1 00:00:01,400 --> 00:00:07,500 this is lecture No. 11 of AV 2 3 2 electromagnetic 15 2 00:00:07,700 --> 00:00:11,400 Today we will study Gaza's Law based on that 3 00:00:11,400 --> 00:00:13,500 we will drive max notification 4 00:00:14,966 --> 00:00:16,600 application of Gauze's Law 5 00:00:16,766 --> 00:00:19,800 and this will color section 4.5 6 00:00:19,800 --> 00:00:22,300 and 4.6 of your textbook 7 00:00:22,466 --> 00:00:25,700 this is a demonstrate No. 6 of your semester 8 00:00:27,000 --> 00:00:29,800 the learning objectives of today's lecture 9 00:00:29,800 --> 00:00:32,100 would be to understand causes 10 00:00:32,100 --> 00:00:35,100 law for determining the electrical 11 00:00:35,700 --> 00:00:39,666 then on to derive the Maxel's first equation 12 00:00:39,966 --> 00:00:40,400 and then 13 00:00:40,400 --> 00:00:43,966 we'll discuss the basic applications of causes law 14 00:00:45,766 --> 00:00:46,766 Gaza's law 15 00:00:47,366 --> 00:00:48,600 states that 16 00:00:48,700 --> 00:00:53,900 the total electric flux to any closed surface 17 00:00:54,266 --> 00:00:54,733 remember 18 00:00:54,733 --> 00:00:59,566 it is total electric flux to any closed surface 19 00:01:00,200 --> 00:01:03,900 is equal to the total charge 20 00:01:03,933 --> 00:01:06,100 and closed by that surface 21 00:01:06,300 --> 00:01:06,600 so 22 00:01:06,600 --> 00:01:09,966 the total number of lines passing to a closed surface 23 00:01:09,966 --> 00:01:11,966 for example it is your closed surface 24 00:01:11,966 --> 00:01:14,500 then total number of electric 25 00:01:14,500 --> 00:01:17,133 the net electric number of electric uh 26 00:01:17,133 --> 00:01:18,800 flux lines passing through surface 27 00:01:18,800 --> 00:01:21,166 would be equal to be charged 28 00:01:21,166 --> 00:01:26,200 which is and closed by this closed surface 29 00:01:26,733 --> 00:01:28,566 so the net flats 30 00:01:29,766 --> 00:01:32,666 Netflix out of this surface 31 00:01:32,666 --> 00:01:36,666 that is size is equal to the total charge 32 00:01:36,666 --> 00:01:39,033 which is enclosed by that surface 33 00:01:39,133 --> 00:01:43,200 so that's why it is size equal to Q and closed 34 00:01:44,766 --> 00:01:46,700 so if you remember that 35 00:01:46,800 --> 00:01:49,800 the total number of electric clocks lines 36 00:01:49,800 --> 00:01:51,800 it is equal to a closed surface 37 00:01:51,800 --> 00:01:53,766 it is equal to the dock 38 00:01:53,766 --> 00:01:56,866 product of your electric clock density 39 00:01:56,866 --> 00:01:58,966 with that surface area 40 00:01:59,266 --> 00:02:03,000 and uh also by the the basic definition of that uh 41 00:02:03,000 --> 00:02:04,966 uh charge if we know we 42 00:02:04,966 --> 00:02:08,300 we know the one in charge density of uh 43 00:02:08,300 --> 00:02:11,400 of that uh at the geomaxi 44 00:02:11,400 --> 00:02:13,766 then we can find out the total charge 45 00:02:13,766 --> 00:02:15,633 and closed by that complete 46 00:02:15,933 --> 00:02:19,000 uh volume by simply integrating uh 47 00:02:19,000 --> 00:02:20,466 integating that uh 48 00:02:20,666 --> 00:02:23,200 volume charge density over that volume 49 00:02:23,200 --> 00:02:28,233 uh volume uh volume surface uh volume uh volume 50 00:02:29,700 --> 00:02:33,600 charge against the object okay 51 00:02:33,600 --> 00:02:39,200 so let's not utilize this basic definition of Moses 52 00:02:39,200 --> 00:02:40,000 that equation 53 00:02:40,000 --> 00:02:43,800 so we just need to equate these two equations 54 00:02:43,800 --> 00:02:46,066 let's create these two questions over here 55 00:02:46,066 --> 00:02:48,533 right beside the that is 56 00:02:48,533 --> 00:02:50,800 your flux is equal to be total charge 57 00:02:50,800 --> 00:02:52,700 and caused by that surface 58 00:02:53,166 --> 00:02:56,233 so this is stated as the first 59 00:02:57,300 --> 00:02:59,766 first equation of Maxwell 60 00:02:59,900 --> 00:03:03,866 so this is Maxwell's first equation in integral form 61 00:03:04,933 --> 00:03:06,200 and it has been dried 62 00:03:06,200 --> 00:03:09,000 by using the definition of causes law 63 00:03:10,066 --> 00:03:10,766 okay now 64 00:03:10,766 --> 00:03:13,700 let's apply the divergent serum that will study that 65 00:03:13,700 --> 00:03:14,866 we can we get 66 00:03:14,866 --> 00:03:18,066 convert a close surface in chiguel into it 67 00:03:18,500 --> 00:03:20,333 and draw volume and draw 68 00:03:20,333 --> 00:03:23,200 and draw open volume integral 69 00:03:23,200 --> 00:03:26,900 by choosing the definition of this divergous over 70 00:03:26,933 --> 00:03:29,033 of that same factor field 71 00:03:29,566 --> 00:03:31,900 so now if you involve this definition 72 00:03:31,900 --> 00:03:35,300 you create these two equations now right 73 00:03:36,300 --> 00:03:40,266 you plug in first of all you plug in this over here 74 00:03:40,466 --> 00:03:41,000 then you 75 00:03:41,000 --> 00:03:44,166 then it will be a equation of these three questions 76 00:03:44,166 --> 00:03:46,233 right and then they are on 77 00:03:46,500 --> 00:03:49,700 you can see that these two onions 78 00:03:49,700 --> 00:03:54,500 they are going to cancel out and then what will be 79 00:03:54,500 --> 00:03:55,366 we will be left over 80 00:03:55,366 --> 00:04:00,000 with that is row V is equal to del