1 00:00:00 --> 00:00:08,082 2 00:00:08,082 --> 00:00:16,406 Today we're going to um work on a whole new concept and that is 3 00:00:16,406 --> 00:00:23,253 the concept of electric flux. We've come a long way. 4 00:00:23,253 --> 00:00:27,684 We started out with Coulomb's law. 5 00:00:27,684 --> 00:00:35,471 We got electric field lines. And now we have electric flux. 6 00:00:35,471 --> 00:00:43,259 Suppose I have an electric field which is like so 7 00:00:43,259 --> 00:00:47,851 and I bring in that electric field a surface, 8 00:00:47,851 --> 00:00:52,86 an open surface like a handkerchief or a piece of 9 00:00:52,86 --> 00:00:55,365 paper. And so here it is. 10 00:00:55,365 --> 00:01:00,583 Something like that. And I carve this surface up in 11 00:01:00,583 --> 00:01:05,384 very small surface elements, each with size DA, 12 00:01:05,384 --> 00:01:09,559 that's the area, teeny weeny little area, 13 00:01:09,559 --> 00:01:15,09 and let this be the normal, N roof, the normal on that 14 00:01:15,09 --> 00:01:20,959 surface. So now the local electric field 15 00:01:20,959 --> 00:01:26,218 say at that location would be for instance this. 16 00:01:26,218 --> 00:01:30,917 It's a vector. The electric flux D phi that 17 00:01:30,917 --> 00:01:37,295 goes through this little surface now is defined as the dot 18 00:01:37,295 --> 00:01:43,56 product of E and the vector perpendicular to this element 19 00:01:43,56 --> 00:01:48,372 which has this as a magnitude DA. 20 00:01:48,372 --> 00:01:52,553 Now our book will always write for NDA simply DA. 21 00:01:52,553 --> 00:01:57,519 So I will do that also although I don't like it but I will 22 00:01:57,519 --> 00:02:02,658 follow the notation of the book. So this vector DA is always 23 00:02:02,658 --> 00:02:07,276 perpendicular to that little element DA and it has the 24 00:02:07,276 --> 00:02:10,934 magnitude DA. And so this since it is a dot 25 00:02:10,934 --> 00:02:15,813 product is the magnitude of E times the area DA times the 26 00:02:15,813 --> 00:02:21,09 cosine of the angle between these two vectors, 27 00:02:21,09 --> 00:02:23,484 theta. And this is scalar. 28 00:02:23,484 --> 00:02:28,559 The number can be larger than zero, smaller than zero, 29 00:02:28,559 --> 00:02:33,059 and it can be zero. And I can calculate the flux 30 00:02:33,059 --> 00:02:38,421 through the entire surface by doing an integral over that 31 00:02:38,421 --> 00:02:42,06 whole surface. The unit of flux follows 32 00:02:42,06 --> 00:02:47,805 immediately from the definition. That is newtons per coulombs 33 00:02:47,805 --> 00:02:54,708 for the units of this flux, is newtons per coulombs times 34 00:02:54,708 --> 00:02:59,02 square meters. But no one ever thinks of it 35 00:02:59,02 --> 00:03:01,69 that way. Just SU SI units. 36 00:03:01,69 --> 00:03:07,85 I can give you a some intuition for this flux by comparing it 37 00:03:07,85 --> 00:03:13,087 first with an airflow. These red arrows that you see 38 00:03:13,087 --> 00:03:19,247 there represent the velocity of air and you see there a black 39 00:03:19,247 --> 00:03:22,346 rectangle three times. 40 00:03:22,346 --> 00:03:26,822 In the first case notice that the normal to the surface of 41 00:03:26,822 --> 00:03:31,377 that area is parallel to the velocity vector of the air and 42 00:03:31,377 --> 00:03:36,325 so if you want to know now what the amount of air is in terms of 43 00:03:36,325 --> 00:03:41,037 cubic meters per second going through this rectangle it would 44 00:03:41,037 --> 00:03:43,393 be V times A, it's very simple. 45 00:03:43,393 --> 00:03:48,026 However, if you rotate this rectangle ninety degrees so that 46 00:03:48,026 --> 00:03:52,738 the normal to that rectangle is perpendicular to the velocity 47 00:03:52,738 --> 00:03:56,668 vector, nothing goes through that 48 00:03:56,668 --> 00:04:01,51 rectangle and so it's zero. And so now the flux -- the air 49 00:04:01,51 --> 00:04:04,738 flux is zero, and if the angle is sixty 50 00:04:04,738 --> 00:04:09,58 degrees then it is of course V times A times the cosine of 51 00:04:09,58 --> 00:04:13,317 sixty degrees. Now think of these red vectors 52 00:04:13,317 --> 00:04:17,479 as electric fields. So now the electric flux going 53 00:04:17,479 --> 00:04:22,746 in the first case through that surface is now simply E times A. 54 00:04:22,746 --> 00:04:27,673 In the second case it's zero. And in the last case it is EA 55 00:04:27,673 --> 00:04:32,223 times the cosine of sixty degrees. 56 00:04:32,223 --> 00:04:36,769 So you can sometimes think of this as airflows. 57 00:04:36,769 --> 00:04:42,698 We also saw that when we dealt with field lines that can come 58 00:04:42,698 --> 00:04:47,936 in sometimes very handy. I now take a surface which is 59 00:04:47,936 --> 00:04:52,679 not open as this one is, this is an open surface. 60 00:04:52,679 --> 00:04:58,213 Can come in from both sides. But now I choose one that is 61 00:04:58,213 --> 00:05:03,997 completely closed. Like a potato bag or a balloon. 62 00:05:03,997 --> 00:05:08,817 I'll draw, put this line in here to give you a feeling 63 00:05:08,817 --> 00:05:12 there's a completely closed surface. 64 00:05:12 --> 00:05:17,456 So you can only get inside if you penetrate that surface from 65 00:05:17,456 --> 00:05:21,094 the outside. And so now I can put up here 66 00:05:21,094 --> 00:05:25,823 and here these normals, DA and there's another normal 67 00:05:25,823 --> 00:05:31,188 here, maybe in this direction DA, in this case by convention 68 00:05:31,188 --> 00:05:36,183 the normal to the surface locally to the surface 69 00:05:36,183 --> 00:05:40,607 is always from the inside of the surface to the outside world. 70 00:05:40,607 --> 00:05:44,523 It's uniquely determined because it's a closed surface. 71 00:05:44,523 --> 00:05:47,061 Here it was not uniquely determined. 72 00:05:47,061 --> 00:05:51,412 I arbitrarily chose this one but I could have flipped it over 73 00:05:51,412 --> 00:05:55,401 a hundred eighty degrees since it's an open surface it's 74 00:05:55,401 --> 00:05:58,301 ill-defined. Here it's never ill-defined. 75 00:05:58,301 --> 00:06:02,507 So the normal is always chosen to go from the inside to the 76 00:06:02,507 --> 00:06:05,046 outside. And now I can calculate the 77 00:06:05,046 --> 00:06:09,241 total flux going through this closed 78 00:06:09,241 --> 00:06:12,442 surface. Locally multiplying E with DA, 79 00:06:12,442 --> 00:06:16,99 dot product over the whole surface, out comes a certain 80 00:06:16,99 --> 00:06:21,539 number, and that is now therefore the integral of E dot 81 00:06:21,539 --> 00:06:26,762 DA integrated over that closed surface and since it is a closed 82 00:06:26,762 --> 00:06:31,816 surface we put a circle here to remind us that it is a closed 83 00:06:31,816 --> 00:06:37,375 integral and here in this case it is a closed surface. 84 00:06:37,375 --> 00:06:42,014 And this now is the total flux through that surface. 85 00:06:42,014 --> 00:06:47,379 It could be larger than zero. It could be smaller than zero. 86 00:06:47,379 --> 00:06:50,289 It's a scalar, it's not a vector. 87 00:06:50,289 --> 00:06:55,382 It could be equal to zero. If it's equal to zero then you 88 00:06:55,382 --> 00:07:00,657 can think of it whatever flows in if you think of it as air 89 00:07:00,657 --> 00:07:04,84 also flows out. If more flows out than flows in 90 00:07:04,84 --> 00:07:11,504 then it is positive. If more flows in than flows out 91 00:07:11,504 --> 00:07:16,939 it is negative. So let's now calculate the flux 92 00:07:16,939 --> 00:07:22,847 for a very simple case where I have a point charge. 93 00:07:22,847 --> 00:07:29,345 So here I have a point charge and I'm going to put a bag 94 00:07:29,345 --> 00:07:35,016 around this point charge and the bag is a sphere. 