1 00:00:00 --> 00:00:05,652 2 00:00:05,652 --> 00:00:09,78 So today no new concepts, no new ideas, 3 00:00:09,78 --> 00:00:16,081 you can release a little bit and I want to discuss with you 4 00:00:16,081 --> 00:00:22,708 the connection between electric potential and electric fields. 5 00:00:22,708 --> 00:00:29,009 Imagine you have an electric field here in space and that I 6 00:00:29,009 --> 00:00:35,093 take a charge Q in my pocket, I start at position A and I 7 00:00:35,093 --> 00:00:38,941 walk around and I return at that 8 00:00:38,941 --> 00:00:41,406 point A. Since these forces are 9 00:00:41,406 --> 00:00:45,185 conservative forces, if the electric field is a 10 00:00:45,185 --> 00:00:49,294 static electric field, there are no moving charges, 11 00:00:49,294 --> 00:00:53,484 but that becomes more difficult, then the forces are 12 00:00:53,484 --> 00:00:58,168 conservative forces and so the work that I do when I march 13 00:00:58,168 --> 00:01:01,947 around and coming back at point A must be zero. 14 00:01:01,947 --> 00:01:08,109 It's clear when you uh look at the equation number three that 15 00:01:08,109 --> 00:01:13,01 the potential difference between point A and point A is 16 00:01:13,01 --> 00:01:17,184 obviously zero. I g- start at point A and I end 17 00:01:17,184 --> 00:01:22,447 at point A and that is the integral in going from A back to 18 00:01:22,447 --> 00:01:26,803 point A of E dot DL and that then has to be zero. 19 00:01:26,803 --> 00:01:32,248 And we normally indicate such an integral with a circle which 20 00:01:32,248 --> 00:01:35,333 means you end up where you started. 21 00:01:35,333 --> 00:01:40,778 This is a line now this is not a closed surface 22 00:01:40,778 --> 00:01:45,027 as we had in equation one. This is a closed line. 23 00:01:45,027 --> 00:01:50,074 And so whenever we deal with static electric fields we can 24 00:01:50,074 --> 00:01:53,615 add now another equation if we like that. 25 00:01:53,615 --> 00:01:58,927 And that is if we have a closed line of E dot DL so we end up 26 00:01:58,927 --> 00:02:02,911 where we started that then has to become zero. 27 00:02:02,911 --> 00:02:07,603 Later in the course we will see that there are special 28 00:02:07,603 --> 00:02:12,473 situations where we don't deal with static 29 00:02:12,473 --> 00:02:15,857 fields when we don't have conservative E fields, 30 00:02:15,857 --> 00:02:18,233 that that is not the case anymore. 31 00:02:18,233 --> 00:02:21,401 But for now it is. So if we know the electric 32 00:02:21,401 --> 00:02:25,865 field everywhere then we -- we can see equation number two then 33 00:02:25,865 --> 00:02:28,169 we know the potential everywhere. 34 00:02:28,169 --> 00:02:32,129 And so if we turn it the other way around if we knew the 35 00:02:32,129 --> 00:02:36,665 potential everywhere you want to know what the electric field is 36 00:02:36,665 --> 00:02:42,282 and that of course is possible. If you look at equation two and 37 00:02:42,282 --> 00:02:46,195 three you see that the potential is the integral of the 38 00:02:46,195 --> 00:02:49,528 electric field. So it is obvious that the field 39 00:02:49,528 --> 00:02:52,354 must be the derivative of the potential. 40 00:02:52,354 --> 00:02:56,557 Now when you have fields being derivative of potentials you 41 00:02:56,557 --> 00:02:59,963 always have to worry about plus and minus signs, 42 00:02:59,963 --> 00:03:04,238 whether um you have to pay MIT twenty-seven thousand dollars 43 00:03:04,238 --> 00:03:08,152 tuition to be here or whether MIT pays you twenty-seven 44 00:03:08,152 --> 00:03:12,354 thousand dollars tuition for coming here is 45 00:03:12,354 --> 00:03:17,493 only a difference of a minus sign but it's a big difference 46 00:03:17,493 --> 00:03:20,948 of course. And so let's work this out in 47 00:03:20,948 --> 00:03:24,492 some detail. I have here a charge plus Q. 48 00:03:24,492 --> 00:03:29,277 And at a distance R at that location P we know what the 49 00:03:29,277 --> 00:03:33,086 electric field is, we've done that a zillion 50 00:03:33,086 --> 00:03:36,276 times. This is the unit vector in the 51 00:03:36,276 --> 00:03:41,503 direction from Q to that point and we know that the electric 52 00:03:41,503 --> 00:03:46,288 field is pointing away from that charge 53 00:03:46,288 --> 00:03:52,433 and we know that the electric field E we have seen that 54 00:03:52,433 --> 00:03:59,375 already the first lecture is Q divided by four pi epsilon zero 55 00:03:59,375 --> 00:04:03,472 R squared in the direction of R roof. 56 00:04:03,472 --> 00:04:09,958 And last lecture we derived what the electric potential is 57 00:04:09,958 --> 00:04:14,965 at that location. The electric potential is Q 58 00:04:14,965 --> 00:04:20,086 divided by four pi epsilon zero R. 59 00:04:20,086 --> 00:04:23,895 This is a vector. This is a scalar. 60 00:04:23,895 --> 00:04:30,168 So the um potential is the integral of the electric field 61 00:04:30,168 --> 00:04:36,888 along a line and now I want to try whether the electric field 62 00:04:36,888 --> 00:04:42,377 can be written as the derivative of the potential. 63 00:04:42,377 --> 00:04:47,53 So let us take DV DR and let's see what we get. 64 00:04:47,53 --> 00:04:53,578 If I take this DV DR I get a minus Q divided by four pi 65 00:04:53,578 --> 00:04:57,106 epsilon zero R squared. 66 00:04:57,106 --> 00:05:01,826 Of course if I want to know what the electric field is I 67 00:05:01,826 --> 00:05:07,233 need a vector so I will multiply both sides, which is completely 68 00:05:07,233 --> 00:05:12,468 legal, there's nothing illegal about that, with unit vector in 69 00:05:12,468 --> 00:05:16,501 the direction R so that turns them into vectors. 70 00:05:16,501 --> 00:05:21,822 And now you see that I'm almost there, this is almost the same, 71 00:05:21,822 --> 00:05:26,285 except for a minus sign. And so the derivative of the 72 00:05:26,285 --> 00:05:29,031 potential is minus E, not plus E. 73 00:05:29,031 --> 00:05:33,447 And so I will write that down here, 74 00:05:33,447 --> 00:05:38,719 that E equals minus DV DR. So they are closely related if 75 00:05:38,719 --> 00:05:44,556 you know what the -- oh I want -- I want this to be a vector so 76 00:05:44,556 --> 00:05:48,51 I put here R roof. Vector on the left side, 77 00:05:48,51 --> 00:05:52,275 you must have a vector on the right side. 78 00:05:52,275 --> 00:05:57,171 And so if you know the potential everywhere in space, 79 00:05:57,171 --> 00:06:02,161 then you can retrieve the electric field. 80 00:06:02,161 --> 00:06:06,581 I mentioned last time that the electric field vectors -- e- 81 00:06:06,581 --> 00:06:10,544 electric field lines, are always perpendicular to the 82 00:06:10,544 --> 00:06:14,659 equipotential surfaces. And that's obvious why that has 83 00:06:14,659 --> 00:06:18,012 to be the case. Imagine that you are in an -- 84 00:06:18,012 --> 00:06:22,128 in space and that you move with a charge in your pocket 85 00:06:22,128 --> 00:06:24,948 perpendicular to electric field lines. 86 00:06:24,948 --> 00:06:29,292 So you purposely move only perpendicular to electric field 87 00:06:29,292 --> 00:06:32,112 lines. So that means that the force on 88 00:06:32,112 --> 00:06:37,107 you and the direction in which you move are always at 89 00:06:37,107 --> 00:06:39,677 ninety-degree angles. So you'll only move 90 00:06:39,677 --> 00:06:41,733 perpendicular to the field lines. 91 00:06:41,733 --> 00:06:44,624 These are the field lines, you move like this. 92 00:06:44,624 --> 00:06:47,515 These are the field lines, you move like this. 93 00:06:47,515 --> 00:06:51,112 So you never do any work. Because the dot product between 94 00:06:51,112 --> 00:06:54,839 DL and E is zero and if you don't do any work the potential 95 00:06:54,839 --> 00:06:57,344 remains the same, that's the definition. 