dot d 81 00:04:00,200 --> 00:04:01,500 and this is known as D 82 00:04:02,500 --> 00:04:05,300 Maxwell's first equation in differential form 83 00:04:05,300 --> 00:04:07,566 so always remember these two equations 84 00:04:07,566 --> 00:04:10,500 from your electrostatic point of view 85 00:04:10,500 --> 00:04:12,633 toward this course labels 86 00:04:13,700 --> 00:04:16,433 and it is a first of four Maxwell's equation 87 00:04:17,566 --> 00:04:20,366 okay volume charge density is the same 88 00:04:20,366 --> 00:04:21,100 what does it state 89 00:04:21,100 --> 00:04:23,666 that the volume charge density will be 90 00:04:23,666 --> 00:04:27,766 is the same as the divergent of electric trucks density 91 00:04:27,766 --> 00:04:29,500 to that close surface 92 00:04:30,400 --> 00:04:34,066 Goss law is an alternative statement of Cooms law 93 00:04:34,066 --> 00:04:36,400 this you can see in your 94 00:04:36,400 --> 00:04:38,466 and problems that how to drive it 95 00:04:39,733 --> 00:04:43,433 drive the Cooms law from the from this or Goss law 96 00:04:45,966 --> 00:04:48,200 this picture shows that 97 00:04:49,533 --> 00:04:53,833 what is the Netflix out of this close close object 98 00:04:53,900 --> 00:04:54,900 close surface 99 00:04:55,933 --> 00:04:58,900 so for example this uh closed surface uh 100 00:04:58,900 --> 00:04:59,700 it is uh 101 00:04:59,700 --> 00:05:04,100 having the charge 10 nanoculum and minus 5 nanoculum 102 00:05:04,133 --> 00:05:06,200 and also in its exterior 103 00:05:06,200 --> 00:05:10,266 there are 20 nanoculum and 15 nanoculum however 104 00:05:10,266 --> 00:05:15,100 the Netflix the Netflix out of this complete surface 105 00:05:15,266 --> 00:05:16,700 would be equal to the 106 00:05:17,766 --> 00:05:20,500 would be equal to discharge and closed by this surface 107 00:05:20,500 --> 00:05:23,333 so for this first uh closed surface 108 00:05:23,333 --> 00:05:26,733 the net plugs leaving this V1 is equal to 10 - 5 109 00:05:26,733 --> 00:05:29,033 that is equal to 5 nano column 110 00:05:29,666 --> 00:05:30,900 and for the second surface 111 00:05:30,900 --> 00:05:33,600 there is nothing and closed by this surface 112 00:05:33,600 --> 00:05:36,700 so that's why this total tax 113 00:05:36,700 --> 00:05:38,333 out of the surface is equal to zero 114 00:05:38,333 --> 00:05:39,900 so why this is equal to zero 115 00:05:40,100 --> 00:05:41,600 because the number of 116 00:05:41,600 --> 00:05:43,766 lines entering due to this surface 117 00:05:43,766 --> 00:05:46,300 for example due to this charge 118 00:05:46,466 --> 00:05:49,100 are equal to the number of lines leaving 119 00:05:49,100 --> 00:05:51,666 this close surface and likewise 120 00:05:51,966 --> 00:05:53,566 the number of line and change 121 00:05:53,566 --> 00:05:56,166 the surface due to this plus 15 coolant 122 00:05:56,533 --> 00:05:58,300 are equal to the number of lines 123 00:05:58,400 --> 00:06:00,466 leaving this close surface to 124 00:06:00,466 --> 00:06:05,066 so at the end there's no Netflix out of this complete 125 00:06:05,300 --> 00:06:08,166 complete closed surface and likewise for this 126 00:06:08,166 --> 00:06:10,400 the first example uh 127 00:06:10,400 --> 00:06:13,133 it is in material or independent offers 128 00:06:13,133 --> 00:06:16,700 uh exterior charges that are 20 nanocolor man 129 00:06:16,700 --> 00:06:19,566 15 nanocolor man and the total flex 130 00:06:19,566 --> 00:06:23,266 out of that surface is only dependent upon the charge 131 00:06:23,266 --> 00:06:25,333 which is enclosed by this surface 132 00:06:25,333 --> 00:06:28,900 that is 10 nanocolor man minus 5 nanocolor 133 00:06:30,800 --> 00:06:33,700 let's not discuss what is the significance of this 134 00:06:33,700 --> 00:06:34,600 God's law 135 00:06:34,700 --> 00:06:39,266 God's law provides and easy means of finding electric 136 00:06:39,266 --> 00:06:42,800 or electric field electric field density 137 00:06:42,800 --> 00:06:44,466 or electrical density 138 00:06:44,500 --> 00:06:46,866 for symmetrical charge distribution 139 00:06:46,866 --> 00:06:49,400 so it is only applicable to these types of 140 00:06:49,400 --> 00:06:52,500 the charge distribution such as point charge 141 00:06:53,166 --> 00:06:55,733 an infinite um inline charge 142 00:06:55,733 --> 00:06:57,300 Australian distribution 143 00:06:57,300 --> 00:07:00,833 or it may be an infinite charge distribution as well 144 00:07:02,533 --> 00:07:03,300 so it is a 145 00:07:03,300 --> 00:07:06,666 applicable if there is a symmetric charge distribution 146 00:07:06,666 --> 00:07:08,933 and for that we construct a mathematical 147 00:07:08,933 --> 00:07:12,366 closed surface of unknown geometric 148 00:07:12,566 --> 00:07:15,500 and this surface would be known as Gaussian surface 149 00:07:16,133 --> 00:07:18,966 and the cavities of this guardian surface are 150 00:07:18,966 --> 00:07:19,533 the d 151 00:07:19,533 --> 00:07:23,000 should be normal or peninsure to that guardian surface 152 00:07:23,266 --> 00:07:26,766 and if it is a normal to that surface 153 00:07:26,933 --> 00:07:30,266 then the maximum flux will be passing to that 154 00:07:30,600 --> 00:07:33,900 because for example if this is your surface right 155 00:07:34,000 --> 00:07:36,666 so what is the the unit vector of