95 00:07:35,016 --> 00:07:42,223 It is a sphere and the sphere has radius capital R. 96 00:07:42,223 --> 00:07:47,941 And let this charge be plus Q. Just for simplicity. 97 00:07:47,941 --> 00:07:52,057 Well, I pick a small element DA here. 98 00:07:52,057 --> 00:07:56,288 And at element DA is radially outward. 99 00:07:56,288 --> 00:07:59,604 DA. This is the normal to that 100 00:07:59,604 --> 00:08:05,778 surface so that this radial. The electric field at that 101 00:08:05,778 --> 00:08:13,096 point is also radial. We have dealt with that before. 102 00:08:13,096 --> 00:08:18,82 So DA and E not only here but anywhere on the surface of this 103 00:08:18,82 --> 00:08:23,303 sphere are parallel. For the cosine of the angle 104 00:08:23,303 --> 00:08:27,119 equals one. I can also introduce here the 105 00:08:27,119 --> 00:08:33,129 unit vector R roof which is the unit vector going from capital Q 106 00:08:33,129 --> 00:08:39,234 to that element where I evaluate the teeny weeny little amount of 107 00:08:39,234 --> 00:08:42,477 flux. So if now I want to know what 108 00:08:42,477 --> 00:08:46,44 the total flux is through this 109 00:08:46,44 --> 00:08:51,971 sphere that's very easy because since this is a sphere the E 110 00:08:51,971 --> 00:08:56,938 vector in magnitude is everywhere the same because the 111 00:08:56,938 --> 00:09:01,062 radius is the same, the same distance to this 112 00:09:01,062 --> 00:09:05,936 charge, and DA and E are parallel, so it's simply the 113 00:09:05,936 --> 00:09:10,435 surface four pi R squared of that sphere times E. 114 00:09:10,435 --> 00:09:14,559 And so now I have that the total 115 00:09:14,559 --> 00:09:21,25 flux through that closed surface is simply four pi R 116 00:09:21,25 --> 00:09:25,317 squared times E. Well what is E? 117 00:09:25,317 --> 00:09:32,663 The electric field at this distance R equals Q divided by 118 00:09:32,663 --> 00:09:38,304 four pi epsilon zero R squared times R roof. 119 00:09:38,304 --> 00:09:46,175 That gives me the direction and so if I know that the flux is 120 00:09:46,175 --> 00:09:51,73 four pi R squared times E I put the four 121 00:09:51,73 --> 00:09:56,323 pi R squared here, I lose the four pi R squared, 122 00:09:56,323 --> 00:10:01,99 and I find that the E vector, at least the magnitude of the 123 00:10:01,99 --> 00:10:06,779 electric field -- uh excuse me, that the flux phi, 124 00:10:06,779 --> 00:10:11,958 that's what I want to calculate, I multiply this by E, 125 00:10:11,958 --> 00:10:15,085 equals Q divided by epsilon zero. 126 00:10:15,085 --> 00:10:20,361 And this is independent of the distance R. 127 00:10:20,361 --> 00:10:24,984 And that's not so surprising because if you think of it as 128 00:10:24,984 --> 00:10:30,093 air flowing out then all the air has to come out somehow whether 129 00:10:30,093 --> 00:10:34,878 I make the sphere this big or whether I make the sphere this 130 00:10:34,878 --> 00:10:37,554 big. So the flux being independent 131 00:10:37,554 --> 00:10:42,095 of the size of my sphere, the flux is given by the charge 132 00:10:42,095 --> 00:10:46,717 which is right here at the center divided by epsilon zero. 133 00:10:46,717 --> 00:10:50,772 Now if I had chosen some other shape, not a sphere, 134 00:10:50,772 --> 00:10:56,868 but I have dented it like this, it's clear that the air that 135 00:10:56,868 --> 00:10:59,825 flows out would be exactly the same. 136 00:10:59,825 --> 00:11:04,556 And so I don't have to take a sphere to find this result. 137 00:11:04,556 --> 00:11:09,54 I could have taken any type of strange closed surface around 138 00:11:09,54 --> 00:11:14,271 this point charge and I would have found exactly the same 139 00:11:14,271 --> 00:11:17,059 result. And if I put more than one 140 00:11:17,059 --> 00:11:22,043 charge inside this potato bag then clearly since I know that 141 00:11:22,043 --> 00:11:27,619 electric fields from different charges can be added, 142 00:11:27,619 --> 00:11:33,427 should be added vectorially, it is clear that the relation 143 00:11:33,427 --> 00:11:39,133 should also hold for any collection of charges inside the 144 00:11:39,133 --> 00:11:45,451 bag and therefore we now arrive at our first milestone in eight 145 00:11:45,451 --> 00:11:48,814 oh two, which we call Gauss's law. 146 00:11:48,814 --> 00:11:54,214 And Gauss's law says that the flux, the electric flux, 147 00:11:54,214 --> 00:11:59,105 going through a closed surface, being 148 00:11:59,105 --> 00:12:05,599 the closed surface of E dot DA is the sum of all charges Q 149 00:12:05,599 --> 00:12:12,321 which are inside the bag that you may choose at any time you 150 00:12:12,321 --> 00:12:16,537 pick that bag divided by epsilon zero. 151 00:12:16,537 --> 00:12:23,258 And this is the first of four equations of Maxwell which are 152 00:12:23,258 --> 00:12:29,638 at the heart of this course. So the electric flux through 153 00:12:29,638 --> 00:12:33,419 any closed surface is always the 154 00:12:33,419 --> 00:12:37,868 charge inside that closed surface divided by epsilon zero. 155 00:12:37,868 --> 00:12:42,551 And if that flux happens to be zero, it means there is no net 156 00:12:42,551 --> 00:12:46,142 charge inside the bag. There could be positive, 157 00:12:46,142 --> 00:12:50,2 there could be negative charges, but the net is zero. 158 00:12:50,2 --> 00:12:54,493 Gauss's law always holds. No matter how weird the charge 159 00:12:54,493 --> 00:12:58,942 distribution inside the bag. No matter how weird the shape 160 00:12:58,942 --> 00:13:02,298 of this bag. It always holds. 161 00:13:02,298 --> 00:13:06,388 But Gauss's law won't help you very much if you don't have a 162 00:13:06,388 --> 00:13:10,13 situation whereby the charges are distributed in a very 163 00:13:10,13 --> 00:13:12,764 symmetric way. Gauss's law holds but it 164 00:13:12,764 --> 00:13:16,922 doesn't do you any good if you want to calculate the electric 165 00:13:16,922 --> 00:13:18,793 field. In order to calculate 166 00:13:18,793 --> 00:13:23,021 successfully the electric field you do need forms of symmetry, 167 00:13:23,021 --> 00:13:27,11 and there are three forms of symmetry that we will deal with 168 00:13:27,11 --> 00:13:30,021 in eight oh two. One is of course spherical 169 00:13:30,021 --> 00:13:32,447 symmetry. Another one is cylindrical 170 00:13:32,447 --> 00:13:37,429 symmetry. And a third one is flat planes 171 00:13:37,429 --> 00:13:41,078 with uniformly charged distributions. 172 00:13:41,078 --> 00:13:45,133 Then we also have situations of symmetry. 173 00:13:45,133 --> 00:13:51,52 And so now I would like to as a first example use an application 174 00:13:51,52 --> 00:13:57,602 of Gauss's law and I will start with a situation of spherical 175 00:13:57,602 --> 00:14:00,846 symmetry. And I use a thin shell, 176 00:14:00,846 --> 00:14:05,206 a hollow sphere, which is thin, 177 00:14:05,206 --> 00:14:09,349 and so this radius is R and I put charge Q on here but it is 178 00:14:09,349 --> 00:14:11,947 uniformly distributed. That's crucial. 179 00:14:11,947 --> 00:14:15,598 If it's not uniformly distributed I have no symmetry, 180 00:14:15,598 --> 00:14:19,32 I can't do the problem. So it's uniformly distributed. 181 00:14:19,32 --> 00:14:23,393 We will learn later in the course that it's very easy to do 182 00:14:23,393 --> 00:14:27,536 this because any conductor of this shape if you bring charge 183 00:14:27,536 --> 00:14:31,187 on it will automatically distribute itself uniformly. 