96 00:06:57,344 --> 00:07:00,878 And so you can see that therefore equipotential surfaces 97 00:07:00,878 --> 00:07:04,925 must always be perpendicular to field lines and field lines must 98 00:07:04,925 --> 00:07:09,233 always be perpendicular to the equipotentials. 99 00:07:09,233 --> 00:07:12,592 And I will show you again the -- the viewgraph, 100 00:07:12,592 --> 00:07:16,755 the overhead projection of uh the nice drawing by Maxwell. 101 00:07:16,755 --> 00:07:20,406 With the plus four charge and the minus one charge. 102 00:07:20,406 --> 00:07:24,642 The same one we saw last time. Only to point out again this 103 00:07:24,642 --> 00:07:27,928 ninety-degree angle. I discussed this in great 104 00:07:27,928 --> 00:07:30,922 detail last lecture so I will not do that. 105 00:07:30,922 --> 00:07:33,551 R- the red lines are really surfaces. 106 00:07:33,551 --> 00:07:37,494 This is three-dimensional, you have to rotate the whole 107 00:07:37,494 --> 00:07:41,903 thing about the vertical. So these are surfaces. 108 00:07:41,903 --> 00:07:45,364 And the red ones are positive potential surfaces and the blue 109 00:07:45,364 --> 00:07:47,44 ones are negative potential surfaces. 110 00:07:47,44 --> 00:07:50,381 That is not important. But the green lines are field 111 00:07:50,381 --> 00:07:52,111 lines. And notice if I take for 112 00:07:52,111 --> 00:07:55,283 instance this field line, it's perpendicular here to the 113 00:07:55,283 --> 00:07:57,878 red, perpendicular there, perpendicular there, 114 00:07:57,878 --> 00:08:00,128 perpendicular there. Perpendicular here. 115 00:08:00,128 --> 00:08:02,088 Perpendicular here. Coming in here, 116 00:08:02,088 --> 00:08:03,703 perpendicular, perpendicular, 117 00:08:03,703 --> 00:08:06,125 perpendicular. Everywhere where you look on 118 00:08:06,125 --> 00:08:09,816 this graph you will see that the field lines are perpendicular to 119 00:08:09,816 --> 00:08:12,517 the equipotentials. 120 00:08:12,517 --> 00:08:16,328 And that is something that we now fully understand. 121 00:08:16,328 --> 00:08:20,902 The situation means then that if you release a charge at zero 122 00:08:20,902 --> 00:08:25,4 speed that it would always start to move perpendicular to an 123 00:08:25,4 --> 00:08:29,974 equipotential surface because it always starts to move in the 124 00:08:29,974 --> 00:08:34,243 direction of a field line, a plus charge in the direction 125 00:08:34,243 --> 00:08:37,749 of the field line, minus charge in the opposite 126 00:08:37,749 --> 00:08:40,722 direction. So if you're in space and you 127 00:08:40,722 --> 00:08:45,754 release a charge at zero speed it always takes off 128 00:08:45,754 --> 00:08:48,489 perpendicular to equipotentials. 129 00:08:48,489 --> 00:08:51,93 You have something similar with gravity. 130 00:08:51,93 --> 00:08:55,018 If you look at maps of mountaineers, 131 00:08:55,018 --> 00:08:58,547 contours of equal altitude, equal height, 132 00:08:58,547 --> 00:09:03,047 if you started skiing and you started at that point, 133 00:09:03,047 --> 00:09:08,076 and you started with zero speed, you would always take off 134 00:09:08,076 --> 00:09:11,164 perpendicular to the equipotentials. 135 00:09:11,164 --> 00:09:15,399 So this is the direction in which 136 00:09:15,399 --> 00:09:19,437 you start to move. If you start off with zero 137 00:09:19,437 --> 00:09:22,465 speed. I now want to give you some 138 00:09:22,465 --> 00:09:28,154 deeper feeling of the connection between potential and electric 139 00:09:28,154 --> 00:09:32,467 fields and I want you to follow me very closely. 140 00:09:32,467 --> 00:09:36,597 Each step that I make I want you to follow me. 141 00:09:36,597 --> 00:09:41,735 So imagine that I am somewhere in space at p- position P. 142 00:09:41,735 --> 00:09:48,618 At that position P there is a potential, one unique potential, 143 00:09:48,618 --> 00:09:50,591 V of P. That's a given. 144 00:09:50,591 --> 00:09:55,795 And there is an electric field at that location where I am. 145 00:09:55,795 --> 00:10:01,088 And now what I'm going to do, I'm going to make an extremely 146 00:10:01,088 --> 00:10:04,138 small step only in the X direction. 147 00:10:04,138 --> 00:10:07,996 Not in Y, not in Z. Only in the X direction. 148 00:10:07,996 --> 00:10:13,378 If I measure no change in the potential over that little step 149 00:10:13,378 --> 00:10:18,313 it means that the component of the electric field in the 150 00:10:18,313 --> 00:10:24,732 direction X is zero. If I do measure a difference in 151 00:10:24,732 --> 00:10:31,08 potential then the component, the X component of the electric 152 00:10:31,08 --> 00:10:37,216 field, the magnitude of that, would be that little sidestep 153 00:10:37,216 --> 00:10:41,554 that I have made, delta X, it would be the 154 00:10:41,554 --> 00:10:47,584 potential difference that I measure divided by that little 155 00:10:47,584 --> 00:10:52,663 sidestep. And I keep Y and Z constant. 156 00:10:52,663 --> 00:10:57,109 And these are magnitudes. But that's why I put these 157 00:10:57,109 --> 00:11:00,945 vertical bars here. Equally if I made a small 158 00:11:00,945 --> 00:11:05,566 sidestep in the Y direction and I measured a potential 159 00:11:05,566 --> 00:11:10,972 difference delta V keeping X and Z constant, that would then be 160 00:11:10,972 --> 00:11:15,68 the component of the electric field in the Y direction. 161 00:11:15,68 --> 00:11:20,301 Earlier we wrote down for E as a unit new- newtons per 162 00:11:20,301 --> 00:11:23,527 coulombs. From now on we almost always 163 00:11:23,527 --> 00:11:27,243 will write down for the unit of 164 00:11:27,243 --> 00:11:32,155 electric field volts per meter. It is exactly the same thing as 165 00:11:32,155 --> 00:11:35,72 newtons over coulombs, there is no difference, 166 00:11:35,72 --> 00:11:39,206 but this gives you a little bit more insight. 167 00:11:39,206 --> 00:11:43,959 You make a little sidestep in meters and you measure how much 168 00:11:43,959 --> 00:11:47,444 the potential changes, it's volts per meters. 169 00:11:47,444 --> 00:11:50,613 It is a potential change over a distance. 170 00:11:50,613 --> 00:11:56,08 So now I can write down the connection between electric 171 00:11:56,08 --> 00:12:01,104 field and potential in Cartesian coordinates. 172 00:12:01,104 --> 00:12:07,956 It looks much more scary than the nice way that I could write 173 00:12:07,956 --> 00:12:13,209 it down up there. When I had only a function of 174 00:12:13,209 --> 00:12:18,005 distance R. And so in Cartesian coordinates 175 00:12:18,005 --> 00:12:24,514 we now get E equals minus, that minus sign we discussed at 176 00:12:24,514 --> 00:12:31,251 length, and now we get DV DX times X roof plus DV DY times Y 177 00:12:31,251 --> 00:12:35,946 roof plus DV DZ times Z roof. 178 00:12:35,946 --> 00:12:40,886 And what you see here, this first term here, 179 00:12:40,886 --> 00:12:46,63 including of course the minus sign, that is E of X. 180 00:12:46,63 --> 00:12:52,834 And this term including the minus sign, that is E of Y. 181 00:12:52,834 --> 00:12:57,544 And so on. And the fact that you see these 182 00:12:57,544 --> 00:13:03,173 curled Ds it means partial derivatives. 183 00:13:03,173 --> 00:13:07,39 That means when you do this derivative you keep Z and Y 184 00:13:07,39 --> 00:13:10,513 constant. When you do this derivative you 185 00:13:10,513 --> 00:13:15,042 do X and Z constant and so on. And so this is the Cartesian 186 00:13:15,042 --> 00:13:19,727 notation for which in eighteen oh two you will learn or maybe 187 00:13:19,727 --> 00:13:24,255 you already have learned we would write this E equals minus 188 00:13:24,255 --> 00:13:27,691 the gradient of G. This is a vector function. 