:31:52,416 if the power supply is connected, and if you change D 464 00:31:52,416 --> 00:31:57,188 -- so your power supply is connected, and you go up in D by 465 00:31:57,188 --> 00:32:01,137 a factor of two -- the power supply is connected, 466 00:32:01,137 --> 00:32:06,073 [unintelligible] one thing now that cannot change throughout, 467 00:32:06,073 --> 00:32:09,446 and that is V. Potential difference cannot 468 00:32:09,446 --> 00:32:13,49 change, because the power supply is 469 00:32:13,49 --> 00:32:16,551 connected. So now, if V cannot change, 470 00:32:16,551 --> 00:32:20,274 and you increase D by a factor of two, E, now, 471 00:32:20,274 --> 00:32:22,838 must go down by a factor of two. 472 00:32:22,838 --> 00:32:27,057 And that's very different from what happened before, 473 00:32:27,057 --> 00:32:30,201 when E remained constant. So it's very, 474 00:32:30,201 --> 00:32:34,585 very different physics. Well, the physics is the same, 475 00:32:34,585 --> 00:32:37,398 but the results are very different. 476 00:32:37,398 --> 00:32:41,534 And I want you do that, you have all the tools now, 477 00:32:41,534 --> 00:32:45,919 you can believe in those equations, 478 00:32:45,919 --> 00:32:51,024 and they should work for you. All right, let's visit Ohm's 479 00:32:51,024 --> 00:32:56,13 Law and maybe look at Kirchoff -- I prefer to stay on this 480 00:32:56,13 --> 00:32:58,997 center board, convenient for you, 481 00:32:58,997 --> 00:33:01,863 and it is also convenient for me. 482 00:33:01,863 --> 00:33:05,446 A very simple network, keep in mind that, 483 00:33:05,446 --> 00:33:10,193 on an exam, all problems are extremely simple and very 484 00:33:10,193 --> 00:33:13,06 fundamental. Nothing complicated. 485 00:33:13,06 --> 00:33:17,359 You don't have the time for that. 486 00:33:17,359 --> 00:33:22,413 I give here a problem in which I actually give numbers, 487 00:33:22,413 --> 00:33:26,437 on the exam you won't see any numbers, even, 488 00:33:26,437 --> 00:33:30,742 not in the sense of distances, ohms, and so on, 489 00:33:30,742 --> 00:33:34,486 because there is no calculator necessary. 490 00:33:34,486 --> 00:33:39,82 But here, you will see some numbers, this is a battery and 491 00:33:39,82 --> 00:33:43,844 this battery has an EMF, which is ten volts. 492 00:33:43,844 --> 00:33:46,371 That's a given. Plus, minus. 493 00:33:46,371 --> 00:33:52,548 And here, the current is going to split into three. 494 00:33:52,548 --> 00:33:57,053 There is R one, which is one ohm, 495 00:33:57,053 --> 00:34:04,936 R two, which is two ohms, and then we have R three -- put 496 00:34:04,936 --> 00:34:13,524 it a little lower -- R three is three ohms, they come together 497 00:34:13,524 --> 00:34:18,17 here. And here I have a resistor R 498 00:34:18,17 --> 00:34:25,433 four, which is four ohms, and I close the loop and go 499 00:34:25,433 --> 00:34:29,744 back to my battery. Just to make it a little bit 500 00:34:29,744 --> 00:34:33,688 more interesting, I will introduce into this 501 00:34:33,688 --> 00:34:38,274 battery an internal resistance which is very small, 502 00:34:38,274 --> 00:34:42,585 which is oh point one ohms. You can't remove it, 503 00:34:42,585 --> 00:34:45,52 it's intrinsic into that battery. 504 00:34:45,52 --> 00:34:52,4 And so the first question that I would ask you in this case is, 505 00:34:52,4 --> 00:34:56,971 what is the total current that is going to flow? 506 00:34:56,971 --> 00:35:00,376 We're going to get a current I here. 507 00:35:00,376 --> 00:35:05,628 Through here you get I one, through here you get I two, 508 00:35:05,628 --> 00:35:10,103 through here you get I three, I comes out here, 509 00:35:10,103 --> 00:35:13,896 I goes through here, through the fourth, 510 00:35:13,896 --> 00:35:17,884 come back, and I goes through the battery. 