184 00:14:31,187 --> 00:14:35,33 So we have the charge plus Q on there uniformly distributed, 185 00:14:35,33 --> 00:14:38,912 that's a must, and I would like to 186 00:14:38,912 --> 00:14:43,151 know now what is the electric field here at a distance R from 187 00:14:43,151 --> 00:14:47,179 the center and what is the electric -- electric field here 188 00:14:47,179 --> 00:14:51,489 at a distance R from the center. In other words I want to know 189 00:14:51,489 --> 00:14:54,74 what is the electric field everywhere in space. 190 00:14:54,74 --> 00:14:58,414 Just due to this charged -- uniformly charged sphere. 191 00:14:58,414 --> 00:15:01,453 And with Gauss's law it just goes like that. 192 00:15:01,453 --> 00:15:04,35 You now have to choose your Gauss surface. 193 00:15:04,35 --> 00:15:08,448 And if you don't choose it in a clever way you get nowhere. 194 00:15:08,448 --> 00:15:12,193 In a case like this I would think it 195 00:15:12,193 --> 00:15:16,113 is rather obvious that the Gauss surface that you would 196 00:15:16,113 --> 00:15:19,67 choose are themselves spheres, concentric spheres. 197 00:15:19,67 --> 00:15:23,953 If you want to know what the electric field is at this point 198 00:15:23,953 --> 00:15:28,018 you choose a sphere with this radius R going through that 199 00:15:28,018 --> 00:15:32,156 point and if you want to know what it here is you choose a 200 00:15:32,156 --> 00:15:36,003 sphere going through that point. All the way enclosed. 201 00:15:36,003 --> 00:15:39,488 It's a concentric sphere. And now you have to use 202 00:15:39,488 --> 00:15:43,045 symmetry arguments. And the symmetry arguments are 203 00:15:43,045 --> 00:15:45,12 the following. 204 00:15:45,12 --> 00:15:48,174 Since this is spherically symmetric, this problem, 205 00:15:48,174 --> 00:15:51,104 if you were here, whatever the electric field is 206 00:15:51,104 --> 00:15:54,844 here in magnitude must be the same as it is there and it must 207 00:15:54,844 --> 00:15:58,397 be the same as it is there. Because of the symmetry of the 208 00:15:58,397 --> 00:16:01,826 problem, it couldn't be any larger here than it could be 209 00:16:01,826 --> 00:16:03,072 here. That's obvious. 210 00:16:03,072 --> 00:16:06,314 That's a symmetry argument because the charge here is 211 00:16:06,314 --> 00:16:09,617 uniformly distributed. That's symmetry argument number 212 00:16:09,617 --> 00:16:11,487 one. Now comes another symmetry 213 00:16:11,487 --> 00:16:13,918 argument. And that is the electric field 214 00:16:13,918 --> 00:16:17,39 if there is an electric field must 215 00:16:17,39 --> 00:16:20,958 be either radially pointing outwards or radially pointing 216 00:16:20,958 --> 00:16:23,188 inwards. So either it has to be like 217 00:16:23,188 --> 00:16:26,247 this or it has to be like this and here the same. 218 00:16:26,247 --> 00:16:30,134 Either like this or like this. Because we already know if this 219 00:16:30,134 --> 00:16:33,893 is a positive charge then it's going to be pointing outward. 220 00:16:33,893 --> 00:16:37,653 It cannot go like this or like this because nature could not 221 00:16:37,653 --> 00:16:41,093 decide in this spherically symmetric problem to go like 222 00:16:41,093 --> 00:16:43,77 this or like this. It can only go radially. 223 00:16:43,77 --> 00:16:47,211 That's the second symmetry argument. 224 00:16:47,211 --> 00:16:51,733 So now if we go to this sphere now and we know that E is 225 00:16:51,733 --> 00:16:55,927 radially outwards apart from a plus or a minus sign, 226 00:16:55,927 --> 00:17:00,532 apart from the fact that the angle between DA and E could 227 00:17:00,532 --> 00:17:04,644 either be zero degrees or a hundred eighty degrees, 228 00:17:04,644 --> 00:17:08,591 we know now that the surface area of that sphere, 229 00:17:08,591 --> 00:17:13,114 which is four pi R squared, times the magnitude of the E 230 00:17:13,114 --> 00:17:16,897 vector right here, I can do that now because DA 231 00:17:16,897 --> 00:17:21,585 and E are either parallel or antiparallel. 232 00:17:21,585 --> 00:17:25,988 That must be equal to Q inside divided by epsilon zero. 233 00:17:25,988 --> 00:17:29,169 There is no Q inside, so E must be zero. 234 00:17:29,169 --> 00:17:33,327 That's an amazing result. You say well uh there's no 235 00:17:33,327 --> 00:17:36,426 charge inside. Still an amazing result. 236 00:17:36,426 --> 00:17:40,993 Because it means that anywhere inside here no matter what 237 00:17:40,993 --> 00:17:45,56 radius you choose the electric field equals exactly zero. 238 00:17:45,56 --> 00:17:50,616 And it means that if some crazy conspiracy of all these charges 239 00:17:50,616 --> 00:17:55,182 that are uniformly distributed here which 240 00:17:55,182 --> 00:18:00,187 each individually contribute to the electric field inside 241 00:18:00,187 --> 00:18:05,729 through Coulomb's law that all those together can for through a 242 00:18:05,729 --> 00:18:10,197 conspiracy make the E field everywhere inside zero. 243 00:18:10,197 --> 00:18:13,325 It's a nontrivial result. All right. 244 00:18:13,325 --> 00:18:17,437 So now we know that the E field inside is zero. 245 00:18:17,437 --> 00:18:20,207 So this is for R smaller than R. 246 00:18:20,207 --> 00:18:25,57 Let's now go R larger than R. Everything I told you holds for 247 00:18:25,57 --> 00:18:29,378 the sphere which is outside this 248 00:18:29,378 --> 00:18:31,813 hollow sphere. Everything holds. 249 00:18:31,813 --> 00:18:36,447 The E field here must be the same everywhere on the surface. 250 00:18:36,447 --> 00:18:39,903 DA and E are either parallel or antiparallel. 251 00:18:39,903 --> 00:18:44,615 So I can write down again that four pi R squared which is the 252 00:18:44,615 --> 00:18:49,092 surface area times the electric field vector must be the Q 253 00:18:49,092 --> 00:18:53,098 inside divided by epsilon zero but this Q is that Q. 254 00:18:53,098 --> 00:18:56,004 It's not zero. There is charge inside. 255 00:18:56,004 --> 00:19:00,638 And so now I know that the electric field E 256 00:19:00,638 --> 00:19:05,43 in terms of its magnitude is Q divided by four pi R squared 257 00:19:05,43 --> 00:19:09,065 epsilon zero. And we know the direction if it 258 00:19:09,065 --> 00:19:13,939 is positive of course it is radially outwards and if this is 259 00:19:13,939 --> 00:19:18,483 negative it's radially inwards. And this is a nontrivial 260 00:19:18,483 --> 00:19:21,126 result. We have seen this before. 261 00:19:21,126 --> 00:19:26 If I have put all the charge right here at the middle at the 262 00:19:26 --> 00:19:30,214 center we would have gotten exactly the same answer. 263 00:19:30,214 --> 00:19:34,096 We've seen that before. In other 264 00:19:34,096 --> 00:19:39,208 words whether the charge is uniformly distributed over a 265 00:19:39,208 --> 00:19:44,971 sphere or whether the charge is all of it exactly at the center 266 00:19:44,971 --> 00:19:48,875 of the sphere, that makes no difference for 267 00:19:48,875 --> 00:19:53,336 the E field as long as you're outside the sphere. 268 00:19:53,336 --> 00:19:58,913 If you plot the electric field as a function of R and if here 269 00:19:58,913 --> 00:20:03,839 is capital R and if this is the -- the field strength, 270 00:20:03,839 --> 00:20:09,694 then you would get that the electric field is zero 271 00:20:09,694 --> 00:20:14,634 inside, jumps to a maximum value and this falls off as one 272 00:20:14,634 --> 00:20:18,88 over R squared proportional to one over R squared. 273 00:20:18,88 --> 00:20:23,733 If I go back to the situation that the charge -- that the 274 00:20:23,733 --> 00:20:29,019 electric field inside is zero, you may say isn't that a little 275 00:20:29,019 --> 00:20:33,005 bit of a cheat. Because yeah there is no charge 276 00:20:33,005 --> 00:20:36,038 inside. But have you really used the 277 00:20:36,038 --> 00:20:40,025 charge outside. And if you have used it how did 278 00:20:40,025 --> 00:20:42,081 you use it. 279 00:20:42,081 --> 00:20:45,489 Well I have used it. I use it through my symmetry 280 00:20:45,489 --> 00:20:48,116 arguments. The symmetry arguments take 281 00:20:48,116 --> 00:20:51,879 into account that the charge is uniformly distributed. 