this surface 156 00:07:36,666 --> 00:07:38,966 that is in this section right 157 00:07:38,966 --> 00:07:41,266 you remember the unit vector of that surface 158 00:07:41,533 --> 00:07:43,400 it isn't this direction now 159 00:07:43,400 --> 00:07:46,266 if your d your electric flux 160 00:07:46,600 --> 00:07:49,666 your electric field is passing to the surface 161 00:07:49,666 --> 00:07:52,366 in a direction which is normal to the surface 162 00:07:52,366 --> 00:07:53,566 so for that case 163 00:07:53,566 --> 00:07:56,566 the angle between the normal to that surface 164 00:07:56,566 --> 00:07:57,800 and this d would be 165 00:07:58,666 --> 00:08:03,200 this angle in between these two will be 0 degree right 166 00:08:04,666 --> 00:08:06,000 but in in other direction 167 00:08:06,000 --> 00:08:08,866 it will be a 1 or 2 degree can it can be 1 or 2 degree 168 00:08:08,866 --> 00:08:13,266 so if it is a 0 degree you know that cause of 0 0 to 1 169 00:08:13,266 --> 00:08:15,100 so this this means that 170 00:08:15,100 --> 00:08:17,133 the maximum flux would be passing to this 171 00:08:17,133 --> 00:08:18,166 so surface 172 00:08:18,366 --> 00:08:22,600 and in case it is a potential like your for example 173 00:08:22,600 --> 00:08:25,466 let's change the color so if it is like this 174 00:08:25,466 --> 00:08:27,133 your d is in this direction 175 00:08:27,133 --> 00:08:28,466 then in this case 176 00:08:28,466 --> 00:08:32,466 the angle would be between them would be 90 degree 177 00:08:33,133 --> 00:08:35,466 and your cause of teacher would be quarter 0 178 00:08:35,466 --> 00:08:37,966 and no flux will be positive to that play 179 00:08:37,966 --> 00:08:39,233 that close surface 180 00:08:40,300 --> 00:08:42,266 so guys in surface is chosen 181 00:08:42,966 --> 00:08:47,233 such that d is constant in the parts of the surface 182 00:08:47,666 --> 00:08:50,400 for which d is normal to the surface 183 00:08:50,533 --> 00:08:52,133 because for the financial 184 00:08:52,133 --> 00:08:54,066 the Netflix would be equal to 0 185 00:08:54,166 --> 00:08:56,533 so this will see in our coming uh 186 00:08:56,533 --> 00:08:57,366 slides as well 187 00:08:57,366 --> 00:09:00,300 that how this d is constant on this complete surface 188 00:09:00,300 --> 00:09:03,966 that we have assumed for the calculation of this uh 189 00:09:04,266 --> 00:09:05,933 uh this uh 190 00:09:05,933 --> 00:09:09,400 flags are the total charging closed 191 00:09:10,400 --> 00:09:11,300 and therefore 192 00:09:11,300 --> 00:09:15,266 we can bring d outside the integration sign and uh 193 00:09:15,266 --> 00:09:17,200 and the solution is uh easier no 194 00:09:17,200 --> 00:09:21,133 so if we treat this complete d as a constant term 195 00:09:21,133 --> 00:09:23,500 then we can definitely create treat it uh 196 00:09:23,666 --> 00:09:26,266 uh we can definitely bring it out of the constant 197 00:09:26,600 --> 00:09:28,933 that integration so that are uh 198 00:09:28,933 --> 00:09:31,866 our mathematical solution is easier 199 00:09:33,666 --> 00:09:36,333 so guys this now we will apply this causes now 200 00:09:36,333 --> 00:09:37,800 that is the electric substance to 201 00:09:37,800 --> 00:09:39,166 is equal to the total charge 202 00:09:39,166 --> 00:09:41,066 and closed by that close surface 203 00:09:41,200 --> 00:09:44,900 and we will utilize this equation right 204 00:09:45,000 --> 00:09:47,900 so for a point charge per point charge 205 00:09:47,900 --> 00:09:50,333 it would be enclosed in a Gaussian surface 206 00:09:50,333 --> 00:09:53,233 then this enclosed the charge would be equal to q 207 00:09:53,533 --> 00:09:56,700 and if it is a inline charge distribution 208 00:09:56,700 --> 00:09:58,700 infinite line charge distribution 209 00:09:58,900 --> 00:10:02,400 and the total charge enclosed by that Guardian surface 210 00:10:02,400 --> 00:10:03,900 if the line charge distribution 211 00:10:03,900 --> 00:10:05,666 has the density of a royale 212 00:10:05,666 --> 00:10:09,266 would be simply equal to the integration of that royale 213 00:10:09,266 --> 00:10:12,100 over the entire land of that Guardian part 214 00:10:12,766 --> 00:10:16,000 and if it is a sheet of infinite 215 00:10:16,000 --> 00:10:17,866 a sheet of charge distribution withdraw 216 00:10:17,866 --> 00:10:20,766 then they with charge and see the portable s 217 00:10:20,766 --> 00:10:21,066 then 218 00:10:21,066 --> 00:10:23,666 we will integrate that to find out the total charge 219 00:10:23,666 --> 00:10:24,766 and caused by that 220 00:10:24,766 --> 00:10:27,200 goes in surface that we consider an outcome 221 00:10:27,200 --> 00:10:30,900 and slide and if it is a walling 222 00:10:36,566 --> 00:10:45,766 charge distribution 223 00:10:49,533 --> 00:10:50,333 so first of all 224 00:10:50,333 --> 00:10:53,733 appoint charge Q is located at the origin 225 00:10:53,733 --> 00:10:55,400 so this is your charge 226 00:10:56,100 --> 00:10:58,300 okay it is located at the region 227 00:10:58,300 --> 00:11:02,733 we have to determine the uh d at uh at point P 228 00:11:02,733 --> 00:11:04,066 for example point P 229 00:11:04,100 --> 00:11:04,800 and in order 230 00:11:04,800 --> 00:11:07,900 in order to media the electric create or uh uh 231 00:11:07,900 --> 00:11:09,900 this d at this