511 00:35:17,884 --> 00:35:22,748 So what is I? In a problem like this, 512 00:35:22,748 --> 00:35:25,762 there are many roads to success. 513 00:35:25,762 --> 00:35:29,554 Not just one. And it's a matter of taste 514 00:35:29,554 --> 00:35:34,319 which one you prefer. If I call this point A and I 515 00:35:34,319 --> 00:35:38,111 call this point D, then what I would do, 516 00:35:38,111 --> 00:35:43,07 I would ask myself the question, if this is point A, 517 00:35:43,07 --> 00:35:47,641 and this is point D, what resistor -- which your 518 00:35:47,641 --> 00:35:52,6 book calls the equivalent resistor -- the equivalent 519 00:35:52,6 --> 00:35:57,86 resistance -- what resistor would I have put 520 00:35:57,86 --> 00:36:02,935 here, instead of these three, for the current I to be exactly 521 00:36:02,935 --> 00:36:07,671 the same as what it is now? So I'm going to replace these 522 00:36:07,671 --> 00:36:11,477 three by one resistor, which is this imaginary 523 00:36:11,477 --> 00:36:14,522 resistor. As you have noticed in your 524 00:36:14,522 --> 00:36:19,851 book, where you undoubtedly read up on, one over R equivalent is 525 00:36:19,851 --> 00:36:25,686 one over R one plus one over R two plus one over R three. 526 00:36:25,686 --> 00:36:29,82 You know all these numbers, and so you will find our 527 00:36:29,82 --> 00:36:32,819 equivalent is oh point five five ohms. 528 00:36:32,819 --> 00:36:36,791 Check this at home, and I hope I didn't goof up on 529 00:36:36,791 --> 00:36:39,628 that one. Notice that this resistor, 530 00:36:39,628 --> 00:36:44,086 this equivalent resistance, is smaller than the smallest 531 00:36:44,086 --> 00:36:47,085 one, which is one ohm. That's obvious. 532 00:36:47,085 --> 00:36:51,299 It has to be that way. Think of these as water flows. 533 00:36:51,299 --> 00:36:54,947 Water flow through this one, through this one, 534 00:36:54,947 --> 00:36:57,942 and through this one. 535 00:36:57,942 --> 00:37:01,626 Remove these two. Water is only flowing through 536 00:37:01,626 --> 00:37:04,749 this one. Now you add these two pipes so 537 00:37:04,749 --> 00:37:08,593 more water can flow, so the equivalent resistance 538 00:37:08,593 --> 00:37:11,155 goes down. Same with electricity. 539 00:37:11,155 --> 00:37:15,88 So the equivalent resistance of parallel resistors is always 540 00:37:15,88 --> 00:37:20,124 lower than the smallest. Now it's trivial to calculate 541 00:37:20,124 --> 00:37:22,527 the current I, I use Ohm's Law. 542 00:37:22,527 --> 00:37:27,332 Ohm's Law says that potential difference that is available b- 543 00:37:27,332 --> 00:37:30,297 by the battery is E, 544 00:37:30,297 --> 00:37:33,024 it's ten volts, is now the current, 545 00:37:33,024 --> 00:37:37,356 the total current times all resistances along the road. 546 00:37:37,356 --> 00:37:40,965 I go once around, I have here R e- equivalent, 547 00:37:40,965 --> 00:37:44,816 then I have R four, because the full current goes 548 00:37:44,816 --> 00:37:48,185 through R four, and then I have this little 549 00:37:48,185 --> 00:37:51,393 stinky R of I. Doesn't going to make much 550 00:37:51,393 --> 00:37:55,163 difference, but it's there. And so you can find, 551 00:37:55,163 --> 00:37:58,452 now, what I is, because you know all other 552 00:37:58,452 --> 00:38:03,427 numbers, and you'll find that what I -- 553 00:38:03,427 --> 00:38:07,472 I is my goal, and I think I found two point 554 00:38:07,472 --> 00:38:10,458 one five amperes, that is right. 555 00:38:10,458 --> 00:38:15,08 Two point one five amperes. So we know what I is. 556 00:38:15,08 --> 00:38:17,392 Is this the only way? No. 557 00:38:17,392 --> 00:38:21,052 But it is one way, it's very effective. 