282 00:20:51,879 --> 00:20:56,352 If the charge on the sphere had not been uniformly distributed I 283 00:20:56,352 --> 00:21:00,47 could not have used the symmetry argument and therefore the 284 00:21:00,47 --> 00:21:04,304 electric field inside would in fact not have been zero. 285 00:21:04,304 --> 00:21:08,635 If there is more charge on the sphere here than there is there 286 00:21:08,635 --> 00:21:11,404 the field inside the sphere is not zero. 287 00:21:11,404 --> 00:21:17,598 So I have used all that charge by using my symmetry argument. 288 00:21:17,598 --> 00:21:22,707 Gauss's law and Coulomb's law in a way are the same law. 289 00:21:22,707 --> 00:21:27,444 They both link the electric field with the charge Q. 290 00:21:27,444 --> 00:21:32,552 Key is the fact that the electric force falls off as one 291 00:21:32,552 --> 00:21:36,639 over R square. If the electric field strength 292 00:21:36,639 --> 00:21:42,026 did not fall off as one over R square Gauss's law would not 293 00:21:42,026 --> 00:21:45,649 even hold. And the electric field inside 294 00:21:45,649 --> 00:21:49,942 this uniformly charged sphere would 295 00:21:49,942 --> 00:21:52,914 not be zero. So it is the immediate 296 00:21:52,914 --> 00:21:58,07 consequence of the fact that electric forces fall off as one 297 00:21:58,07 --> 00:22:02,003 over R squared. Gravitational forces also fall 298 00:22:02,003 --> 00:22:06,897 off as one over R squared. Therefore if you take a planet 299 00:22:06,897 --> 00:22:12,054 if it existed which is a hollow spherically spherical planet 300 00:22:12,054 --> 00:22:17,036 with hollow inside it means there would be no 301 00:22:17,036 --> 00:22:20,709 gravitational field inside that hollow planet. 302 00:22:20,709 --> 00:22:25,363 So if you were there there would be no gravitational force 303 00:22:25,363 --> 00:22:27,486 on you. If it is spherical. 304 00:22:27,486 --> 00:22:32,221 If that planet were a cubical planet then the gravitational 305 00:22:32,221 --> 00:22:36,548 field inside would not be zero. You say well big deal, 306 00:22:36,548 --> 00:22:41,284 with eight oh one we always take a planet and then it's not 307 00:22:41,284 --> 00:22:45,938 as far as we're outside the planet we put all the mass and 308 00:22:45,938 --> 00:22:49,04 we consider it as a point. Yeah indeed. 309 00:22:49,04 --> 00:22:52,711 It's not a big deal for you and it 310 00:22:52,711 --> 00:22:56,557 is not a big deal for me but it was a big deal for Newton. 311 00:22:56,557 --> 00:23:00,538 He intuitively sensed that it was correct that if you have a 312 00:23:00,538 --> 00:23:03,71 planet of uniform mass distribution that you can 313 00:23:03,71 --> 00:23:07,488 consider it as a point mass as long as you're outside the 314 00:23:07,488 --> 00:23:10,052 planet. But it took him twenty years to 315 00:23:10,052 --> 00:23:13,089 prove it and he finally published his results. 316 00:23:13,089 --> 00:23:15,45 It would take us now thirty seconds. 317 00:23:15,45 --> 00:23:17,88 He didn't have access to Gauss's law. 318 00:23:17,88 --> 00:23:21,318 Came about a hundred years later. 319 00:23:21,318 --> 00:23:26,164 But the net result is that you see here in front of you that if 320 00:23:26,164 --> 00:23:29,135 you have uniformly charge distribution, 321 00:23:29,135 --> 00:23:32,418 and you can draw the parallel with gravity, 322 00:23:32,418 --> 00:23:36,874 that it's you get the same electric field outside that you 323 00:23:36,874 --> 00:23:41,096 would have gotten if all the charge is at one location. 324 00:23:41,096 --> 00:23:44,301 At the center. This is spherical symmetry, 325 00:23:44,301 --> 00:23:47,271 number one. That's the easiest symmetry 326 00:23:47,271 --> 00:23:51,805 that we have in eight oh two. Now I will present you with a 327 00:23:51,805 --> 00:23:56,033 second form of symmetry which is a 328 00:23:56,033 --> 00:24:00,648 flat horizontal plane. And I want you to work out most 329 00:24:00,648 --> 00:24:04,915 of it but I'll help you a little bit to set it up. 330 00:24:04,915 --> 00:24:09,094 Suppose we have a plane which is very very large. 331 00:24:09,094 --> 00:24:12,49 Think of it for now as infinitely large. 332 00:24:12,49 --> 00:24:16,409 That doesn't exist of course infinitely large. 333 00:24:16,409 --> 00:24:21,634 And I put on this plane charge. And I put a certain amount of 334 00:24:21,634 --> 00:24:25,814 charge density which I call sigma. 335 00:24:25,814 --> 00:24:28,829 Sigma is an amount of charge Q per area A. 336 00:24:28,829 --> 00:24:32,802 So it is a certain number of coulombs per square meter. 337 00:24:32,802 --> 00:24:37,215 And it's uniformly distributed, so the whole plane everywhere 338 00:24:37,215 --> 00:24:40,746 has the same number of coulombs per square meter. 339 00:24:40,746 --> 00:24:43,1 Or microcoulombs or nanocoulombs, 340 00:24:43,1 --> 00:24:46,63 whatever you prefer. And this is a plane which is 341 00:24:46,63 --> 00:24:50,603 huge and you are being asked what is the electric field 342 00:24:50,603 --> 00:24:54,207 anywhere in space, just like we before we ask what 343 00:24:54,207 --> 00:24:57,681 is the electric field anywhere 344 00:24:57,681 --> 00:25:01,45 inside the sphere and anywhere outside the sphere. 345 00:25:01,45 --> 00:25:06,141 Now I want to know what it is anywhere in the vicinity of this 346 00:25:06,141 --> 00:25:08,91 plane. If now you pick a clever Gauss 347 00:25:08,91 --> 00:25:11,987 surface the answer pops out very quickly. 348 00:25:11,987 --> 00:25:16,678 If you would choose a sphere as a Gauss surface you're dead in 349 00:25:16,678 --> 00:25:20,908 the waters, you get nowhere because there's no spherical 350 00:25:20,908 --> 00:25:23,6 symmetry. I will define for you that 351 00:25:23,6 --> 00:25:28,966 Gauss surface but I want you to work out at 352 00:25:28,966 --> 00:25:32,75 home how you get the electric field. 353 00:25:32,75 --> 00:25:39,454 Suppose I want to know what the electric field is at a distance 354 00:25:39,454 --> 00:25:44,644 D above the plane. What I do now is I choose this 355 00:25:44,644 --> 00:25:48,644 as my Gauss surface. Watch me closely. 356 00:25:48,644 --> 00:25:52,861 This is the intersection with the plane. 357 00:25:52,861 --> 00:25:59,178 This is my Gauss surface. It is a closed surface. 358 00:25:59,178 --> 00:26:05,424 Three conditions have to be met for you to be able to calculate 359 00:26:05,424 --> 00:26:09,353 what the E vector is at that location D. 360 00:26:09,353 --> 00:26:15,598 The first one is that this is a flat plane here and this is the 361 00:26:15,598 --> 00:26:20,333 same flat plane. Must be parallel to this plane. 362 00:26:20,333 --> 00:26:24,766 That's a must. If you don't do that you can't 363 00:26:24,766 --> 00:26:30,508 use Gauss's law. The second one is that these 364 00:26:30,508 --> 00:26:35,526 vertical walls that you have here are indeed perpendicular to 365 00:26:35,526 --> 00:26:38,454 that plane. In other words these are 366 00:26:38,454 --> 00:26:42,135 parallel and these are ver- exactly vertical. 367 00:26:42,135 --> 00:26:47,154 If you don't make them vertical if you do this you're dead in 368 00:26:47,154 --> 00:26:50,249 the waters. Can't use Gauss's law very 369 00:26:50,249 --> 00:26:53,511 effectively. And then the third argument 370 00:26:53,511 --> 00:26:58,446 which is very important that this flat surface is a distance 371 00:26:58,446 --> 00:27:03,633 D above the plane and that this flat surface is 372 00:27:03,633 --> 00:27:06,476 exactly the same distance below the plane. 