189 00:13:27,691 --> 00:13:31,517 This is a scalar function. And this is just a s- a 190 00:13:31,517 --> 00:13:36,046 different notation, just a matter of words, 191 00:13:36,046 --> 00:13:42,517 for this mathematical recipe. And you'll get that with 192 00:13:42,517 --> 00:13:49,356 eighteen in eighteen oh two if you haven't seen that yet. 193 00:13:49,356 --> 00:13:56,438 So I now want a straightforward example whereby we assume a 194 00:13:56,438 --> 00:14:03,643 certain dependence on X and I give you it is a given that V, 195 00:14:03,643 --> 00:14:08,894 the potential, is ten to the fifth 196 00:14:08,894 --> 00:14:11,946 times X. So that is a given. 197 00:14:11,946 --> 00:14:18,729 And this holds between X equals zero and say ten to the minus 198 00:14:18,729 --> 00:14:23,477 two meters. So it holds over a space of one 199 00:14:23,477 --> 00:14:27,434 centimeter. So the potential changes 200 00:14:27,434 --> 00:14:33,538 linearly with distance. What now is the electric field? 201 00:14:33,538 --> 00:14:38,399 In that space? Well, the electric field I go 202 00:14:38,399 --> 00:14:46,276 back to my description there. There's only a component in the 203 00:14:46,276 --> 00:14:50,616 X direction. So the first derivative becomes 204 00:14:50,616 --> 00:14:56,57 minus ten to the fifth times X roof and the others are zero. 205 00:14:56,57 --> 00:15:02,423 So EY is zero and EZ is zero. So you may say well yeah whoa 206 00:15:02,423 --> 00:15:06,965 nice mathematics but we don't see any physics. 207 00:15:06,965 --> 00:15:10,598 This is more physical than you think. 208 00:15:10,598 --> 00:15:15,038 Imagine that I have here a plate 209 00:15:15,038 --> 00:15:19,22 which is charged, it's positive charge. 210 00:15:19,22 --> 00:15:25,823 And the plate is at location X and I have another plate here, 211 00:15:25,823 --> 00:15:31,985 it's say at location zero. I call this plate A and I call 212 00:15:31,985 --> 00:15:37,377 this plate B and this plate is charged negatively. 213 00:15:37,377 --> 00:15:45,191 So X goes into this direction. So I can put the electric field 214 00:15:45,191 --> 00:15:50,193 inside here according to the recipe minus ten to the fifth 215 00:15:50,193 --> 00:15:55,459 and it is in the direction of minus X roof so e- X roof is in 216 00:15:55,459 --> 00:15:59,233 this direction, the electric field is in the 217 00:15:59,233 --> 00:16:03,358 opposite direction, and it's the same everywhere 218 00:16:03,358 --> 00:16:07,22 and that is very physical. We discussed that. 219 00:16:07,22 --> 00:16:12,398 When we discussed the electric field near very large planes, 220 00:16:12,398 --> 00:16:16,698 that the electric field inside was a 221 00:16:16,698 --> 00:16:20,183 constant, remember, and the electric field inside 222 00:16:20,183 --> 00:16:24,249 was sigma divided by epsilon zero if sigma is the surface 223 00:16:24,249 --> 00:16:27,008 charge density on each of these plates. 224 00:16:27,008 --> 00:16:31,582 And we argued that the electric field outside was about zero and 225 00:16:31,582 --> 00:16:35,285 that the electric field outside here was about zero. 226 00:16:35,285 --> 00:16:39,278 So it's extremely physical. This is exactly what you see 227 00:16:39,278 --> 00:16:41,892 here. The electric field minus ten to 228 00:16:41,892 --> 00:16:47,192 the fifth times e- so the magnitude of this electric field 229 00:16:47,192 --> 00:16:53,658 here, the magnitude, is ten to the fifth volts per 230 00:16:53,658 --> 00:16:57,749 meter. What now is the potential 231 00:16:57,749 --> 00:17:03,028 difference? Well, VA minus VB minus VB is 232 00:17:03,028 --> 00:17:08,966 the integral in going from A to B of E dot DL. 233 00:17:08,966 --> 00:17:17,148 Well I go from here to here so I write down for DL I wrote down 234 00:17:17,148 --> 00:17:24,01 DX of course. Because I called that the X 235 00:17:24,01 --> 00:17:28,471 direction now. So I will write down here dot 236 00:17:28,471 --> 00:17:31,997 DX. And so this is minus ten to the 237 00:17:31,997 --> 00:17:38,013 fifth times the integral in going from A to B of X roof dot 238 00:17:38,013 --> 00:17:40,814 DX. It looks scary but it is 239 00:17:40,814 --> 00:17:46,622 trivial, the X roof is the unit vector in this direction. 240 00:17:46,622 --> 00:17:51,394 And DX is a little vector DX in this direction. 241 00:17:51,394 --> 00:17:56,677 So they're both in the same direction. 242 00:17:56,677 --> 00:18:01,442 So the cosine, the angle between the two is 243 00:18:01,442 --> 00:18:05,299 one. So I can forget about vectors, 244 00:18:05,299 --> 00:18:11,652 I can forget about the dot. And so this becomes minus ten 245 00:18:11,652 --> 00:18:18,572 to the fifth times the integral in going from A to B of DX and 246 00:18:18,572 --> 00:18:23,791 that is trivial. That is minus ten to the fifth 247 00:18:23,791 --> 00:18:29,956 times the location. I have to do the integral 248 00:18:29,956 --> 00:18:34,936 between A and B. So I get here X of B minus X of 249 00:18:34,936 --> 00:18:38,433 A. And if this is ten to the minus 250 00:18:38,433 --> 00:18:44,048 two meters, to go from here to here is one centimeter, 251 00:18:44,048 --> 00:18:48,71 I must multiply this by ten to the minus two, 252 00:18:48,71 --> 00:18:53,584 so I get that this is minus one thousand volts. 253 00:18:53,584 --> 00:18:57,504 So A is a thousand volts lower than B. 254 00:18:57,504 --> 00:19:03,31 That's what it means. And that's something that's 255 00:19:03,31 --> 00:19:06,806 very physical. Notice that if you go from left 256 00:19:06,806 --> 00:19:10,068 to right that the potential grows linearly, 257 00:19:10,068 --> 00:19:14,03 this is lower than that, and if you in your head use 258 00:19:14,03 --> 00:19:17,526 planes like this parallel to the other planes, 259 00:19:17,526 --> 00:19:21,255 each one of those planes would be equipotentials, 260 00:19:21,255 --> 00:19:24,285 they everywhere have the same potential. 261 00:19:24,285 --> 00:19:28,79 And gradually when you move it up your potential increases, 262 00:19:28,79 --> 00:19:32,674 but notice the electric field goes 263 00:19:32,674 --> 00:19:35,981 from plus to minus in the opposite direction. 264 00:19:35,981 --> 00:19:40,64 That's always the reason that's behind that -- that minus sign. 265 00:19:40,64 --> 00:19:45,148 Well clearly I'm always free to choose where I choose my zero 266 00:19:45,148 --> 00:19:48,004 potential. We discussed that last time. 267 00:19:48,004 --> 00:19:51,836 You don't always have to choose infinitely far zero. 268 00:19:51,836 --> 00:19:55,142 So I could choose this arbitrarily to be zero 269 00:19:55,142 --> 00:19:57,772 potential. This would then be plus a 270 00:19:57,772 --> 00:20:00,628 thousand. And so you then find that the 271 00:20:00,628 --> 00:20:05,813 potential V is then simply ten to the fifth times X, 272 00:20:05,813 --> 00:20:10,287 when X is zero you find the potential to be zero, 273 00:20:10,287 --> 00:20:15,694 and when X is one centimeter you find the potential to be a 274 00:20:15,694 --> 00:20:21,194 thousand volts and that then goes together with the electric 275 00:20:21,194 --> 00:20:26,135 field equals minus ten to the fifth in this direction. 276 00:20:26,135 --> 00:20:29,212 And so this is extremely physical. 