given point 232 00:11:09,900 --> 00:11:12,500 we need to consider our Guardian surface 233 00:11:12,733 --> 00:11:17,000 and for this point if we consider in a threed geometry 234 00:11:17,000 --> 00:11:21,300 then we can see that it can be a very good surface 235 00:11:21,500 --> 00:11:23,400 that can contain this evaluation 236 00:11:23,400 --> 00:11:24,666 point b and 237 00:11:24,666 --> 00:11:27,933 this will also sacrifice the symmetry of this 238 00:11:27,933 --> 00:11:30,100 complete problem right 239 00:11:30,100 --> 00:11:33,366 so you because uh it's very good symmetric object 240 00:11:34,200 --> 00:11:36,966 so this is everywhere normal to the surface 241 00:11:36,966 --> 00:11:40,200 since this is uh moving away from the electrical lines 242 00:11:40,200 --> 00:11:41,666 and moving away from the charge 243 00:11:41,666 --> 00:11:42,933 so if you see over here 244 00:11:42,933 --> 00:11:45,666 so at every point on the surface 245 00:11:45,933 --> 00:11:49,466 this electric field or your in other words 246 00:11:49,466 --> 00:11:53,966 the electric pregnancy is normal to this surface 247 00:11:53,966 --> 00:11:55,000 like this 248 00:11:55,700 --> 00:11:56,500 okay 249 00:11:58,866 --> 00:12:01,466 so if it is a normal to that surface 250 00:12:02,466 --> 00:12:03,333 surface 251 00:12:03,333 --> 00:12:06,866 so that you know that it is in the radiant direction 252 00:12:06,866 --> 00:12:09,033 it is also in the radiant direction 253 00:12:09,066 --> 00:12:11,733 so both are having the TikTok 254 00:12:11,733 --> 00:12:14,200 this is your AR direction right 255 00:12:14,500 --> 00:12:15,466 this is your Dr 256 00:12:16,533 --> 00:12:18,400 so both are this is your surface 257 00:12:18,400 --> 00:12:22,766 like this surface and surface normal will be like this 258 00:12:22,766 --> 00:12:24,200 the end so 259 00:12:24,200 --> 00:12:25,200 both are having 260 00:12:25,566 --> 00:12:29,600 the angular distribution of displacement of 0 degree 261 00:12:29,600 --> 00:12:31,500 so call of 0 will be one 262 00:12:32,566 --> 00:12:34,933 now you know who we apply that to Gaussian's Law 263 00:12:34,933 --> 00:12:38,333 that is the electric clock is equal to the charging 264 00:12:38,333 --> 00:12:41,266 close by that surface so first of all 265 00:12:41,266 --> 00:12:42,400 did I said uh 266 00:12:42,400 --> 00:12:43,800 first of all this uh 267 00:12:44,466 --> 00:12:47,300 that your side we know that side 268 00:12:47,300 --> 00:12:49,700 you could close and check lovely dot BS 269 00:12:49,766 --> 00:12:51,400 since we are here we are here 270 00:12:51,400 --> 00:12:54,600 you can see that on since it is a symmetric geometric 271 00:12:54,600 --> 00:12:58,500 so at all points at all points of the surface 272 00:12:58,800 --> 00:13:00,800 the value of this 273 00:13:01,900 --> 00:13:04,566 the value of this electric flux density 274 00:13:04,566 --> 00:13:06,500 it is going to be staying 275 00:13:06,500 --> 00:13:07,866 is going to stay the same 276 00:13:09,600 --> 00:13:13,733 okay so if we solve this complete integration 277 00:13:13,733 --> 00:13:18,266 we can treat this electric looks 10C as a constant 278 00:13:18,266 --> 00:13:20,800 we can take it out of this integration 279 00:13:20,900 --> 00:13:23,766 and then we can solve this integration 280 00:13:23,766 --> 00:13:24,600 now observe that 281 00:13:24,600 --> 00:13:27,100 since it is as radical partner system problem 282 00:13:27,100 --> 00:13:30,466 so DS is equal to Oscars and pwpw five 283 00:13:30,766 --> 00:13:33,666 and since in the Sperical Partner System 284 00:13:33,700 --> 00:13:35,700 the feature approaches from 0 to pi 285 00:13:35,766 --> 00:13:37,700 and this 5 is 286 00:13:37,700 --> 00:13:40,833 the emperor is approaching from 0 to 2 pi 287 00:13:40,966 --> 00:13:43,566 and since Dr is constant 288 00:13:43,566 --> 00:13:45,700 it can be taken out of the integration 289 00:13:45,700 --> 00:13:47,466 and we know that this this 290 00:13:47,466 --> 00:13:48,333 if you either 291 00:13:48,333 --> 00:13:50,966 you call this integration or from the institution 292 00:13:50,966 --> 00:13:53,466 since it is a complete sphere 293 00:13:53,466 --> 00:13:56,566 so it's area the surface area of this complete space 294 00:13:56,566 --> 00:13:59,300 report to 4 by R square or also 295 00:13:59,300 --> 00:14:01,766 you can find this in the same answer 296 00:14:01,766 --> 00:14:03,100 the same for by R square 297 00:14:03,100 --> 00:14:06,300 by swallowing this integration in your own time 298 00:14:06,900 --> 00:14:10,200 now the total charge and close by this guardian circles 299 00:14:10,200 --> 00:14:12,166 not all guardian circles for the sphere 300 00:14:12,300 --> 00:14:13,266 now the total charge 301 00:14:13,266 --> 00:14:16,400 and close by this guardian circle is equal to Q 302 00:14:16,466 --> 00:14:19,700 right now you can create this LED 303 00:14:19,700 --> 00:14:22,433 this left right this right hand side of this equation 304 00:14:23,500 --> 00:14:25,666 and you just rearrange this thing 305 00:14:25,666 --> 00:14:26,966 you can find out your d 306 00:14:26,966 --> 00:14:30,066 R is equal to q divided by 4 by r square 307 00:14:30,066 --> 00:14:31,300 and in vector notation 308 00:14:31,300 --> 00:14:33,066 since it is in the