373 00:27:06,476 --> 00:27:09,736 And you can already smell why that is important. 374 00:27:09,736 --> 00:27:13,759 Because if you ever want to use a symmetry argument if this 375 00:27:13,759 --> 00:27:17,851 plane is uniformly charged the electric field vector here in 376 00:27:17,851 --> 00:27:21,735 terms of magnitude obviously must be the same as there in 377 00:27:21,735 --> 00:27:24,509 terms of magnitude, maybe not in terms of 378 00:27:24,509 --> 00:27:27,977 direction, as long as this D is the same as that D. 379 00:27:27,977 --> 00:27:31,93 So that's why it's important that the two Ds are the same. 380 00:27:31,93 --> 00:27:37,272 And the only charge that you have inside when you apply 381 00:27:37,272 --> 00:27:41,457 Gauss's law is the charge which is of course here. 382 00:27:41,457 --> 00:27:45,3 That's the only charge inside that closed box. 383 00:27:45,3 --> 00:27:50,425 If you work this out at home you will find an amazing result. 384 00:27:50,425 --> 00:27:55,378 You will find that the electric flux through these vertical 385 00:27:55,378 --> 00:27:59,051 walls is zero. Nothing comes out through the 386 00:27:59,051 --> 00:28:01,613 vertical walls. Think about it. 387 00:28:01,613 --> 00:28:05,884 Why that is. Use symmetry arguments. 388 00:28:05,884 --> 00:28:09,65 But something comes out here or comes in here if it is a 389 00:28:09,65 --> 00:28:12,595 negative charge and something goes out here. 390 00:28:12,595 --> 00:28:16,43 And so you only have two contributions from those two end 391 00:28:16,43 --> 00:28:18,758 plates. You'll work on that and you 392 00:28:18,758 --> 00:28:22,662 will find perhaps to your amazing result that the electric 393 00:28:22,662 --> 00:28:26,771 field equals sigma divided by two epsilon zero and that it is 394 00:28:26,771 --> 00:28:29,921 independent of how far you are from that plane. 395 00:28:29,921 --> 00:28:34,03 Whether you're very far away or whether you're close it's the 396 00:28:34,03 --> 00:28:36,496 same. So if this is that plane and if 397 00:28:36,496 --> 00:28:40,378 the plane is positively charged 398 00:28:40,378 --> 00:28:45,569 then E would be like this here and E would be like this here 399 00:28:45,569 --> 00:28:51,024 and it would be independent of distance and if it is negatively 400 00:28:51,024 --> 00:28:56,038 charged E would be like so and it would be like -- like so 401 00:28:56,038 --> 00:29:01,493 pointing towards the plane and in all cases would the magnitude 402 00:29:01,493 --> 00:29:04,66 be sigma divided by two epsilon zero. 403 00:29:04,66 --> 00:29:08,708 Does it mean if I go very far away 404 00:29:08,708 --> 00:29:12,465 from that plane that it is still independent of the 405 00:29:12,465 --> 00:29:14,794 distance? Yeah, if that plane is 406 00:29:14,794 --> 00:29:18,101 infinitely large. But if the plane is only as 407 00:29:18,101 --> 00:29:22,685 large as the lecture hall here then clearly it would hold very 408 00:29:22,685 --> 00:29:27,044 accurately as long as I stay relatively close to the plane. 409 00:29:27,044 --> 00:29:31,477 In other words if my distance to the plane is small compared 410 00:29:31,477 --> 00:29:35,61 to the linear size of the plane. But if I go miles away, 411 00:29:35,61 --> 00:29:40,044 well of course then that plane is charged looks like a point 412 00:29:40,044 --> 00:29:43,567 charge if I'm five miles away from 413 00:29:43,567 --> 00:29:47,489 twenty-six one hundred if the plane is only as large as this 414 00:29:47,489 --> 00:29:51,677 lecture hall then it looks like a point charge and obviously the 415 00:29:51,677 --> 00:29:55,333 electric field will then fall off as one over R squared. 416 00:29:55,333 --> 00:29:59,056 So when I say the E field doesn't change with distance it 417 00:29:59,056 --> 00:30:02,911 means of course that you have to be relatively close to the 418 00:30:02,911 --> 00:30:06,302 surface relative to the linear size of that surface. 419 00:30:06,302 --> 00:30:10,423 So you are going to prove this and I'm going to use this now to 420 00:30:10,423 --> 00:30:16,183 calculate for you a much more complicated configuration of 421 00:30:16,183 --> 00:30:20,138 two charged planes but I use that result. 422 00:30:20,138 --> 00:30:25,279 That's very important. And suppose I have here a -- a 423 00:30:25,279 --> 00:30:30,025 plate, very large, nothing is infinitely large of 424 00:30:30,025 --> 00:30:35,858 course, and it has a surface charge density plus sigma and I 425 00:30:35,858 --> 00:30:41,296 have here a plate which has surface charge density minus 426 00:30:41,296 --> 00:30:46,634 sigma and the separation between these two 427 00:30:46,634 --> 00:30:51,53 plates happens to be D. And the question now is what is 428 00:30:51,53 --> 00:30:54,794 the electric field anywhere in space. 429 00:30:54,794 --> 00:30:58,965 Here, here and here. And we'll think of them as 430 00:30:58,965 --> 00:31:02,048 being infinitely large, each plate. 431 00:31:02,048 --> 00:31:05,765 And I now use the superposition principle. 432 00:31:05,765 --> 00:31:09,119 I say to myself aha. This plate alone, 433 00:31:09,119 --> 00:31:13,29 forget this one, this plate alone would give me 434 00:31:13,29 --> 00:31:17,732 an E vector oh stick to my colors, 435 00:31:17,732 --> 00:31:22,618 give me an E vector like so and that is sigma divided by two 436 00:31:22,618 --> 00:31:26,178 epsilon zero, this one is also pointing away 437 00:31:26,178 --> 00:31:29,821 from this, sigma divided by two epsilon zero, 438 00:31:29,821 --> 00:31:34,044 and here it's also sigma divided by two epsilon zero 439 00:31:34,044 --> 00:31:38,515 because it's independent of the distance to this plate. 440 00:31:38,515 --> 00:31:41,247 What is the negative charge doing? 441 00:31:41,247 --> 00:31:46,133 Well, the negative charge has E vectors pointing towards it. 442 00:31:46,133 --> 00:31:50,272 So here I have an E vector which is 443 00:31:50,272 --> 00:31:52,53 sigma divided by two epsilon zero. 444 00:31:52,53 --> 00:31:56,636 Here I have one that is sigma divided by two epsilon zero and 445 00:31:56,636 --> 00:31:59,715 I have one that is pointing towards the plate, 446 00:31:59,715 --> 00:32:02,588 which is sigma divided by two epsilon zero. 447 00:32:02,588 --> 00:32:06,146 I use the superposition principle, I can add electric 448 00:32:06,146 --> 00:32:10,183 vectors and when I do that I find that these two cancel each 449 00:32:10,183 --> 00:32:13,194 other out so the electric field here is zero. 450 00:32:13,194 --> 00:32:17,025 The electric field here is sigma divided by epsilon zero. 451 00:32:17,025 --> 00:32:20,583 The two support each other. They are both in the same 452 00:32:20,583 --> 00:32:25,155 direction. And the electric field here is 453 00:32:25,155 --> 00:32:28,692 again zero. And that is an amazing result. 454 00:32:28,692 --> 00:32:34,126 Of course it's only accurate if these plates are extraordinarily 455 00:32:34,126 --> 00:32:39,474 large and so if I have to draw the field lines in the situation 456 00:32:39,474 --> 00:32:43,528 like this then the field lines would be like so. 457 00:32:43,528 --> 00:32:48,703 If the upper plate is positive and the field in here would be 458 00:32:48,703 --> 00:32:52,584 the same everywhere, would be outside zero and 459 00:32:52,584 --> 00:32:57,419 outside zero here. Now clearly this cannot be true 460 00:32:57,419 --> 00:33:01,367 if you get into this area here where you are near the end of 461 00:33:01,367 --> 00:33:03,642 these plates. That is not possible. 