277 00:20:29,212 --> 00:20:34,619 This is something that you would have whenever we deal with 278 00:20:34,619 --> 00:20:38,721 parallel plates. As long as there's no charge 279 00:20:38,721 --> 00:20:42,773 moving, and we're dealing with solid 280 00:20:42,773 --> 00:20:45,991 conductors, so we have static electric fields, 281 00:20:45,991 --> 00:20:49,995 the charges are not heavily moving, then the field inside 282 00:20:49,995 --> 00:20:54,285 the conductor is always zero. Not the case in a nonconductor. 283 00:20:54,285 --> 00:20:58,003 It's only in a conductor because conductors have free 284 00:20:58,003 --> 00:21:02,078 electrons to move and if these free electrons see electric 285 00:21:02,078 --> 00:21:06,154 fields inside which they may they start to move until they 286 00:21:06,154 --> 00:21:10,372 experience no longer a force, thereby they kill the electric 287 00:21:10,372 --> 00:21:12,524 field inside. 288 00:21:12,524 --> 00:21:18,098 So the charge in a conductor always rearranges itself so that 289 00:21:18,098 --> 00:21:23,951 the electric field becomes zero. If the field is a static field, 290 00:21:23,951 --> 00:21:28,596 not rapidly changing. And so now I want to evaluate 291 00:21:28,596 --> 00:21:33,612 with you the situation that I'm going to charge a solid 292 00:21:33,612 --> 00:21:39,001 conductor and ask myself the question where does the charge 293 00:21:39,001 --> 00:21:42,066 go. In honor of um Valentine's Day 294 00:21:42,066 --> 00:21:46,726 let's take a solid heart, steel heart, 295 00:21:46,726 --> 00:21:49,575 it's solid all the way throughout. 296 00:21:49,575 --> 00:21:54,497 So this is a solid conductor and I bring on this conductor 297 00:21:54,497 --> 00:21:57,779 charge from the outside. Plus or minus, 298 00:21:57,779 --> 00:22:02,701 let's just take plus for now. And so the question that I'm 299 00:22:02,701 --> 00:22:05,723 asking you now, this is a conductor, 300 00:22:05,723 --> 00:22:10,213 this is not an insulator, the story for insulators is 301 00:22:10,213 --> 00:22:15,653 totally different, this has free moving electrons 302 00:22:15,653 --> 00:22:20,861 inside, I'm asking you now if I touch this conducting heart -- 303 00:22:20,861 --> 00:22:25,897 by the way, your heart is a very good conductor -- um if you 304 00:22:25,897 --> 00:22:30,848 touch this conducting heart where would this charge end up? 305 00:22:30,848 --> 00:22:34,86 Where would it go to? And I leave you with three 306 00:22:34,86 --> 00:22:38,104 choices. And we'll have a vote on that. 307 00:22:38,104 --> 00:22:42,884 The first choice is that the plus charges would uniformly 308 00:22:42,884 --> 00:22:47,493 distribute for throughout. A possibility. 309 00:22:47,493 --> 00:22:50,244 The second possibility, less likely, 310 00:22:50,244 --> 00:22:54,565 I think, that all the charge will go to one place there. 311 00:22:54,565 --> 00:22:58,416 I don't know which place that would be, but maybe. 312 00:22:58,416 --> 00:23:03,052 And then the third possibility is that maybe the charge will 313 00:23:03,052 --> 00:23:07,845 uniformly distribute itself only on the outer surface and then 314 00:23:07,845 --> 00:23:11,224 the fourth possibility is none of the above. 315 00:23:11,224 --> 00:23:14,288 All these suggestions I made were wrong. 316 00:23:14,288 --> 00:23:19,6 Who think that the charge might uniformly distribute it 317 00:23:19,6 --> 00:23:23,189 throughout the conductor? I see one or two hands. 318 00:23:23,189 --> 00:23:26,255 That's good. Don't feel ashamed of raising 319 00:23:26,255 --> 00:23:29,396 your hands, in the worst case you're wrong, 320 00:23:29,396 --> 00:23:33,209 I've been so many times wrong when it comes to this. 321 00:23:33,209 --> 00:23:37,471 Don't feel bad about that. Who thinks that the charge will 322 00:23:37,471 --> 00:23:39,864 all go to one point in the heart? 323 00:23:39,864 --> 00:23:43,453 You have the courage? You think it will go to one 324 00:23:43,453 --> 00:23:45,772 point? Charge repels each other, 325 00:23:45,772 --> 00:23:49,634 right, so that doesn't seem likely. 326 00:23:49,634 --> 00:23:53,541 Who thinks that it will uniformly distribute itself on 327 00:23:53,541 --> 00:23:57,007 the outer surface? Who thinks none of the above? 328 00:23:57,007 --> 00:23:59,956 Very good. Well, those who suggested that 329 00:23:59,956 --> 00:24:04,158 it might be uniform on the outside I would still give them 330 00:24:04,158 --> 00:24:07,181 a B but it's not uniform, as you will see. 331 00:24:07,181 --> 00:24:10,204 But it will go exclusively to the outside. 332 00:24:10,204 --> 00:24:12,564 And I will prove that now to you. 333 00:24:12,564 --> 00:24:18,019 Let us first look for that ridiculous possibility that the 334 00:24:18,019 --> 00:24:21,602 charge would somehow end up in the conductor itself. 335 00:24:21,602 --> 00:24:25,536 I take here a Gaussian surface which is a closed surface. 336 00:24:25,536 --> 00:24:29,681 I know inside the conductor if we have electrostatic fields, 337 00:24:29,681 --> 00:24:33,193 not fastly moving charges, but it's a static field, 338 00:24:33,193 --> 00:24:36,916 I know that the E field everywhere must be zero on the 339 00:24:36,916 --> 00:24:40,71 surface, this is a closed surface, so the integral of E 340 00:24:40,71 --> 00:24:44,855 dot DA equation one is zero. That means the charge inside my 341 00:24:44,855 --> 00:24:49,351 sphere is zero and so there cannot be any charge. 342 00:24:49,351 --> 00:24:53,11 So Gauss's law immediately kills the possibility that there 343 00:24:53,11 --> 00:24:55,767 would be any charge inside this conductor. 344 00:24:55,767 --> 00:24:59,526 So that's out of the question. So that leaves you only with 345 00:24:59,526 --> 00:25:02,378 one choice, that is on the -- at the surface. 346 00:25:02,378 --> 00:25:04,711 So the charge must be at the surface. 347 00:25:04,711 --> 00:25:08,276 And later in a later lecture I will discuss with you the 348 00:25:08,276 --> 00:25:11,646 details why that charge is not uniformly distributed. 349 00:25:11,646 --> 00:25:14,887 It would be uniformly distributed at the surface if 350 00:25:14,887 --> 00:25:17,933 this were a sphere. But not if it has this funny 351 00:25:17,933 --> 00:25:21,917 shape. But it will be at the surface. 352 00:25:21,917 --> 00:25:26,092 Now I'm going to make this heart a very special heart, 353 00:25:26,092 --> 00:25:29,085 more like a real heart, it's open here, 354 00:25:29,085 --> 00:25:32,63 but it is solid here, so this is a conducting, 355 00:25:32,63 --> 00:25:35,466 the heart muscle, and here it's open, 356 00:25:35,466 --> 00:25:39,405 there's nothing here. And again I'm going to charge 357 00:25:39,405 --> 00:25:41,847 it. Bring charge on the outside. 358 00:25:41,847 --> 00:25:46,337 So now it's obvious that we don't expect that there is any 359 00:25:46,337 --> 00:25:49,488 charge that will be inside the conductor. 360 00:25:49,488 --> 00:25:53,347 That's clear, the same argument 361 00:25:53,347 --> 00:25:55,856 holds with the Gauss's law argument. 362 00:25:55,856 --> 00:26:00,298 But now is it perhaps possible that some of the positive charge 363 00:26:00,298 --> 00:26:04,669 will go on the inside of this surface and some on the outside? 364 00:26:04,669 --> 00:26:08,396 Who thinks that maybe some will now go on the inside, 365 00:26:08,396 --> 00:26:11,62 because now the situation is different, right, 366 00:26:11,62 --> 00:26:14,558 there is now, it's now a hollow conductor. 367 00:26:14,558 --> 00:26:18,571 Anyone in favor of some of that charge maybe going on the 368 00:26:18,571 --> 00:26:20,147 inside? I see one hand, 369 00:26:20,147 --> 00:26:24,304 two hands, who says no it's not possible, 370 00:26:24,304 --> 00:26:27,595 it will not go to the inside? It will still go to the 371 00:26:27,595 --> 00:26:29,748 outside. Well, most of you are very 372 00:26:29,748 --> 00:26:32,47 careful now, you don't want to vote anymore. 