558 00:38:21,052 --> 00:38:26,349 So now I want to know what I one, I two, and I three is. 559 00:38:26,349 --> 00:38:31,839 Well, if I know the potential difference between A and D, 560 00:38:31,839 --> 00:38:36,751 V A minus V D, that must be, 561 00:38:36,751 --> 00:38:39,839 according to Ohm's Law, I one R one. 562 00:38:39,839 --> 00:38:43,987 If I go this route. But since we're dealing with 563 00:38:43,987 --> 00:38:48,046 conservative forces, I can also go through this 564 00:38:48,046 --> 00:38:53,253 path, the path doesn't matter, I must get the same potential 565 00:38:53,253 --> 00:38:56,959 difference. So it's also I two times R two, 566 00:38:56,959 --> 00:39:00,842 and so it must also be I three times R three. 567 00:39:00,842 --> 00:39:04,902 So if I only could find I two, then, of course, 568 00:39:04,902 --> 00:39:09,7 I would know the potential difference, 569 00:39:09,7 --> 00:39:13,725 out pops, immediately, I one and out pops, 570 00:39:13,725 --> 00:39:15,689 immediately, I three. 571 00:39:15,689 --> 00:39:21,188 And now I'm going to apply Kirchoff's First Rule in order 572 00:39:21,188 --> 00:39:24,723 to find I two. I couldn't find I one, 573 00:39:24,723 --> 00:39:30,418 but I just decided on I two. And Kirchoff's First Rule says 574 00:39:30,418 --> 00:39:35,328 that the closed loop integral of E dot D L -- this, 575 00:39:35,328 --> 00:39:38,47 now, is a closed loop -- is zero. 576 00:39:38,47 --> 00:39:43,569 In the future, you will see situations that 577 00:39:43,569 --> 00:39:45,492 it's not zero. Here, zero. 578 00:39:45,492 --> 00:39:49,26 I don't know why Kirchoff got the credit for this, 579 00:39:49,26 --> 00:39:52,952 this was long known before him, but nevertheless, 580 00:39:52,952 --> 00:39:55,49 it's called Kirchoff's First Rule. 581 00:39:55,49 --> 00:39:59,489 So I go with closed path, and this is the closed path 582 00:39:59,489 --> 00:40:04,027 that I have decided to take. And that closed loop integral E 583 00:40:04,027 --> 00:40:07,795 dot D L must be zero. Any other path would also be 584 00:40:07,795 --> 00:40:11,795 zero. I chose the one through R two, 585 00:40:11,795 --> 00:40:14,372 because my goal is to find I two. 586 00:40:14,372 --> 00:40:17,756 Once I have I two, I want an I three follow 587 00:40:17,756 --> 00:40:20,737 immediately from this. I have decided, 588 00:40:20,737 --> 00:40:24,603 very arbitrarily, that if I go down in potential, 589 00:40:24,603 --> 00:40:29,195 then I will call that -- I will give that a negative sign, 590 00:40:29,195 --> 00:40:33,948 and if I go down in potential, I will give that a plus sign. 591 00:40:33,948 --> 00:40:37,654 You can reverse that. That makes no difference, 592 00:40:37,654 --> 00:40:42,327 because the sum of them is going to be zero 593 00:40:42,327 --> 00:40:44,806 anyhow. So I will stick to my 594 00:40:44,806 --> 00:40:49,235 convention for now, that if I go down in potential, 595 00:40:49,235 --> 00:40:53,841 I will give it a minus sign, if I go up in potential, 596 00:40:53,841 --> 00:40:55,789 plus sign. I go, first, 597 00:40:55,789 --> 00:40:58,092 from A to D, through R two. 598 00:40:58,092 --> 00:41:02,078 The current is I two. The resistance is R two. 599 00:41:02,078 --> 00:41:07,215 And I go down in potential. So I get my first term is minus 600 00:41:07,215 --> 00:41:09,961 I two times R two. I'm now at D. 601 00:41:09,961 --> 00:41:12,441 I started at A, I'm now at D. 602 00:41:12,441 --> 00:41:17,125 I went through two. I come out here, 603 00:41:17,125 --> 00:41:19,516 and I go through four, R four. 604 00:41:19,516 --> 00:41:23,802 I go down in potential. The current through R four is 605 00:41:23,802 --> 00:41:25,945 I. So I get minus I R four. 606 00:41:25,945 --> 00:41:29,737 I go up, and I see this battery in front of me, 607 00:41:29,737 --> 00:41:32,622 and I have to climb up in potential. 