462 00:33:03,642 --> 00:33:05,716 Why not? Well you can't use your 463 00:33:05,716 --> 00:33:09,73 symmetry arguments so Gauss's law is not going to help you if 464 00:33:09,73 --> 00:33:13,678 you get anywhere near this area. And it is very difficult to 465 00:33:13,678 --> 00:33:17,626 calculate the electric field configuration when you are near 466 00:33:17,626 --> 00:33:20,838 the edges, which we call the -- the fringe field. 467 00:33:20,838 --> 00:33:24,919 Maxwell of course was a clever man and he knew how to do that. 468 00:33:24,919 --> 00:33:29,816 Today we can also do that very easily with uh computers. 469 00:33:29,816 --> 00:33:33,384 But I'll show you from Maxwell's original publications 470 00:33:33,384 --> 00:33:37,692 that in a situation like that he was already perfectly capable of 471 00:33:37,692 --> 00:33:41,732 calculating these electric field lines and you have these two 472 00:33:41,732 --> 00:33:45,03 horizontal plates, which one is plus and which one 473 00:33:45,03 --> 00:33:48,733 is minus doesn't matter, he doesn't put arrows in there, 474 00:33:48,733 --> 00:33:52,166 and what you see is an extremely strong field inside 475 00:33:52,166 --> 00:33:55,128 the two plates, and remember that the density 476 00:33:55,128 --> 00:33:58,831 of field lines tells you something about the strength of 477 00:33:58,831 --> 00:34:02,207 the [inaudible] very strong field 478 00:34:02,207 --> 00:34:06,2 but when you get near the edge the field is not really zero. 479 00:34:06,2 --> 00:34:10,329 The field strength drops very rapidly because look the density 480 00:34:10,329 --> 00:34:12,427 is very low. But it is not zero. 481 00:34:12,427 --> 00:34:16,555 And the electric field is not zero here either and is not zero 482 00:34:16,555 --> 00:34:18,585 there. In our assumption in our 483 00:34:18,585 --> 00:34:22,511 simplification we have however assumed that the plate is so 484 00:34:22,511 --> 00:34:26,436 large that we don't have to worry about any end effects and 485 00:34:26,436 --> 00:34:30,971 in that case the electric field is only existent in 486 00:34:30,971 --> 00:34:34,218 between the plates but not anywhere else. 487 00:34:34,218 --> 00:34:38,927 I now want to demonstrate to you some of the things that we 488 00:34:38,927 --> 00:34:42,986 have learned today. And the first thing that I want 489 00:34:42,986 --> 00:34:47,533 to demonstrate is that the electric field outside a large 490 00:34:47,533 --> 00:34:52,566 plane is more or less constant. Doesn't matter how far away you 491 00:34:52,566 --> 00:34:55,083 are. Now the way I'm going to do 492 00:34:55,083 --> 00:34:59,467 that is of course I don't have an infinite large plane, 493 00:34:59,467 --> 00:35:03,445 the plane that you're going to see 494 00:35:03,445 --> 00:35:05,328 only a few square meters in size. 495 00:35:05,328 --> 00:35:08,152 And so with only something like one by one meter, 496 00:35:08,152 --> 00:35:11,859 then it would only be true that the electric field is very close 497 00:35:11,859 --> 00:35:14,565 to constant if I stay very close to that plane. 498 00:35:14,565 --> 00:35:18,037 The moment that I go out as far as a meter of course it's no 499 00:35:18,037 --> 00:35:20,214 longer true. So it's very qualitative, 500 00:35:20,214 --> 00:35:23,45 what I'm going to show you. But you're going to see very 501 00:35:23,45 --> 00:35:25,332 shortly there a very large plane. 502 00:35:25,332 --> 00:35:27,45 I'm going to get it in a few minutes. 503 00:35:27,45 --> 00:35:30,451 And let's assume that we look at that plane edge on. 504 00:35:30,451 --> 00:35:33,158 So here is that plane. Look at it from edge on, 505 00:35:33,158 --> 00:35:37,22 it will be put here. It will block your view, 506 00:35:37,22 --> 00:35:39,392 that's why we don't have it up now. 507 00:35:39,392 --> 00:35:42,777 And what I will do now is I will connect that with the 508 00:35:42,777 --> 00:35:46,673 VandeGraaff which is behind it. If you wait a few minutes then 509 00:35:46,673 --> 00:35:49,547 class will pay attention to me and not to you. 510 00:35:49,547 --> 00:35:53,251 Uh here is the VandeGraaff, we're going to attach it to the 511 00:35:53,251 --> 00:35:57,338 VandeGraaff and then we use this interesting fishing rod which is 512 00:35:57,338 --> 00:36:01,298 a small Mylar balloon which we will charge with the same charge 513 00:36:01,298 --> 00:36:05,449 as the VandeGraaff, the same charge as the plate, 514 00:36:05,449 --> 00:36:08,87 and we will hold that in front of the plane. 515 00:36:08,87 --> 00:36:12,051 And then of course there will be a force. 516 00:36:12,051 --> 00:36:15,631 So here is my glass rod. This is the vertical. 517 00:36:15,631 --> 00:36:19,608 And because there will be a repelling force on this 518 00:36:19,608 --> 00:36:22,949 air-filled balloon, there will be an angle. 519 00:36:22,949 --> 00:36:27,721 There's an electric force on it because the two have the same 520 00:36:27,721 --> 00:36:30,426 charge. And this is the angle theta 521 00:36:30,426 --> 00:36:35,119 that I will show you projected on that wall. 522 00:36:35,119 --> 00:36:38,977 And when I move this away from this plane you will see that the 523 00:36:38,977 --> 00:36:42,587 angle theta becomes smaller. Yes of course because look how 524 00:36:42,587 --> 00:36:45,512 small that plane is. No matter what I do if I go 525 00:36:45,512 --> 00:36:49,308 from twenty to forty centimeters you can't really say that the 526 00:36:49,308 --> 00:36:52,731 plane is infinitely large compared to forty centimeters. 527 00:36:52,731 --> 00:36:56,216 But you will see that the angle of theta will change very 528 00:36:56,216 --> 00:36:58,394 slowly. And then we will remove that 529 00:36:58,394 --> 00:37:01,879 plane and then we will do exactly the same experiment but 530 00:37:01,879 --> 00:37:06,297 we will use only the VandeGraaff which produces now an 531 00:37:06,297 --> 00:37:09,333 electric field. And that electric field now 532 00:37:09,333 --> 00:37:13,887 falls off as one over R squared. It's not constant as a function 533 00:37:13,887 --> 00:37:17,5 of distance but it falls off as one over R squared. 534 00:37:17,5 --> 00:37:21,331 This is a hollow sphere. So you can think of it as all 535 00:37:21,331 --> 00:37:24,945 the charge right at the center. As we demonstrated, 536 00:37:24,945 --> 00:37:27,33 it's on the blackboard still here. 537 00:37:27,33 --> 00:37:30,004 You know, you get that amazing result. 538 00:37:30,004 --> 00:37:34,124 And so now if I place this -- if I place this fishing rod, 539 00:37:34,124 --> 00:37:37,666 this balloon, near the spherical 540 00:37:37,666 --> 00:37:43,091 VandeGraaff you will see that this angle theta drops very fast 541 00:37:43,091 --> 00:37:45,937 when I start moving my hand away. 542 00:37:45,937 --> 00:37:50,562 Extraordinarily fast. If I double the distance to the 543 00:37:50,562 --> 00:37:55,899 center the force on that little object will become four times 544 00:37:55,899 --> 00:37:58,567 smaller. It's inverse R square. 545 00:37:58,567 --> 00:38:04,082 So let's first do the plane and then we'll try to do the -- the 546 00:38:04,082 --> 00:38:08,618 uh single VandeGraaff. And we'll try to optimize the 547 00:38:08,618 --> 00:38:11,561 light conditions. 548 00:38:11,561 --> 00:38:16,534 We have a projection here. There's a carbon arc. 549 00:38:16,534 --> 00:38:22,566 Which will hopefully produce some light in that direction. 