373 00:26:32,47 --> 00:26:35,698 It cannot go to the inside. Why can it not go to the 374 00:26:35,698 --> 00:26:37,977 inside? Let this be my Gauss surface. 375 00:26:37,977 --> 00:26:39,94 Closed surface. Think of this as 376 00:26:39,94 --> 00:26:42,788 three-dimensional. Everywhere on that line the 377 00:26:42,788 --> 00:26:46,46 electric field is zero because you're inside the conductor. 378 00:26:46,46 --> 00:26:48,739 So the surface integral is also zero. 379 00:26:48,739 --> 00:26:52,348 So Gauss's law says there cannot be any charge inside that 380 00:26:52,348 --> 00:26:54,5 box. And so again the charge has to 381 00:26:54,5 --> 00:26:57,733 go to the outer surface and 382 00:26:57,733 --> 00:27:00,208 nothing will go to the inner surface. 383 00:27:00,208 --> 00:27:04,334 And so the conclusion then is that the electric field is zero 384 00:27:04,334 --> 00:27:08,048 in the conductor but the electric field is also zero in 385 00:27:08,048 --> 00:27:11,073 this opening. There's never any charge there. 386 00:27:11,073 --> 00:27:14,374 And so the whole heart including the cavity is an 387 00:27:14,374 --> 00:27:17,193 equipotential. There is never any electric 388 00:27:17,193 --> 00:27:19,944 field anywhere. The only electric field's 389 00:27:19,944 --> 00:27:23,176 outside the heart. And there are field lines and 390 00:27:23,176 --> 00:27:27,508 these field lines everywhere are perpendicular to the surface of 391 00:27:27,508 --> 00:27:31,974 the heart because the heart is an 392 00:27:31,974 --> 00:27:35,953 equipotential. So here you get very funny 393 00:27:35,953 --> 00:27:41,822 field lines that go like this. They have to be perpendicular 394 00:27:41,822 --> 00:27:45,701 locally where they reach the heart wall. 395 00:27:45,701 --> 00:27:51,669 Earlier in my lectures I showed that a uniformly solid sphere 396 00:27:51,669 --> 00:27:57,737 has electric field zero inside and I even showed to you that a 397 00:27:57,737 --> 00:28:02,711 hollow conducting sphere also has zero 398 00:28:02,711 --> 00:28:06,697 electric field inside. Today I have demonstrated that 399 00:28:06,697 --> 00:28:10,915 it doesn't have to be a sphere. You don't need spherical 400 00:28:10,915 --> 00:28:13,751 symmetry. That any shape provided that 401 00:28:13,751 --> 00:28:17,508 it's a hollow conductor, it has to be a conductor, 402 00:28:17,508 --> 00:28:21,802 any shape will give you an electric field of zero inside. 403 00:28:21,802 --> 00:28:24,562 And I first want to demonstrate that. 404 00:28:24,562 --> 00:28:27,783 I have here something that is not a sphere. 405 00:28:27,783 --> 00:28:31,233 It's a paint can. It has some aluminum on top. 406 00:28:31,233 --> 00:28:35,067 It has an opening there. It's not 407 00:28:35,067 --> 00:28:37,627 perfect. It's not really closed like 408 00:28:37,627 --> 00:28:40,261 this is. So the electric field inside 409 00:28:40,261 --> 00:28:43,993 will not be exactly zero. But it will be very close. 410 00:28:43,993 --> 00:28:47,432 I must have an opening because I want to get in. 411 00:28:47,432 --> 00:28:51,236 I want to get charge, see whether there is any charge 412 00:28:51,236 --> 00:28:54,017 on the inside. So I must be able to get 413 00:28:54,017 --> 00:28:56,87 through. So I'm going to charge this one 414 00:28:56,87 --> 00:29:01,333 and then I will take some charge off the outside and take some 415 00:29:01,333 --> 00:29:07,114 charge off the inside and use the electroscope and see whether 416 00:29:07,114 --> 00:29:11,882 we can demonstrate that indeed there is charge on the outside 417 00:29:11,882 --> 00:29:14,584 but there is nothing on the inside. 418 00:29:14,584 --> 00:29:18,876 I will use the same method that I used last time when I 419 00:29:18,876 --> 00:29:22,293 challenged you to figure out how this works. 420 00:29:22,293 --> 00:29:26,903 This is this crazy method which we call electrophorus elec- 421 00:29:26,903 --> 00:29:30,241 electrophorus, it's difficult to pronounce. 422 00:29:30,241 --> 00:29:33,499 Electrophorus. We have here a glass plate. 423 00:29:33,499 --> 00:29:36,917 I rub it with cat fur. Think about it again. 424 00:29:36,917 --> 00:29:41,079 It's a little problem inside the 425 00:29:41,079 --> 00:29:42,947 problem. Metal plate. 426 00:29:42,947 --> 00:29:45,469 I put it on top. I touch it. 427 00:29:45,469 --> 00:29:48,271 I get a shock. I touch it here. 428 00:29:48,271 --> 00:29:51,727 I touch it again. I get again a shock. 429 00:29:51,727 --> 00:29:55,276 And I charge this up. I touch it again. 430 00:29:55,276 --> 00:29:59,105 I get another shock. And I touch it again. 431 00:29:59,105 --> 00:30:02,187 Let's get a little bit more on it. 432 00:30:02,187 --> 00:30:08,164 The charge on this plate is positive by the way. 433 00:30:08,164 --> 00:30:12,472 That I create on the glass. I touch it. 434 00:30:12,472 --> 00:30:17,459 The charge on here is negative. Not positive. 435 00:30:17,459 --> 00:30:20,293 Put it on again. Touch it. 436 00:30:20,293 --> 00:30:23,466 OK. So I should have negative 437 00:30:23,466 --> 00:30:28,794 charge on there now. Here is little test sphere. 438 00:30:28,794 --> 00:30:34,121 It's a conductor. I'll take some charge off from 439 00:30:34,121 --> 00:30:36,274 this side. Touch it. 440 00:30:36,274 --> 00:30:40,241 Boy. There's charge. 441 00:30:40,241 --> 00:30:42,685 There's no question. We agree, right? 442 00:30:42,685 --> 00:30:43,907 There's charge. OK. 443 00:30:43,907 --> 00:30:47,438 Now I touch the inside, let's hope that no sparks fly 444 00:30:47,438 --> 00:30:48,524 over. I touch it. 445 00:30:48,524 --> 00:30:49,678 Nothing. See that? 446 00:30:49,678 --> 00:30:52,868 Absolutely nothing. So there's no charge inside, 447 00:30:52,868 --> 00:30:56,399 the charge is on the outside. Which is what I've just 448 00:30:56,399 --> 00:30:59,386 demonstrated. You see it in front of your own 449 00:30:59,386 --> 00:31:01,49 eyes. All the charge goes to the 450 00:31:01,49 --> 00:31:03,595 outside. Not so intuitive but an 451 00:31:03,595 --> 00:31:07,6 immediate consequence of the fact that it's a conductor that 452 00:31:07,6 --> 00:31:12,624 the electrons will move freely so that the electric 453 00:31:12,624 --> 00:31:17,427 field in the conductor itself is zero and we have argued that 454 00:31:17,427 --> 00:31:21,429 no charge can ever go on the inside of the surface. 455 00:31:21,429 --> 00:31:25,752 It all stays on the outside. So when I touch the inside 456 00:31:25,752 --> 00:31:29,355 there was no charge. So if you are inside that 457 00:31:29,355 --> 00:31:33,037 conductor, if your house is a conducting house, 458 00:31:33,037 --> 00:31:37,68 and someone in the outside world charges your house up when 459 00:31:37,68 --> 00:31:41,203 you're inside, you have no knowledge of that. 460 00:31:41,203 --> 00:31:44,665 It's quite amazing, isn't it? 461 00:31:44,665 --> 00:31:48,941 You are electrically shielded from the outside world. 462 00:31:48,941 --> 00:31:53,628 Now I'm going to make the situation even more complicated. 463 00:31:53,628 --> 00:31:58,48 I now take a conducting object, doesn't have to be a sphere. 464 00:31:58,48 --> 00:32:03,003 And I bring that conducting object hollow in an external 465 00:32:03,003 --> 00:32:06,621 electric field. So someone outside your house 466 00:32:06,621 --> 00:32:11,062 is turning on a VandeGraaff creating an electric field. 467 00:32:11,062 --> 00:32:16,316 What now is going to happen? Well, due to induction, 468 00:32:16,316 --> 00:32:21,091 you're going to get some charge polarization in the conductor. 