608 00:41:32,622 --> 00:41:37,238 How much do I have to climb up? That EMF, of the battery. 609 00:41:37,238 --> 00:41:41,854 But that dinky toy little resistance R of I makes me down 610 00:41:41,854 --> 00:41:45,234 a little bit, and so I get another minus I 611 00:41:45,234 --> 00:41:48,788 times R of I, and that, 612 00:41:48,788 --> 00:41:52,871 now, is zero. And that's one equation with 613 00:41:52,871 --> 00:41:56,058 only one unknown, which is I two, 614 00:41:56,058 --> 00:42:01,535 because we already have I. That's the two point one five 615 00:42:01,535 --> 00:42:04,223 amperes. And so you'll find, 616 00:42:04,223 --> 00:42:08,605 now, that I two becomes oh point six amperes. 617 00:42:08,605 --> 00:42:13,982 And so you know now that V A minus V D is the potential 618 00:42:13,982 --> 00:42:20,056 difference, which was I two R two, is now going 619 00:42:20,056 --> 00:42:24,676 to be one point two volts. Because I two is oh point six 620 00:42:24,676 --> 00:42:29,296 amperes, but R two -- there's a two here -- is two ohms. 621 00:42:29,296 --> 00:42:34,168 So it's one point two volts. And so the one point two volts 622 00:42:34,168 --> 00:42:38,368 is also I one R one, and it's also I three R three, 623 00:42:38,368 --> 00:42:41,392 so you get I one and you get I three. 624 00:42:41,392 --> 00:42:45,004 What is the power delivered by this battery? 625 00:42:45,004 --> 00:42:48,448 That is the EMF times the total current I. 626 00:42:48,448 --> 00:42:51,884 We know the EMF, ten volts. 627 00:42:51,884 --> 00:42:55,795 The total current is two point one five amperes. 628 00:42:55,795 --> 00:42:58,957 So this is twenty one point five watts. 629 00:42:58,957 --> 00:43:01,786 How does that show up, that energy? 630 00:43:01,786 --> 00:43:04,948 Well, it comes out in the form of heat. 631 00:43:04,948 --> 00:43:07,361 Heat in R four, heat in R one, 632 00:43:07,361 --> 00:43:09,941 R two, and R three, and a teeny, 633 00:43:09,941 --> 00:43:15,1 weeny little bit of heat inside that battery because of that oh 634 00:43:15,1 --> 00:43:17,846 point one ohm internal resistance. 635 00:43:17,846 --> 00:43:23,088 How much power comes out in resistance R two? 636 00:43:23,088 --> 00:43:26,343 Well, that is, of course, the potential 637 00:43:26,343 --> 00:43:30,626 difference over R two, which was that V A minus V D 638 00:43:30,626 --> 00:43:34,48 times the current through R two. That's power, 639 00:43:34,48 --> 00:43:38,163 power is potential difference times current. 640 00:43:38,163 --> 00:43:42,274 This is the total power delivered by the battery. 641 00:43:42,274 --> 00:43:47,156 That is the total potential difference available times the 642 00:43:47,156 --> 00:43:49,554 total current. But of course, 643 00:43:49,554 --> 00:43:55,742 R two only sees a potential difference which is one point 644 00:43:55,742 --> 00:44:00,746 two volts, and it has an I two which is only oh point six 645 00:44:00,746 --> 00:44:05,123 amperes, so this is only oh point seven two watts. 646 00:44:05,123 --> 00:44:10,484 So that is the number of joules per second, in terms of heat, 647 00:44:10,484 --> 00:44:15,487 that is produced in R two. I think you'll believe me when 648 00:44:15,487 --> 00:44:18,614 I say that the top is still running. 649 00:44:18,614 --> 00:44:24,242 And the clue I will give you is that the answer lies in eight oh 650 00:44:24,242 --> 00:44:28,068 two. Give it some thought, 651 00:44:28,068 --> 00:44:33,452 it's a very cute t- top. All right, let's talk about 652 00:44:33,452 --> 00:44:39,153 kinetic energy increase due to charges that move over a 653 00:44:39,153 --> 00:44:43,693 potential difference. I have two conductors, 654 00:44:43,693 --> 00:44:47,916 very funny in shape, but they are equally 655 00:44:47,916 --> 00:44:55,413 potentials, there's no current running inside the conductors. 