550 00:38:22,566 --> 00:38:28,174 If the carbon arc works. [laughter] Marcos oh I forgot 551 00:38:28,174 --> 00:38:31,454 to turn on the power. Thank you. 552 00:38:31,454 --> 00:38:37,698 So this carbon arc is coming on now and you'll see there the 553 00:38:37,698 --> 00:38:42,565 shadows on that wall. See my hand, 554 00:38:42,565 --> 00:38:45,978 here is that plane, and it is far from infinitely 555 00:38:45,978 --> 00:38:49,249 large, that plane. If I were this far away from 556 00:38:49,249 --> 00:38:52,52 it, four centimeters, very close approximation, 557 00:38:52,52 --> 00:38:56,643 it would be infinitely large. But if I'm here and there and 558 00:38:56,643 --> 00:39:00,412 that's where I will be, of course it is not infinitely 559 00:39:00,412 --> 00:39:02,9 large anymore. So let's uh start the 560 00:39:02,9 --> 00:39:05,744 VandeGraaff. You can see that I turned it 561 00:39:05,744 --> 00:39:07,237 on. It's rotating now. 562 00:39:07,237 --> 00:39:11,006 I have to put charge on here so I'll touch it with the 563 00:39:11,006 --> 00:39:15,13 VandeGraaff and so this is now charged. 564 00:39:15,13 --> 00:39:17,657 It has the same charge as the plane. 565 00:39:17,657 --> 00:39:21,556 The plane is being charged. And here you see the angle. 566 00:39:21,556 --> 00:39:25,095 Try to remember that angle. It's hard to estimate, 567 00:39:25,095 --> 00:39:28,922 maybe fifteen degrees. You see the vertical and if now 568 00:39:28,922 --> 00:39:33,038 I -- it's about um thirty centimeters away from the plane. 569 00:39:33,038 --> 00:39:37,443 And if now I go back to fifty centimeters, which is where I am 570 00:39:37,443 --> 00:39:40,837 now, you see the angle hasn't changed very much. 571 00:39:40,837 --> 00:39:43,797 If I go further out, to sixty centimeters, 572 00:39:43,797 --> 00:39:47,552 yeah, the angle goes down a little. 573 00:39:47,552 --> 00:39:50,251 Of course it does. But not very much. 574 00:39:50,251 --> 00:39:53,85 And if I go far away, all the way to Mass Avenue, 575 00:39:53,85 --> 00:39:58,273 of course the force on this little object would be inversely 576 00:39:58,273 --> 00:40:02,996 R squared because then the whole plane would behave like a point 577 00:40:02,996 --> 00:40:05,545 source. So I've shown you that very 578 00:40:05,545 --> 00:40:09,818 close to this plane the electric field stays approximately 579 00:40:09,818 --> 00:40:12,367 constant. So if now we remove this, 580 00:40:12,367 --> 00:40:16,265 Marcos if you can yeah, you'll have to take this also 581 00:40:16,265 --> 00:40:19,356 off. Thank you very much. 582 00:40:19,356 --> 00:40:22,086 So now we have the VandeGraaff alone. 583 00:40:22,086 --> 00:40:26,18 So now we know that the electric field falls off as one 584 00:40:26,18 --> 00:40:29,592 over R squared. It's a very good approximation 585 00:40:29,592 --> 00:40:32,095 now. We can think of the charge as 586 00:40:32,095 --> 00:40:36,341 being right at the center. I will give it a little bit of 587 00:40:36,341 --> 00:40:38,843 charge. Oh, it is already charged. 588 00:40:38,843 --> 00:40:41,876 [laughter] OK. So look at the projection. 589 00:40:41,876 --> 00:40:46,426 The uh the balloon is now uh oh maybe thirty centimeters away 590 00:40:46,426 --> 00:40:49,989 from the center, maybe forty, 591 00:40:49,989 --> 00:40:53,371 boy, the angle is almost forty-five degrees. 592 00:40:53,371 --> 00:40:56,124 And now I go, I double the distance, 593 00:40:56,124 --> 00:40:59,978 I go to about ninety centimeters, and look at that 594 00:40:59,978 --> 00:41:03,281 angle theta. The angle theta is now down to 595 00:41:03,281 --> 00:41:07,056 oh maybe ten degrees. I will go back where I was. 596 00:41:07,056 --> 00:41:09,651 This angle is about forty degrees. 597 00:41:09,651 --> 00:41:14,213 And now it's very small and when I go here which is about a 598 00:41:14,213 --> 00:41:18,853 meter-and-a-half you can hardly see that there is any angle. 599 00:41:18,853 --> 00:41:24,441 It's only a few degrees. And so I've shown you only 600 00:41:24,441 --> 00:41:29,896 qualitatively that the electric field falls off very rapidly. 601 00:41:29,896 --> 00:41:34,624 In the vicinity of a hollow uniformly charged sphere. 602 00:41:34,624 --> 00:41:40,079 And that it doesn't fall off very fast if you are in the near 603 00:41:40,079 --> 00:41:44,716 vicinity of a plane. The second thing I want to show 604 00:41:44,716 --> 00:41:50,08 you has to do with the fact that the electric field inside a 605 00:41:50,08 --> 00:41:54,005 uniformly charged sphere is zero. 606 00:41:54,005 --> 00:41:57,729 Here I have a sphere which is not entirely closed. 607 00:41:57,729 --> 00:42:02,136 I can't make it closed because I want to demonstrate to you 608 00:42:02,136 --> 00:42:06,24 that there is no electric field inside when I charge it 609 00:42:06,24 --> 00:42:09,279 uniformly. And since I have to get inside 610 00:42:09,279 --> 00:42:12,927 I need an opening. There's nothing I can do about 611 00:42:12,927 --> 00:42:15,13 it. Since there is an opening, 612 00:42:15,13 --> 00:42:18,55 the electric field is not exactly zero inside. 613 00:42:18,55 --> 00:42:22,881 It's only true if this is a complete closed surface and if 614 00:42:22,881 --> 00:42:26,985 the charge is uniformly distributed. 615 00:42:26,985 --> 00:42:31,156 But it's a good approximation. The opening is quite small. 616 00:42:31,156 --> 00:42:35,474 And what I'm going to do is I'm going to charge this sphere. 617 00:42:35,474 --> 00:42:39,719 I'm putting charge outside. I use a device that we have not 618 00:42:39,719 --> 00:42:44,184 used before, but that's not so important, but here is now that 619 00:42:44,184 --> 00:42:47,112 hollow sphere. I'm going to put charge on 620 00:42:47,112 --> 00:42:49,6 there. Let's suppose it is positive 621 00:42:49,6 --> 00:42:52,015 charge. So this will be positively 622 00:42:52,015 --> 00:42:54,723 charged. Since it is a -- a conductor, 623 00:42:54,723 --> 00:42:58,519 as we will learn I think the next 624 00:42:58,519 --> 00:43:02,565 lecture or at least this week, that the charge will 625 00:43:02,565 --> 00:43:06,936 automatically distribute uniformly, only does that on a 626 00:43:06,936 --> 00:43:11,225 conductor, and now to demonstrate to you that there is 627 00:43:11,225 --> 00:43:14,786 an electric field here, I will use induction. 628 00:43:14,786 --> 00:43:19,319 I have two Ping Pong balls painted with conducting paint. 629 00:43:19,319 --> 00:43:22,961 They touch each other. Under influence of this 630 00:43:22,961 --> 00:43:27,493 electric field this one will become negative and this one 631 00:43:27,493 --> 00:43:32,434 will become positive, we have discussed that last 632 00:43:32,434 --> 00:43:36,182 time, you create a dipole. Not important that it is a 633 00:43:36,182 --> 00:43:38,057 dipole. I separate the two. 634 00:43:38,057 --> 00:43:41,877 I have negative charge here and positive charge there. 635 00:43:41,877 --> 00:43:45,986 I will touch any one of these two balls, it doesn't matter 636 00:43:45,986 --> 00:43:48,293 which one, with the electroscope. 637 00:43:48,293 --> 00:43:51,392 And you will see that there is charge there. 638 00:43:51,392 --> 00:43:55,573 So I have demonstrated then that there is an electric field 639 00:43:55,573 --> 00:43:59,178 outside that sphere. Now I will do exactly the same 640 00:43:59,178 --> 00:44:01,845 demonstration, but now I put these two 641 00:44:01,845 --> 00:44:04,891 conducting balls inside, 642 00:44:04,891 --> 00:44:06,929 so here they are. I touch them, 643 00:44:06,929 --> 00:44:10,869 you just have to t- trust me that I really will touch them, 644 00:44:10,869 --> 00:44:15,013 and then I will take them out. And if I didn't make a mistake, 645 00:44:15,013 --> 00:44:18,953 if I didn't touch the rim by accident, then I will show you 646 00:44:18,953 --> 00:44:22,961 that there is no electric field inside, it means there is no 647 00:44:22,961 --> 00:44:26,222 induction, so these balls did not pick up charge. 648 00:44:26,222 --> 00:44:29,551 I show you with the electroscope that indeed there 649 00:44:29,551 --> 00:44:34,103 is no charge on it. So that is the way I want to do 650 00:44:34,103 --> 00:44:37,391 this. So there is the electroscope. 