469 00:32:21,091 --> 00:32:25,553 One side may end up negative and the other side may end up 470 00:32:25,553 --> 00:32:28,685 positive. But what happens on the inside? 471 00:32:28,685 --> 00:32:31,268 Nothing. The electric field in the 472 00:32:31,268 --> 00:32:36,044 conductor must stay everywhere zero if it is a static electric 473 00:32:36,044 --> 00:32:38,705 field. And so no charge will -- can 474 00:32:38,705 --> 00:32:43,246 accumulate here and no charge can accumulate on the inside. 475 00:32:43,246 --> 00:32:47,082 And so as you bring this electric 476 00:32:47,082 --> 00:32:51,73 field on the outside you may get negative and positive charge 477 00:32:51,73 --> 00:32:54,675 on the outside, maybe negative here and 478 00:32:54,675 --> 00:32:57,309 positive there, but inside nothing. 479 00:32:57,309 --> 00:33:01,803 You are inside electrically shielded from the outside world 480 00:33:01,803 --> 00:33:06,374 in the same way that you were when someone was trying to put 481 00:33:06,374 --> 00:33:10,326 charge onto your house, now someone is trying to zap 482 00:33:10,326 --> 00:33:14,355 you with electric fields, nothing will happen inside. 483 00:33:14,355 --> 00:33:18,98 You will never see an electric field inside. 484 00:33:18,98 --> 00:33:23,97 I will show you a interesting drawing, interesting figure, 485 00:33:23,97 --> 00:33:27,298 which is a conducting box. It's closed. 486 00:33:27,298 --> 00:33:32,551 The cup that you see open is just to allow you to look inside 487 00:33:32,551 --> 00:33:37,717 but it is closed from all sides. And there are some negative 488 00:33:37,717 --> 00:33:42,007 charges here and there are positive charges in the 489 00:33:42,007 --> 00:33:48,398 foreground which you don't see. The red field lines come from 490 00:33:48,398 --> 00:33:51,073 positive charges, end up on the box, 491 00:33:51,073 --> 00:33:55,583 and the negative field lines go from the box to the negative 492 00:33:55,583 --> 00:33:58,334 charges. There is clear polarization. 493 00:33:58,334 --> 00:34:02,538 The box itself is neutral. I started with a neutral box. 494 00:34:02,538 --> 00:34:06,588 But because of this electric field I get polarization. 495 00:34:06,588 --> 00:34:10,028 I end up with negative charge on the box here, 496 00:34:10,028 --> 00:34:13,543 only on the outside, positive charge on the box 497 00:34:13,543 --> 00:34:17,823 here, only on the outside. Inside electric field is zero. 498 00:34:17,823 --> 00:34:23,141 No charge anywhere inside. Due to this crazy electric 499 00:34:23,141 --> 00:34:27,83 field the free moving charges in the conductor will rearrange 500 00:34:27,83 --> 00:34:32,128 themselves in such a way that the electric field is zero 501 00:34:32,128 --> 00:34:36,348 everywhere in the conductor, is zero inside the cavity, 502 00:34:36,348 --> 00:34:40,411 and that the closed loop integral of E dot DL is zero 503 00:34:40,411 --> 00:34:43,303 everywhere if these are static fields. 504 00:34:43,303 --> 00:34:47,679 And it is clearly impossible for us to ever calculate how 505 00:34:47,679 --> 00:34:52,289 that charge configuration at the surface will 506 00:34:52,289 --> 00:34:55,786 have to be in order to meet all those conditions. 507 00:34:55,786 --> 00:34:58,335 But nature can do this effortlessly. 508 00:34:58,335 --> 00:35:02,341 And it can do it extremely fast, obeying all the laws of 509 00:35:02,341 --> 00:35:04,745 physics. It puts very quickly plus 510 00:35:04,745 --> 00:35:07,221 charge here and minus charge there. 511 00:35:07,221 --> 00:35:11,082 Make sure that there is no charge on the inside of the 512 00:35:11,082 --> 00:35:13,922 surface. It makes sure that the electric 513 00:35:13,922 --> 00:35:18,511 field is everywhere zero inside and in the box and it also makes 514 00:35:18,511 --> 00:35:24,119 sure that the integral E dot DL is zero everywhere in space. 515 00:35:24,119 --> 00:35:27,695 And therefore the box and everything inside becomes an 516 00:35:27,695 --> 00:35:31,204 equipotential so it also arranges matters so that the 517 00:35:31,204 --> 00:35:34,645 field lines wherever they intersect with the box are 518 00:35:34,645 --> 00:35:38,558 always perpendicular to the box. And all of that is done in 519 00:35:38,558 --> 00:35:42,539 almost no time at all by nature. It is an amazing thing that 520 00:35:42,539 --> 00:35:46,587 this happens and something that as I said would be impossible 521 00:35:46,587 --> 00:35:50,298 for us to calculate because the field configurations are 522 00:35:50,298 --> 00:35:53,807 extraordinarily difficult. So if you were inside this 523 00:35:53,807 --> 00:35:57,181 metal box, no matter what happens 524 00:35:57,181 --> 00:36:00,325 on the outside, you would be electrically 525 00:36:00,325 --> 00:36:05,278 isolated from the outside world. You would not notice that there 526 00:36:05,278 --> 00:36:09,601 is a strong electric field outside, nor would you notice 527 00:36:09,601 --> 00:36:13,218 that people are trying to charge up your house. 528 00:36:13,218 --> 00:36:17,463 We call that electrostatic shielding and we give that a 529 00:36:17,463 --> 00:36:21,865 name, that house of yours would be called a Faraday cage. 530 00:36:21,865 --> 00:36:25,403 It's called after the great physicist Faraday. 531 00:36:25,403 --> 00:36:30,905 You will learn a lot more about him during this course. 532 00:36:30,905 --> 00:36:34,813 Before I demonstrate this I want to address an issue which 533 00:36:34,813 --> 00:36:38,926 is related to problem two-one. Which is your next assignment. 534 00:36:38,926 --> 00:36:42,285 And I want to urge you, I make myself no illusion, 535 00:36:42,285 --> 00:36:46,536 but I want to urge you to start working on that assignment this 536 00:36:46,536 --> 00:36:50,169 weekend, not next week. These assignments are not just 537 00:36:50,169 --> 00:36:53,323 baby assignments. These are MIT assignments and 538 00:36:53,323 --> 00:36:57,642 you got to put in a lot of work to do them, so please start this 539 00:36:57,642 --> 00:37:02,715 weekend not to do me a favor but to do yourself that favor. 540 00:37:02,715 --> 00:37:05,596 But let's talk about problem two-one. 541 00:37:05,596 --> 00:37:10,078 In other words I will help you with that problem two-one. 542 00:37:10,078 --> 00:37:15,04 I said several times that it is not possible to get an electric 543 00:37:15,04 --> 00:37:19,922 field inside a hollow conductor. Well, suppose I go inside the 544 00:37:19,922 --> 00:37:22,164 conductor. I go inside there. 545 00:37:22,164 --> 00:37:25,365 And I put sneakily a charge in my pocket. 546 00:37:25,365 --> 00:37:28,486 And I sit inside there and you close it. 547 00:37:28,486 --> 00:37:32,969 Then there is a charge inside, there's nothing you can do 548 00:37:32,969 --> 00:37:37,009 about it. And if there is a charge 549 00:37:37,009 --> 00:37:39,785 inside, there is an electric field. 550 00:37:39,785 --> 00:37:44,846 So now we have a situation and since it is post Valentine's Day 551 00:37:44,846 --> 00:37:48,112 my heart has evolved into a sphere again. 552 00:37:48,112 --> 00:37:51,623 So now we take a spherical conductor, solid, 553 00:37:51,623 --> 00:37:55,95 this is solid material, and somehow I'm sitting inside 554 00:37:55,95 --> 00:38:00,44 here with a charge plus Q. Can make it minus if you want 555 00:38:00,44 --> 00:38:02,807 to. That's exactly the problem 556 00:38:02,807 --> 00:38:06,563 two-one is about. Now clearly there is positive 557 00:38:06,563 --> 00:38:11,84 charge inside. So clearly there has to be an 558 00:38:11,84 --> 00:38:15,883 electric field. But the electric field inside 559 00:38:15,883 --> 00:38:19,834 the conductor, that means the electric field 560 00:38:19,834 --> 00:38:22,315 anywhere here, must be zero. 561 00:38:22,315 --> 00:38:27,645 If it's not zero the electrons will keep moving until it is 562 00:38:27,645 --> 00:38:30,493 zero. So the conducting material 563 00:38:30,493 --> 00:38:35,731 itself has no electric field. What does that mean now with 564 00:38:35,731 --> 00:38:41,061 respect to any charge on the inside surface? 