656 00:44:55,413 --> 00:44:59,295 And so conductor A is at a potential V of A, 657 00:44:59,295 --> 00:45:02,906 this is conductor A, potential is V of A, 658 00:45:02,906 --> 00:45:07,24 and this is conductor be, ahs a potential V of B, 659 00:45:07,24 --> 00:45:11,213 and let's assume that V A is larger than V B. 660 00:45:11,213 --> 00:45:15,276 If you want to change that later, that's fine. 661 00:45:15,276 --> 00:45:20,061 We put the whole thing in vacuum, because I'm going to 662 00:45:20,061 --> 00:45:24,847 release a charge here, plus Q, and the charge will now 663 00:45:24,847 --> 00:45:26,794 go to B. 664 00:45:26,794 --> 00:45:29,843 Electric field configuration is a zoo. 665 00:45:29,843 --> 00:45:33,387 I don't even want to think about what it is. 666 00:45:33,387 --> 00:45:36,436 One way or another, if this is vacuum, 667 00:45:36,436 --> 00:45:39,651 than this charge will finds it way to B. 668 00:45:39,651 --> 00:45:43,195 Let's say this is the routing that it takes. 669 00:45:43,195 --> 00:45:46,491 It ends up on B, and the question now is, 670 00:45:46,491 --> 00:45:51,354 what is the speed at which it reaches B if I release it here 671 00:45:51,354 --> 00:45:54,98 at zero speed? So the electric field is going 672 00:45:54,98 --> 00:45:58,762 to do work on this charge, 673 00:45:58,762 --> 00:46:01,915 and the work, in going from A to B, 674 00:46:01,915 --> 00:46:05,347 integral A to B, is the force dot D L. 675 00:46:05,347 --> 00:46:09,15 That is the electric force on that charge. 676 00:46:09,15 --> 00:46:14,437 Since it a conservative field, it doesn't matter what your 677 00:46:14,437 --> 00:46:19,631 routing is, you will always get the same answer for this. 678 00:46:19,631 --> 00:46:25,382 That electric force is also the force times the electric field, 679 00:46:25,382 --> 00:46:29,927 at any location along the line. And 680 00:46:29,927 --> 00:46:33,728 so you see, you get here a Q times E D L. 681 00:46:33,728 --> 00:46:39,431 But the integral of E D L is the potential difference between 682 00:46:39,431 --> 00:46:42,282 the two. And so the net result, 683 00:46:42,282 --> 00:46:47,51 therefore, is that the work done by the electric fields, 684 00:46:47,51 --> 00:46:53,308 when this charge finally ends up here, that work is the charge 685 00:46:53,308 --> 00:46:57,68 Q times the potential difference V A minus V B, 686 00:46:57,68 --> 00:47:02,907 regardless of which path it chooses to go. 687 00:47:02,907 --> 00:47:06,829 Let's take a practical case, we have a proton which has a 688 00:47:06,829 --> 00:47:10,68 mass one point seven times ten to the minus twenty seven 689 00:47:10,68 --> 00:47:13,41 kilograms. The charge of the pro- proton 690 00:47:13,41 --> 00:47:17,261 is the same of that of the electron, but it is positive. 691 00:47:17,261 --> 00:47:21,042 One point six times ten to the minus nineteen Coulombs. 692 00:47:21,042 --> 00:47:24,963 And let's suppose that the potential difference between A 693 00:47:24,963 --> 00:47:29,375 and B, this is now the potential difference, is a million volts. 694 00:47:29,375 --> 00:47:34,797 Uh, let me put a delta here, because I don't want Vs on both 695 00:47:34,797 --> 00:47:37,52 sides. But that's the difference V A 696 00:47:37,52 --> 00:47:40,321 minus V B. So what is now the kinetic 697 00:47:40,321 --> 00:47:43,355 energy with which this proton reaches B? 698 00:47:43,355 --> 00:47:47,322 So that kinetic energy must be Q times the potential 699 00:47:47,322 --> 00:47:51,834 difference, so that is one point six times ten to the minus 700 00:47:51,834 --> 00:47:56,58 nineteen, times ten to the six, so that is one point six times 701 00:47:56,58 --> 00:47:59,069 ten to the minus thirteen joules. 