radio direction 309 00:14:33,066 --> 00:14:36,066 so it is equal to q divided by 4 by r square 310 00:14:36,066 --> 00:14:38,166 into a unichrector a r 311 00:14:38,500 --> 00:14:40,866 and from there you can find out the electrical field 312 00:14:40,866 --> 00:14:43,900 by using this constituted relationship 313 00:14:43,900 --> 00:14:46,200 that is B is equal to optional not E 314 00:14:46,200 --> 00:14:48,500 so E is equal to divide B divided by optional 315 00:14:48,500 --> 00:14:50,966 not for free space medium 316 00:14:51,000 --> 00:14:55,366 and your electric field is going to be 2 / 4 317 00:14:55,366 --> 00:14:58,066 by astronaut Harshcare er 318 00:14:58,066 --> 00:15:01,600 and this is the scene that we found out using your uh 319 00:15:01,733 --> 00:15:04,066 uh in our previous section as well 320 00:15:04,066 --> 00:15:08,333 that the electric field generated by this uh uh 321 00:15:08,333 --> 00:15:11,000 by this point charge with the help of Coolum's Law 322 00:15:11,000 --> 00:15:13,033 we found out that it is equal to 323 00:15:13,066 --> 00:15:14,166 to divide therefore 324 00:15:14,166 --> 00:15:15,833 perhaps we're not asking er 325 00:15:18,366 --> 00:15:19,566 now the next uh 326 00:15:19,566 --> 00:15:23,266 symmetric charge distribution is infinite line charge 327 00:15:23,266 --> 00:15:24,900 of uniform charge distribution 328 00:15:24,900 --> 00:15:26,000 roll L and it 329 00:15:26,000 --> 00:15:28,333 is in this case we have taken a simple case 330 00:15:28,333 --> 00:15:31,200 that it is oriented along the zxs 331 00:15:31,200 --> 00:15:32,200 like this 332 00:15:32,800 --> 00:15:35,066 so this is your roll L 333 00:15:36,700 --> 00:15:42,466 Auckland L and note to determine a d at point P 334 00:15:43,066 --> 00:15:46,666 for example it is your evaluation point here here 335 00:15:46,900 --> 00:15:49,266 here here 336 00:15:49,866 --> 00:15:51,566 here or here 337 00:15:52,333 --> 00:15:56,566 we have considered a cylindrical surface containing B 338 00:15:57,866 --> 00:16:00,400 to satisfy this symmetry condition 339 00:16:00,400 --> 00:16:04,000 so here this symmetry is being observed over here right 340 00:16:04,400 --> 00:16:06,600 and this what is that symmetry 341 00:16:06,600 --> 00:16:09,000 that symmetry is of cylindrical shape 342 00:16:10,000 --> 00:16:11,100 so your line chart 343 00:16:11,100 --> 00:16:13,800 it is also exhibiting that person intical shape 344 00:16:13,800 --> 00:16:17,066 your Gaussian surface it is also exhibiting that 345 00:16:17,066 --> 00:16:18,900 persometry of symmetrical shape 346 00:16:19,000 --> 00:16:21,500 and uh we know here we here we 347 00:16:21,500 --> 00:16:23,800 we can see that d is constant 348 00:16:23,800 --> 00:16:25,800 and normal to this cynical surface 349 00:16:25,800 --> 00:16:28,333 so again the electric plugs density 350 00:16:28,333 --> 00:16:32,300 it is imminating from this line chart density in this 351 00:16:32,300 --> 00:16:35,666 in this area direction like this one 352 00:16:35,666 --> 00:16:39,000 and here if this is your a N right 353 00:16:40,133 --> 00:16:42,366 this is your a roll then 354 00:16:42,366 --> 00:16:44,900 you can see that these two are parallel to each other 355 00:16:44,900 --> 00:16:45,733 and also these 356 00:16:45,733 --> 00:16:50,333 these perpendicular disk to this normal to disk in god 357 00:16:50,333 --> 00:16:51,166 in surface 358 00:16:53,366 --> 00:16:56,866 so Q and closed is equal to uh 359 00:16:56,866 --> 00:17:01,866 so this is your uh sign so sign is equal to B dot DS 360 00:17:01,866 --> 00:17:04,933 and here d is in the row direction 361 00:17:04,933 --> 00:17:08,633 this is the radial direction 362 00:17:08,933 --> 00:17:10,466 for this clinical partner system 363 00:17:10,466 --> 00:17:11,766 and these constant on 364 00:17:11,766 --> 00:17:14,600 and normal on these clinical partnering surface 365 00:17:14,600 --> 00:17:17,066 and for the rest of the since it is a closed surface 366 00:17:17,066 --> 00:17:20,700 so we have to take care of all these uh three surfaces 367 00:17:20,700 --> 00:17:23,466 that is your top bottom end 368 00:17:23,466 --> 00:17:28,200 this side surfaces so if you see this geometry 369 00:17:28,300 --> 00:17:29,766 then you can observe that 370 00:17:29,766 --> 00:17:31,666 for the top and bottom surface 371 00:17:31,800 --> 00:17:35,500 your a n is a n is in this direction right 372 00:17:36,366 --> 00:17:40,333 your this arrow is in this direction right 373 00:17:40,333 --> 00:17:43,433 so the angle between them is 90 degree 374 00:17:44,133 --> 00:17:44,933 right 375 00:17:45,200 --> 00:17:50,400 so for this case you are um that uh electric prestency 376 00:17:50,400 --> 00:17:54,133 it is a tendential to that surface so for that case 377 00:17:54,133 --> 00:17:56,966 the Peter between the normal to that surface 378 00:17:56,966 --> 00:17:58,600 and this electric prestency 379 00:17:58,600 --> 00:17:59,966 it is going to be zero 380 00:17:59,966 --> 00:18:03,600 so these two things are going to be nullify 381 00:18:03,900 --> 00:18:05,200 so it is zero no 382 00:18:05,200 --> 00:18:08,300 clock signs are passing to this top and bottom surface 383 00:18:08,533 --> 00:18:10,066 and for the side surface 384 00:18:10,700 --> 00:18:12,066 for the side