565 00:38:41,061 --> 00:38:44,563 Now there must be charge on the inside surface. 566 00:38:44,563 --> 00:38:48,066 Because now if I made this my Gaussian surface, 567 00:38:48,066 --> 00:38:51,874 which is now a spherical surface, a closed surface, 568 00:38:51,874 --> 00:38:54,083 Mr. Gauss says that the closed 569 00:38:54,083 --> 00:38:58,499 surface integral of E dot DA over this surface must be zero 570 00:38:58,499 --> 00:39:01,774 because the electric field is zero anywhere. 571 00:39:01,774 --> 00:39:06,191 That's the same as all the charge inside divided by epsilon 572 00:39:06,191 --> 00:39:08,704 zero. So this -- the charge inside 573 00:39:08,704 --> 00:39:12,054 must be zero. Since there can be no charge in 574 00:39:12,054 --> 00:39:17,007 the conductor itself, negative charge must now 575 00:39:17,007 --> 00:39:20,184 accumulate on the inside of that surface. 576 00:39:20,184 --> 00:39:24,156 So that the net charge inside this surface is zero. 577 00:39:24,156 --> 00:39:28,922 So now we do get charge on the inside, and how much charge do 578 00:39:28,922 --> 00:39:31,941 you get on the inside? Exactly minus Q. 579 00:39:31,941 --> 00:39:34,641 So that the sum of the two is zero. 580 00:39:34,641 --> 00:39:37,898 Now this conductor originally was neutral. 581 00:39:37,898 --> 00:39:41,949 It had no net charge. So therefore on the surface of 582 00:39:41,949 --> 00:39:45,365 the conductor we must now see charge plus Q. 583 00:39:45,365 --> 00:39:50,909 Because the minus charge on the inside came from the 584 00:39:50,909 --> 00:39:55,259 induct- conductor itself, and so the sum must be zero. 585 00:39:55,259 --> 00:39:59,937 So now you get a peculiar situation that the plus Q charge 586 00:39:59,937 --> 00:40:04,944 inside which creates an E field inside creates negative charge 587 00:40:04,944 --> 00:40:07,899 on the inside, the same in magnitude, 588 00:40:07,899 --> 00:40:11,264 opposite in sign, and plus Q charge on the 589 00:40:11,264 --> 00:40:13,89 outside. And the electric fields, 590 00:40:13,89 --> 00:40:17,583 they're very complicated. The electric fields, 591 00:40:17,583 --> 00:40:22,735 let me try to put them in, I would imagine that if this 592 00:40:22,735 --> 00:40:26,206 charge Q is closer to this wall than to this wall that the 593 00:40:26,206 --> 00:40:29,677 negative charge here will be larger in density than there. 594 00:40:29,677 --> 00:40:31,565 It's really an induction effect. 595 00:40:31,565 --> 00:40:34,244 The negative charge wants to go to this plus. 596 00:40:34,244 --> 00:40:37,775 That's really what's happening. And so since this charge is 597 00:40:37,775 --> 00:40:41,611 closer to this wall than to that wall it will be able to attract 598 00:40:41,611 --> 00:40:45,447 more electrons and so it's clear that the density of charge here 599 00:40:45,447 --> 00:40:48,857 should be higher than there and so the field lines always 600 00:40:48,857 --> 00:40:52,085 perpendicular to the equipotential, 601 00:40:52,085 --> 00:40:56,268 so they must be always perpendicular to the wall, 602 00:40:56,268 --> 00:41:00,451 sort of like this. So I put in a few field lines. 603 00:41:00,451 --> 00:41:04,46 But here the field will be stronger than there. 604 00:41:04,46 --> 00:41:08,73 So there is a field inside. What now is the charge 605 00:41:08,73 --> 00:41:13,524 distribution on the outside? That is the hardest of all. 606 00:41:13,524 --> 00:41:18,578 And by no means so obvious. It turns out that the charge on 607 00:41:18,578 --> 00:41:23,475 the outside on this sphere, because it is a sphere, 608 00:41:23,475 --> 00:41:27,156 will be uniformly distributed. And it is not intuitive and it 609 00:41:27,156 --> 00:41:29,794 is not obvious. Nature must obey all laws of 610 00:41:29,794 --> 00:41:32,002 physics. The conductor must become an 611 00:41:32,002 --> 00:41:34,701 equipotential. There can be no electric field 612 00:41:34,701 --> 00:41:37,891 inside the conductor. Electric field lines have to be 613 00:41:37,891 --> 00:41:40,284 everywhere perpendicular to the surface. 614 00:41:40,284 --> 00:41:43,964 The closed loop integral of E dot DL must be zero everywhere. 615 00:41:43,964 --> 00:41:47,399 And the only way that nature can do that is by making the 616 00:41:47,399 --> 00:41:49,976 charge distribution on the surface uniform. 617 00:41:49,976 --> 00:41:53,289 And that is amazing when you think 618 00:41:53,289 --> 00:41:55,47 of that. It's independent of the 619 00:41:55,47 --> 00:41:58,074 position of that charge plus Q inside. 620 00:41:58,074 --> 00:42:02,367 So if you start to move around with that charge plus Q inside, 621 00:42:02,367 --> 00:42:04,549 the outside world will not know. 622 00:42:04,549 --> 00:42:08,561 The outside world only knows that there is a charge plus Q 623 00:42:08,561 --> 00:42:12,713 uniformly distributed on the outside because it is a sphere. 624 00:42:12,713 --> 00:42:15,88 That would not be the case if it were a heart. 625 00:42:15,88 --> 00:42:19,751 But the outside world has no way of knowing that you are 626 00:42:19,751 --> 00:42:23,269 moving that charge inside around. 627 00:42:23,269 --> 00:42:26,981 So I'm sitting inside there and suppose I crawl inside there 628 00:42:26,981 --> 00:42:28,994 with a rubber rod and with a cat. 629 00:42:28,994 --> 00:42:32,516 And I use the rubber rod on the cat creating positive and 630 00:42:32,516 --> 00:42:34,277 negative charge, same amount. 631 00:42:34,277 --> 00:42:37,863 The outside world will not know because I don't change the 632 00:42:37,863 --> 00:42:40,505 charge inside. Only if there's plus Q in my 633 00:42:40,505 --> 00:42:42,832 pocket. The fact that I create plus on 634 00:42:42,832 --> 00:42:46,355 the cat and maybe minus on myself, the outside world will 635 00:42:46,355 --> 00:42:49,626 never know because the sum of the charges is still Q. 636 00:42:49,626 --> 00:42:54,28 They may hear the cat scream, that's all they can hear. 637 00:42:54,28 --> 00:42:59,36 But they have no way of knowing that I'm fooling around there 638 00:42:59,36 --> 00:43:03,086 with charges. And so the outside world has no 639 00:43:03,086 --> 00:43:06,557 way of knowing what happens on the inside. 640 00:43:06,557 --> 00:43:09,943 And we call that electrostatic shielding. 641 00:43:09,943 --> 00:43:12,907 That's the effect of a Faraday cage. 642 00:43:12,907 --> 00:43:17,817 I want to demonstrate when I bring this can in the electric 643 00:43:17,817 --> 00:43:22,474 field, it's a conducting hollow object, I bring it in an 644 00:43:22,474 --> 00:43:26,538 electric field of the VandeGraaff, the thing is a 645 00:43:26,538 --> 00:43:31,328 conductor, I will show you that because of 646 00:43:31,328 --> 00:43:36,176 the induction you're going to get like you see on that figure 647 00:43:36,176 --> 00:43:39,973 you're going to get negative charge on one side, 648 00:43:39,973 --> 00:43:44,498 positive charge on the other side, and zero charge on the 649 00:43:44,498 --> 00:43:46,76 inside. That's quite amazing, 650 00:43:46,76 --> 00:43:49,265 isn't it? If this were positive, 651 00:43:49,265 --> 00:43:52,577 let's assume it is, then this side becomes 652 00:43:52,577 --> 00:43:57,263 negative, this side becomes positive, the whole thing is an 653 00:43:57,263 --> 00:43:59,768 equipotential, no charge inside. 