702 00:47:59,069 --> 00:48:03,815 Almost no physicist would call this one point six times ten to 703 00:48:03,815 --> 00:48:07,043 the minus thirteen joules, 704 00:48:07,043 --> 00:48:10,073 but we would say, the kinetic energy of that 705 00:48:10,073 --> 00:48:13,385 proton is one M E V, one million electron volts. 706 00:48:13,385 --> 00:48:17,331 And the reason why we do that is, an electron volt is the 707 00:48:17,331 --> 00:48:21,347 energy that an electron gains if it moves over a potential 708 00:48:21,347 --> 00:48:24,941 difference of one volt. That's the definition of one 709 00:48:24,941 --> 00:48:28,112 electron volt. Though the charge of the proton 710 00:48:28,112 --> 00:48:31,987 is that same of that of an electron, and it moves over a 711 00:48:31,987 --> 00:48:37,152 distance of one million volts, and so the energy is one 712 00:48:37,152 --> 00:48:41,161 million electron volts. But this is not an SI unit, 713 00:48:41,161 --> 00:48:43,485 so be careful. If you work SI, 714 00:48:43,485 --> 00:48:47,895 you've got to use this number. But we would say that's a 715 00:48:47,895 --> 00:48:52,865 proton with a kinetic energy of one M E V, one million electron 716 00:48:52,865 --> 00:48:55,59 volts. So what is the speed with at 717 00:48:55,59 --> 00:48:58,075 which it finally, then, arrives? 718 00:48:58,075 --> 00:49:02,965 Well, one-half M P squared is this number, this is the mass of 719 00:49:02,965 --> 00:49:07,856 the proton, this is the speed that the proton 720 00:49:07,856 --> 00:49:11,967 arrives at point B, and when you use that number, 721 00:49:11,967 --> 00:49:16,934 the mass, you will find that the velocity of that proton is 722 00:49:16,934 --> 00:49:22,159 about one point four times ten to the seven meters per second, 723 00:49:22,159 --> 00:49:26,356 which is about five percent of the speed of light. 724 00:49:26,356 --> 00:49:29,439 In other words, we don't have to make 725 00:49:29,439 --> 00:49:33,807 relativistic corrections. This answer is believable. 726 00:49:33,807 --> 00:49:38,09 Several students have sent me e-mail, 727 00:49:38,09 --> 00:49:40,768 and they have asked me for practice exams. 728 00:49:40,768 --> 00:49:44,165 I am equally surprised, as you are, that the previous 729 00:49:44,165 --> 00:49:47,365 lecturers of eight oh two did not list on the web, 730 00:49:47,365 --> 00:49:48,868 on their website, exams. 731 00:49:48,868 --> 00:49:51,089 They didn't. I was hoping they did, 732 00:49:51,089 --> 00:49:53,31 but they didn't. Professor Belcher, 733 00:49:53,31 --> 00:49:56,772 however, mentioned one particular practice exam to me, 734 00:49:56,772 --> 00:50:00,756 and when you visit the eight oh two website, you can find that 735 00:50:00,756 --> 00:50:03,565 practice exam, but there are no solutions on 736 00:50:03,565 --> 00:50:06,048 that exam. So you can discuss this exam 737 00:50:06,048 --> 00:50:09,96 with your instructors, with your tutors, 738 00:50:09,96 --> 00:50:13,078 I am available if you want to discuss it with me, 739 00:50:13,078 --> 00:50:17,039 I have no problems with that -- it's a little difficult for me 740 00:50:17,039 --> 00:50:20,676 to help six hundred students, but I can help quite a few. 741 00:50:20,676 --> 00:50:24,507 But my advice is that you take the study guide if you really 742 00:50:24,507 --> 00:50:28,079 want to do more practice, and go over some problems that 743 00:50:28,079 --> 00:50:31,066 have worked-out solutions. And I wish you luck, 744 00:50:31,066 --> 50:36 and I'll see you next Friday.