surface no 385 00:18:12,066 --> 00:18:14,566 we have to solve this integration 386 00:18:15,266 --> 00:18:17,666 50 has no component along the a Z 387 00:18:17,766 --> 00:18:20,933 also you can see that there's no component along a Z 388 00:18:20,933 --> 00:18:24,400 so this integration is going to be the zero as well 389 00:18:24,533 --> 00:18:27,333 so d is a tenantial to the top and bottom 390 00:18:27,333 --> 00:18:27,900 so that's why 391 00:18:27,900 --> 00:18:31,033 these two innovations are also coming out to be zero 392 00:18:31,300 --> 00:18:33,566 the d is only having the radial component 393 00:18:33,566 --> 00:18:36,200 so that is another factor of these two terms to be 394 00:18:36,200 --> 00:18:37,433 coming out to be zero 395 00:18:37,866 --> 00:18:41,233 and now we have to solve this on the integration 396 00:18:41,266 --> 00:18:43,666 and again since it is constant 397 00:18:43,666 --> 00:18:46,600 to be constant on the anti causing surface 398 00:18:46,600 --> 00:18:48,800 it can be taken out of this integration 399 00:18:48,800 --> 00:18:50,900 and if we solve this integration 400 00:18:51,300 --> 00:18:53,233 this integration for this 401 00:18:54,566 --> 00:18:56,433 symmetrical object 402 00:18:56,766 --> 00:19:00,300 you know that the DS is equal to road defy 403 00:19:02,166 --> 00:19:04,066 defy D 404 00:19:06,366 --> 00:19:12,066 so you're uh this this one is equal to D5D Z right 405 00:19:12,500 --> 00:19:14,866 because your aero is a normal 406 00:19:15,566 --> 00:19:19,000 okay so if you solve this complete integration you get 407 00:19:19,000 --> 00:19:20,766 so Royce taken as constant 408 00:19:20,766 --> 00:19:24,300 integration of this D5 is equal to 2 409 00:19:24,300 --> 00:19:26,100 5 for this complete slender 410 00:19:26,333 --> 00:19:29,733 and what is the integration of will start to d Z 411 00:19:29,733 --> 00:19:34,933 so that is your length that is your L complete plan 412 00:19:34,933 --> 00:19:38,300 remember that so the so the integration over this 413 00:19:38,300 --> 00:19:40,433 the surface area you can also take that 414 00:19:40,466 --> 00:19:42,266 the surface area of this complete slender 415 00:19:42,266 --> 00:19:44,300 it is equal to to pyro at 416 00:19:45,766 --> 00:19:47,733 okay now you can uh uh 417 00:19:47,733 --> 00:19:50,133 you can uh use the basic definition of the charge 418 00:19:50,133 --> 00:19:51,766 so the charge uh 419 00:19:51,766 --> 00:19:55,133 which is enclosed or which is possessed by this uh 420 00:19:55,133 --> 00:19:58,366 infinite uh uh line charge distribution 421 00:19:58,366 --> 00:19:59,600 it is equal to 422 00:20:00,000 --> 00:20:04,266 it is equal to row L times B complete length of this 423 00:20:05,866 --> 00:20:09,066 infinite line charge distribution so plug in over here 424 00:20:09,733 --> 00:20:12,166 use a take out of this basic equation 425 00:20:12,666 --> 00:20:14,800 that your size equivalent closed 426 00:20:14,900 --> 00:20:17,266 use uh simplify this equation 427 00:20:17,700 --> 00:20:20,166 then you can find out as your DSP to do 428 00:20:20,166 --> 00:20:22,100 and do a bit to Pyro 0 429 00:20:22,100 --> 00:20:24,133 and it is the same thing that will derive 430 00:20:24,133 --> 00:20:25,500 in our section 431 00:20:26,200 --> 00:20:27,966 now section that is your 432 00:20:28,200 --> 00:20:32,600 uh 4.3 and there we took help of that uh 433 00:20:32,600 --> 00:20:35,033 uh your uh 434 00:20:35,533 --> 00:20:39,466 uh that basic definitions of uh your coolance law 435 00:20:42,866 --> 00:20:43,900 okay now let's uh 436 00:20:43,900 --> 00:20:46,133 take the case of this infinite uh 437 00:20:46,133 --> 00:20:49,100 sheet of charge switch your card distribution 438 00:20:49,266 --> 00:20:52,500 and it is for this case we have considered that 439 00:20:52,500 --> 00:20:54,966 it is aligned on your extra plane 440 00:20:54,966 --> 00:20:58,300 so that is easy for zero plane to determine 441 00:20:58,300 --> 00:21:00,000 a d at a point P 442 00:21:00,566 --> 00:21:03,900 we have consider a rectangular box as a garden surface 443 00:21:03,900 --> 00:21:05,766 in this case so this is your 444 00:21:05,866 --> 00:21:08,233 this box is 3D boxes here 445 00:21:08,533 --> 00:21:10,166 and causing circus in this case 446 00:21:10,266 --> 00:21:13,166 and it is uh cutting uh this uh 447 00:21:13,166 --> 00:21:17,000 sheet of charge magically into two halves and has uh 448 00:21:17,000 --> 00:21:19,866 two of its faces parallel to this sheet 449 00:21:19,866 --> 00:21:22,966 so these two faces are apparently to this sheet 450 00:21:25,533 --> 00:21:30,300 okay now d is normal to the sheet like right 451 00:21:30,300 --> 00:21:31,866 if this d 452 00:21:31,866 --> 00:21:34,666 if electrical is being generated by that sheet 453 00:21:34,700 --> 00:21:37,400 for it would be generated either in this direction 454 00:21:37,400 --> 00:21:41,666 or in this direction so it is the normal to the sheet 455 00:21:41,666 --> 00:21:43,533 and it is in the d Z direction 456 00:21:43,533 --> 00:21:49,800 so d has no component along this a y and a X 457 00:21:49,800 --> 00:21:52,900 so it has no component along these two things 458 00:21:53,400 --> 00:21:54,800 these two unitractors 459 00:21:56,000 --> 00:21:56,666 now we if 460 