654 00:43:59,768 --> 00:44:04,413 Quite amazing. So let's turn on this, 655 00:44:04,413 --> 00:44:08,344 the VandeGraaff. So we create that electric 656 00:44:08,344 --> 00:44:11,526 field. We turn on the electroscope. 657 00:44:11,526 --> 00:44:17,048 Here is my little Ping Pong ball conducting and I'm going to 658 00:44:17,048 --> 00:44:21,821 touch first the can on your side, on your left side, 659 00:44:21,821 --> 00:44:25,752 there we go, and I bring this charge on the 660 00:44:25,752 --> 00:44:28,56 electroscope. Boy, nice charge. 661 00:44:28,56 --> 00:44:33,708 I now touch the other side, and I will approach the -- I 662 00:44:33,708 --> 00:44:37,752 heard a spark, sparks are always bad. 663 00:44:37,752 --> 00:44:41,273 I approach that electroscope, and if the reading of the 664 00:44:41,273 --> 00:44:44,142 electroscope, if the deflection becomes less, 665 00:44:44,142 --> 00:44:47,729 as we discussed earlier, it means that the polarity that 666 00:44:47,729 --> 00:44:51,902 I have on here is different from the polarity on the electroscope 667 00:44:51,902 --> 00:44:55,359 and you clearly see that. The deflection becomes less. 668 00:44:55,359 --> 00:44:59,206 So the charge that I took off from this side has a different 669 00:44:59,206 --> 00:45:02,728 polarity than the charge that I took on from that side. 670 00:45:02,728 --> 00:45:06,64 But yet, it's an equipotential. All that strange polarization 671 00:45:06,64 --> 00:45:09,621 of charges takes place at the 672 00:45:09,621 --> 00:45:11,749 surface. And now I will try to get 673 00:45:11,749 --> 00:45:14,071 inside. To see whether I can get some 674 00:45:14,071 --> 00:45:17,295 charge from the inside, and there shouldn't be any, 675 00:45:17,295 --> 00:45:20,197 ooh, I have to take this charge off of course, 676 00:45:20,197 --> 00:45:22,648 and I touch this and you see no charge. 677 00:45:22,648 --> 00:45:25,936 So you've seen three things, which is quite amazing. 678 00:45:25,936 --> 00:45:29,612 That the charge on this side has a different polarity from 679 00:45:29,612 --> 00:45:33,095 the charge on that side, and that everything happens on 680 00:45:33,095 --> 00:45:35,803 the surface, nothing happens on the inside. 681 00:45:35,803 --> 00:45:40,317 I could not get any charge from the inside. 682 00:45:40,317 --> 00:45:45,926 Now we're going to experiment with more dangerous stuff and 683 00:45:45,926 --> 00:45:51,342 that is with the VandeGraaff. Here you see a Faraday cage 684 00:45:51,342 --> 00:45:56,661 with hat -- has some openings, it's not solid conductor, 685 00:45:56,661 --> 00:46:01,496 but it has small openings, which doesn't change the 686 00:46:01,496 --> 00:46:05,268 situation too much, maybe only a little, 687 00:46:05,268 --> 00:46:11,264 and I'm going to go inside that cage, this would also be a nice 688 00:46:11,264 --> 00:46:15,657 Faraday cage but it's very hard for me 689 00:46:15,657 --> 00:46:19,201 to crawl in there. And if I go in there with a 690 00:46:19,201 --> 00:46:23,69 radio just like the radio in your car, then you may not be 691 00:46:23,69 --> 00:46:27,706 able to hear the radio, even though radio waves is a 692 00:46:27,706 --> 00:46:32,195 difficult story because the shielding that we discussed is 693 00:46:32,195 --> 00:46:37,157 only electrostatic shielding and radio waves are electromagnetic 694 00:46:37,157 --> 00:46:40,307 radiation which strongly changing fields. 695 00:46:40,307 --> 00:46:45,898 So it may not be as perfect a shielding as you may think. 696 00:46:45,898 --> 00:46:49,8 But we all know if someone breaks off the antenna of your 697 00:46:49,8 --> 00:46:52,796 car which happens in Cambridge all the time, 698 00:46:52,796 --> 00:46:56,837 you have no reception inside. Because your car is a Faraday 699 00:46:56,837 --> 00:46:58,718 cage. And so what I will do, 700 00:46:58,718 --> 00:47:02,55 I will go into the cage, I will first show you that when 701 00:47:02,55 --> 00:47:05,894 we charge that cage, that we bring it up to a few 702 00:47:05,894 --> 00:47:09,517 hundred thousand volts, I'll just hold some tinsel in 703 00:47:09,517 --> 00:47:13,488 my hand, to convince you that yes indeed this cage will be 704 00:47:13,488 --> 00:47:17,796 charged by the VandeGraaff provided there 705 00:47:17,796 --> 00:47:21,437 is contact here. And we'll see that yes Marcos 706 00:47:21,437 --> 00:47:25,724 give me the full blast, let's just look at the tinsel, 707 00:47:25,724 --> 00:47:30,335 you see this tinsel clearly indicates like an electroscope 708 00:47:30,335 --> 00:47:33,894 that I'm being charged now. So I'll jump off, 709 00:47:33,894 --> 00:47:37,373 if you can discharge. Then I will go inside. 710 00:47:37,373 --> 00:47:42,307 I will have the tinsel with me. So I will show you that inside 711 00:47:42,307 --> 00:47:47,485 there when they charge that cage that the tinsels will not spread 712 00:47:47,485 --> 00:47:51,063 out, and I will bring with me this 713 00:47:51,063 --> 00:47:54,647 wonderful radio and -- Radio: Said a woman who opposes 714 00:47:54,647 --> 00:47:58,637 embalming is a suspect in the murders -- [laughter] I didn't 715 00:47:58,637 --> 00:48:01,882 plan that, believe me. Radio: [unintelligible] -- 716 00:48:01,882 --> 00:48:04,722 self-proclaimed prophet, Catherine Padilla, 717 00:48:04,722 --> 00:48:08,036 grandmother of ten, denies the charges and tells a 718 00:48:08,036 --> 00:48:12,296 reporter she's not quote -- I'll first go in without any charge. 719 00:48:12,296 --> 00:48:15,609 But don't do anything. Radio: [unintelligible] And 720 00:48:15,609 --> 00:48:19,599 she's one that -- when she calls her own -- 721 00:48:19,599 --> 00:48:23,092 [unintelligible] Nothing. I'm shielded. 722 00:48:23,092 --> 00:48:28,331 However, there is a problem. You can still hear me and I'm 723 00:48:28,331 --> 00:48:33,663 wearing a transmitter and the receiver is somewhere outside 724 00:48:33,663 --> 00:48:37,156 this box. So why can you still hear me? 725 00:48:37,156 --> 00:48:41,568 That means that the kind of radio waves that I am 726 00:48:41,568 --> 00:48:46,532 transmitting are very high frequency, it's not a static 727 00:48:46,532 --> 00:48:51,313 field, so somehow they can get through. 728 00:48:51,313 --> 00:48:56,257 So the shielding is not perfect for fast changing electric 729 00:48:56,257 --> 00:48:59,207 fields. But it's good enough for AM 730 00:48:59,207 --> 00:49:02,417 radios. So now I'll go in and I'll try 731 00:49:02,417 --> 00:49:06,494 to be brave and he's going to try to zap me now, 732 00:49:06,494 --> 00:49:10,572 to electrocute me. But since I've taken eight oh 733 00:49:10,572 --> 00:49:13,781 two, I'm not afraid. I burned my hand, 734 00:49:13,781 --> 00:49:16,21 that's a different story. OK. 735 00:49:16,21 --> 00:49:18,553 Marcos. Do the best you can. 736 00:49:18,553 --> 00:49:22,804 Here are the tinsels. Run it up a hundred thousand 737 00:49:22,804 --> 00:49:27,314 volts. Two hundred thousand volts. 738 00:49:27,314 --> 00:49:31,823 I feel as happy like a clam at high tide inside here. 739 00:49:31,823 --> 00:49:35,639 Nothing is happening. I'm not worried at all. 740 00:49:35,639 --> 00:49:39,108 If lightning were to strike me who cares? 741 00:49:39,108 --> 00:49:43,618 I'm in a Faraday cage. Not going to spoil my weekend. 742 00:49:43,618 --> 00:49:47,867 I can touch the inside. There's no charge anywhere 743 00:49:47,867 --> 00:49:50,729 here. My weekend won't be spoiled. 744 00:49:50,729 --> 00:49:55,498 And I hope that the new assignment is not going to spoil 745 00:49:55,498 --> 50:00 yours either. See you next Tuesday.