651 00:44:37,391 --> 00:44:42,227 Here is the sphere. And the way I'm going to charge 652 00:44:42,227 --> 00:44:46,579 it has a nice name, it's called electrophorus, 653 00:44:46,579 --> 00:44:51,222 hard to pronounce, I first rub a glass plate with 654 00:44:51,222 --> 00:44:54,51 cat fur. Then I take a metal plate. 655 00:44:54,51 --> 00:44:58,669 I put on top. And I touch it with my finger. 656 00:44:58,669 --> 00:45:04,472 And now I transfer charge and you think about it why that is. 657 00:45:04,472 --> 00:45:10,966 I put it on here again, touch it again with my finger. 658 00:45:10,966 --> 00:45:14,347 I'm again charging it. Put it on top, 659 00:45:14,347 --> 00:45:19,42 touch it again with my finger. I want a little bit more 660 00:45:19,42 --> 00:45:22,519 charge. So I'm rubbing this again. 661 00:45:22,519 --> 00:45:26,277 Put this on top. Touch it with my finger. 662 00:45:26,277 --> 00:45:30,222 Every time I do that I feel a little shock. 663 00:45:30,222 --> 00:45:33,979 Put it on there. Touch it with my finger. 664 00:45:33,979 --> 00:45:38,018 OK. Let's hope that's enough. 665 00:45:38,018 --> 00:45:40,702 So now comes demonstration number one. 666 00:45:40,702 --> 00:45:44,329 These two spheres conducting completely discharged, 667 00:45:44,329 --> 00:45:46,723 I bring them close to this sphere. 668 00:45:46,723 --> 00:45:48,972 There they are. I separate them. 669 00:45:48,972 --> 00:45:51,801 And now they must have picked up charge. 670 00:45:51,801 --> 00:45:56,008 Shall I use this one or this one to touch the electroscope? 671 00:45:56,008 --> 00:45:59,272 The same to me. My right hand or my left hand, 672 00:45:59,272 --> 00:46:01,52 who wants right? Who wants left? 673 00:46:01,52 --> 00:46:04,567 The right ones have it. There's the charge. 674 00:46:04,567 --> 00:46:08,411 So I've shown you that there is an 675 00:46:08,411 --> 00:46:12,115 electric field there. Through induction I have 676 00:46:12,115 --> 00:46:16,311 created charge on here. Now I'll do the same inside. 677 00:46:16,311 --> 00:46:21,167 It's always tricky because if I hit -- if I hit the rim then 678 00:46:21,167 --> 00:46:24,541 it's not zero. This one has to go in first 679 00:46:24,541 --> 00:46:27,174 because the opening is too small. 680 00:46:27,174 --> 00:46:29,972 Then the second one has to come in. 681 00:46:29,972 --> 00:46:33,346 Now I have to touch them, and I really do. 682 00:46:33,346 --> 00:46:36,473 I wouldn't cheat on you. Not this time. 683 00:46:36,473 --> 00:46:41,164 They are now in contact with each other. 684 00:46:41,164 --> 00:46:45,605 And now I take one out. And I take the other out. 685 00:46:45,605 --> 00:46:50,787 Which one shall I touch it, there shouldn't be any charge 686 00:46:50,787 --> 00:46:55,414 on either one of them. We had left before or we had 687 00:46:55,414 --> 00:46:58,745 right before? Well let's do this one. 688 00:46:58,745 --> 00:47:01,059 This one? Who is for left? 689 00:47:01,059 --> 00:47:04,298 Who is for right? The lefts have it. 690 00:47:04,298 --> 00:47:06,889 Oh. [laughter] What happened? 691 00:47:06,889 --> 00:47:11,886 I must have touched the side. There's no way around it. 692 00:47:11,886 --> 00:47:16,697 I'll make sure that there is enough 693 00:47:16,697 --> 00:47:22,882 charge on it. I'll charge it once more. 694 00:47:22,882 --> 00:47:25,811 Discharge them. OK. 695 00:47:25,811 --> 00:47:30,368 We'll do it again. Go inside. 696 00:47:30,368 --> 00:47:34,111 Go inside. I touch them. 697 00:47:34,111 --> 00:47:38,017 Take it out. Take it out. 698 00:47:38,017 --> 00:47:40,621 Nothing. Nothing. 699 00:47:40,621 --> 00:47:50,263 Maybe a teeny weeny little bit, well, the electric field inside 700 00:47:50,263 --> 00:47:54,825 is not necessarily exactly zero. But it's extremely close. 701 00:47:54,825 --> 00:47:59,467 The last thing I want to show you has to do with the fringe 702 00:47:59,467 --> 00:48:04,269 field that we have seen here. I have here two parallel plates 703 00:48:04,269 --> 00:48:09,071 which I'm going to charge with an instrument that we have not 704 00:48:09,071 --> 00:48:12,993 seen before which is called the uh -- a Wimshurst. 705 00:48:12,993 --> 00:48:18,596 If I rotate this crank I can produce positive and negative 706 00:48:18,596 --> 00:48:21,63 charge. And this plate becomes 707 00:48:21,63 --> 00:48:27,805 positively charged and the other plate automatically becomes 708 00:48:27,805 --> 00:48:32,828 negatively charged. And I'm going to show this to 709 00:48:32,828 --> 00:48:36,176 you right there. That's the idea. 710 00:48:36,176 --> 00:48:38,688 Yeah. We will make it uh. 711 00:48:38,688 --> 00:48:42,141 So there you see these two plates. 712 00:48:42,141 --> 00:48:48,106 And you see a Ping Pong ball. And this Ping Pong ball is a 713 00:48:48,106 --> 00:48:52,496 conductor, we put conducting paint on it. 714 00:48:52,496 --> 00:48:55,865 And remember when I did the demonstration with the balloon 715 00:48:55,865 --> 00:48:59,293 which bounced between my head and the VandeGraaff and every 716 00:48:59,293 --> 00:49:02,484 time that it bounced on the VandeGraaff it took all the 717 00:49:02,484 --> 00:49:05,912 charge of the VandeGraaff and when it bounced on my head it 718 00:49:05,912 --> 00:49:08,572 took my charge, and so it went back and forth, 719 00:49:08,572 --> 00:49:11,704 along the field lines. And that is what I want to show 720 00:49:11,704 --> 00:49:13,891 you now. That this Ping Pong ball will 721 00:49:13,891 --> 00:49:17,26 start to probe that field first outside the capacitor or I 722 00:49:17,26 --> 00:49:21,685 shouldn't use the word capacitor with these plates, 723 00:49:21,685 --> 00:49:26,502 and then I will bring the Ping Pong ball inside and then you 724 00:49:26,502 --> 00:49:30,258 will see that the field is much stronger there. 725 00:49:30,258 --> 00:49:33,442 So let's first get some charge on there. 726 00:49:33,442 --> 00:49:37,85 And listen to the sounds. Every time that it h- hits it 727 00:49:37,85 --> 00:49:40,789 bangs. So it's following almost those 728 00:49:40,789 --> 00:49:44,3 field lines. And in doing that it's actually 729 00:49:44,3 --> 00:49:49,035 transferring charge every time from one plate to the other. 730 00:49:49,035 --> 00:49:53,117 It's nicely going around in an arc 731 00:49:53,117 --> 00:49:57,247 the way you see it there. So it's clear that there is an 732 00:49:57,247 --> 00:50:00,775 electric field outside. I've proven that to you. 733 00:50:00,775 --> 00:50:04,004 Otherwise it would never do what it's doing. 734 00:50:04,004 --> 00:50:07,682 So the electric field outside is not exactly zero, 735 00:50:07,682 --> 00:50:10,836 of course not. This plate is not infinitely 736 00:50:10,836 --> 00:50:13,538 large. And now I will bring this Ping 737 00:50:13,538 --> 00:50:16,917 Pong ball inside, I have to open up the -- the 738 00:50:16,917 --> 00:50:19,92 gap a little, and I will bring it inside. 739 00:50:19,92 --> 00:50:24,063 And you see the field is much stronger. 740 00:50:24,063 --> 00:50:28,657 Now it's going back and forth between those very high-density 741 00:50:28,657 --> 00:50:31,643 field lines, very strong electric field, 742 00:50:31,643 --> 00:50:36,313 going back and forth each time that it hits the plate it c- it 743 00:50:36,313 --> 00:50:40,448 changes polarity and this is not too different from the 744 00:50:40,448 --> 00:50:44,659 experiment I did with the balloon when I bounced it back 745 00:50:44,659 --> 00:50:49,176 from the VandeGraaff to my head and back to the VandeGraaff. 746 00:50:49,176 --> 00:50:51,014 OK. Start working on that 747 00:50:51,014 --> 00:50:53,924 assignment. It's not an easy assignment 748 00:50:53,924 --> 50:59 this week. See you Wednesday.