00:21:56,666 --> 00:22:00,366 if we apply the closing to the definition of this 461 00:22:00,366 --> 00:22:01,400 electric plugs 462 00:22:01,666 --> 00:22:04,600 so these are BS for the anti guardian surface 463 00:22:05,066 --> 00:22:10,866 and we see that for this side for this side 464 00:22:10,933 --> 00:22:14,766 that is these four sides that is this one 465 00:22:14,766 --> 00:22:17,533 this one the back one 466 00:22:17,533 --> 00:22:20,633 the left one so turn back right and back side 467 00:22:20,700 --> 00:22:25,300 for this side the angle between the normal to this side 468 00:22:25,400 --> 00:22:27,166 that is for example a N 469 00:22:27,766 --> 00:22:30,433 right this you can treat it like a X or a y 470 00:22:30,533 --> 00:22:33,033 so the angle between a X and a y 471 00:22:34,466 --> 00:22:36,066 you can treat it like this one 472 00:22:36,300 --> 00:22:40,066 the dot product of this a X and a C 473 00:22:40,400 --> 00:22:44,100 and the dot product of this a y and a Z 474 00:22:44,466 --> 00:22:50,366 it is equal to 0 because they are normal to each other 475 00:22:50,533 --> 00:22:52,500 and for this these four services 476 00:22:52,500 --> 00:22:55,400 be normal to the circuits without the accent e 477 00:22:55,400 --> 00:22:59,766 y and the d it is only having the Z component 478 00:22:59,766 --> 00:23:02,100 so that's why these these four integrals 479 00:23:02,100 --> 00:23:04,300 they are going to be zero 480 00:23:04,400 --> 00:23:04,766 and now 481 00:23:04,766 --> 00:23:07,900 we only have to solve this top and bottom integral 482 00:23:09,300 --> 00:23:10,266 okay for this 483 00:23:10,266 --> 00:23:12,366 for this top for this top 484 00:23:13,100 --> 00:23:19,100 we know that it is a n is equal to its a n is equal to 485 00:23:21,900 --> 00:23:23,500 easy right 486 00:23:25,300 --> 00:23:28,166 and for this bottom this a N is equal to 487 00:23:29,666 --> 00:23:30,633 minus easy 488 00:23:31,933 --> 00:23:32,733 right 489 00:23:33,966 --> 00:23:35,800 and for this DZ 490 00:23:35,933 --> 00:23:40,266 this direction will be positive AZ for the top one 491 00:23:40,300 --> 00:23:43,866 and this negative AZ and d for this 492 00:23:46,500 --> 00:23:48,900 bottom surface so this 19 active day 493 00:23:48,900 --> 00:23:51,500 they will be basically cancelled out with each other 494 00:23:51,500 --> 00:23:55,233 and at the end it would be simply a two times the 495 00:23:56,766 --> 00:24:00,466 better solution of integral on of one side 496 00:24:01,400 --> 00:24:04,933 okay so if we simplify this integration 497 00:24:04,933 --> 00:24:06,833 we simplify the other side 498 00:24:06,900 --> 00:24:08,666 what is the other side of the causes 499 00:24:08,666 --> 00:24:08,800 well 500 00:24:08,800 --> 00:24:11,433 that is the charge and close by this complete circle 501 00:24:12,100 --> 00:24:15,733 now you see that the charge and close by the surface is 502 00:24:15,733 --> 00:24:17,366 let's change this marker 503 00:24:18,966 --> 00:24:22,766 charge and prove that his surface is this one right 504 00:24:23,533 --> 00:24:26,233 and this surface area is equal to a 505 00:24:26,666 --> 00:24:30,433 that is this right this thing 506 00:24:30,533 --> 00:24:33,933 sector failure the integral of this thing 507 00:24:33,933 --> 00:24:35,366 you can assume over here 508 00:24:35,366 --> 00:24:39,833 the integral of this DS equal to a right 509 00:24:41,166 --> 00:24:47,466 and also the uh you can solve the estate 510 00:24:47,466 --> 00:24:50,333 solve the side and side using the same Assumption 511 00:24:50,333 --> 00:24:52,933 that this integral is equal to a 512 00:24:52,933 --> 00:24:54,266 this integral is equal to a 513 00:24:54,266 --> 00:24:56,866 so it is easy taken as common 514 00:24:56,866 --> 00:25:03,800 so it is two times a right two times 2 times d Z 515 00:25:05,366 --> 00:25:10,266 a and I use just a real for this a 516 00:25:10,266 --> 00:25:12,266 it is going to cancel out 517 00:25:12,866 --> 00:25:14,133 you just rearrange that 518 00:25:14,133 --> 00:25:15,700 so that your d is equal to Royce 519 00:25:15,700 --> 00:25:17,033 to Abbot to a Z 520 00:25:17,333 --> 00:25:19,366 and from there you can find out the electric grid 521 00:25:19,366 --> 00:25:21,766 that is your d to Abbot Astronaut 522 00:25:21,766 --> 00:25:23,200 and for the free space medium 523 00:25:23,200 --> 00:25:25,900 and it is equal to Royce to Abbot to astronaut a Z 524 00:25:25,933 --> 00:25:29,066 and it is the same that we drive in at section 4.3 525 00:25:30,733 --> 00:25:32,800 let's consider the last case 526 00:25:32,800 --> 00:25:35,166 that is the uniformly chart sphere 527 00:25:35,300 --> 00:25:37,500 consider sphere of radius a 528 00:25:37,500 --> 00:25:40,433 so this is your sphere a 529 00:25:40,700 --> 00:25:44,266 the uniform charger distribution or not 530 00:25:44,266 --> 00:25:45,733 and we have to find out the electric 531 00:25:45,733 --> 00:25:49,466 field inside and outside the outside 532 00:25:49,466 --> 00:25:51,700 the outside of that sphere 533 00:25:52,900 --> 00:25:54,100 okay and for that 534 00:25:54,100 --> 00:25:56,400 we have to take help of two Gaussian circles 535 00:25:56,400 --> 00:25:57,600 for the first one 536 00:25:57,600 --> 00:26:00,133 the Gaussian surface will be having a radius less than 537 00:26:00,133 --> 00:26:04,500 or equal to this radius of this Jas 538 00:26:04,700 --> 00:26:06,133 medical chart distribution 539 00:26:06,133 --> 00:26:07,066 and for the second one 540 00:26:07,066 --> 00:26:09,800 this r will be greater or equal to this E 541 00:26:11,533 --> 00:26:13,000 now from the first case uh 542 00:26:13,000 --> 00:26:15,200 we take that uh uh 543 00:26:15,200 --> 00:26:19,666 for this uh left hand side that is your cue enclosed 544 00:26:19,666 --> 00:26:22,433 we think cue enclosed is equal to Rovi 545 00:26:22,500 --> 00:26:24,000 and Rovi is constant 546 00:26:24,000 --> 00:26:26,666 so that's why it is taken out of the integration 547 00:26:26,700 --> 00:26:29,500 and now we have to solve this integration right 548 00:26:29,766 --> 00:26:32,966 and here uh the inside we 549 00:26:32,966 --> 00:26:35,766 if we saw this inside the uh 550 00:26:35,766 --> 00:26:40,400 that uh inside Guardian Surface integration 551 00:26:40,400 --> 00:26:45,000 so we can see that the total charge imposed by this in 552 00:26:45,000 --> 00:26:48,700 by this Guardian Surface uh would be equal 553 00:26:48,900 --> 00:26:50,866 the radius of that god in surface 554 00:26:50,866 --> 00:26:53,800 will be equal to the same um 555 00:26:53,800 --> 00:26:57,766 uh same radius that that is the of the god in surface 556 00:26:57,766 --> 00:26:59,800 so the total charge 557 00:27:00,266 --> 00:27:05,533 total charge is radical as radical distributions 558 00:27:05,533 --> 00:27:08,400 radius would be equal to the radius of that 559 00:27:08,900 --> 00:27:11,800 causing surface so that is the catch in this case 560 00:27:12,300 --> 00:27:16,500 so your so that your your radius of integration 561 00:27:16,500 --> 00:27:19,166 it is equal to going to stay the same 562 00:27:19,166 --> 00:27:20,866 that is of your Guardian surface 563 00:27:20,866 --> 00:27:22,300 in this particular case 564 00:27:22,400 --> 00:27:26,000 because everything closed by this surface 565 00:27:26,000 --> 00:27:27,600 the total charge closed 566 00:27:27,600 --> 00:27:29,866 by this surface will be equal to that 567 00:27:29,866 --> 00:27:34,633 to the same radical charge distribution 568 00:27:36,066 --> 00:27:38,633 radius that is off Guardian surface 569 00:27:39,166 --> 00:27:42,066 okay now you can simply write this integration 570 00:27:42,066 --> 00:27:44,866 this integration is equal to the same as 571 00:27:44,866 --> 00:27:47,933 the volume of your spherical object 572 00:27:47,933 --> 00:27:49,733 with radius R also 573 00:27:49,733 --> 00:27:51,200 you can solve this integration 574 00:27:51,200 --> 00:27:53,133 and you will come out that it is equal to 575 00:27:53,133 --> 00:27:55,500 the volume of a sphere with radius 576 00:27:55,500 --> 00:27:56,766 is equal to R 577 00:27:57,600 --> 00:28:01,566 unless all the fluxide of this guardian law 578 00:28:01,566 --> 00:28:02,966 so that is a close interglow 579 00:28:02,966 --> 00:28:06,666 d dot BS is equal to again this electrical 580 00:28:06,900 --> 00:28:09,366 it is eminating in the radiant directions 581 00:28:09,366 --> 00:28:13,000 that is the er so d R can be taken out as a constant 582 00:28:13,000 --> 00:28:15,700 so this integration is same as the 583 00:28:16,066 --> 00:28:17,200 as these surfaces 584 00:28:17,200 --> 00:28:21,300 you are that here you can also follow this integration 585 00:28:21,466 --> 00:28:23,466 now you can equate these two sides 586 00:28:23,466 --> 00:28:24,900 these two sides right 587 00:28:24,900 --> 00:28:28,866 these two pins you can equate using the causes law 588 00:28:29,800 --> 00:28:30,933 you can rearrange 589 00:28:30,933 --> 00:28:34,700 then you will find that is equal to r divided by c rona 590 00:28:34,700 --> 00:28:38,266 p r where r is the radius of your Gaussian surface 591 00:28:38,266 --> 00:28:41,400 so remember that here the integration 592 00:28:41,400 --> 00:28:45,600 to find out the total charge enclosed by that cost 593 00:28:45,600 --> 00:28:48,633 Gaussian surface would be the r 594 00:28:50,000 --> 00:28:50,666 now let's uh 595 00:28:50,666 --> 00:28:53,266 consider the second case where the causing surface 596 00:28:53,266 --> 00:28:56,166 we need to find out the elector field outside of this 597 00:28:56,166 --> 00:28:58,566 uh assertical chart distribution 598 00:28:58,566 --> 00:29:00,066 for this case again 599 00:29:00,066 --> 00:29:03,233 we are going to integrate the one in charge density 600 00:29:03,400 --> 00:29:06,266 run out his constant however in this case 601 00:29:08,466 --> 00:29:09,566 your radius 602 00:29:10,200 --> 00:29:12,466 your radius it is your radius 603 00:29:12,466 --> 00:29:15,666 it is not going to be the radius of that 604 00:29:17,166 --> 00:29:21,033 your god in surface rather rather it is US mission 605 00:29:21,800 --> 00:29:25,766 that's mission from 0 to A+ 606 00:29:27,300 --> 00:29:31,900 a to R so it is that mission of 2 integration 607 00:29:32,166 --> 00:29:36,366 and for this a to R the d is equal to 608 00:29:36,600 --> 00:29:41,933 it is charged and closed by this uh 609 00:29:41,933 --> 00:29:45,300 by this surface is equal to 0 610 00:29:45,800 --> 00:29:49,133 so the total charge enclosed by this guardian surface 611 00:29:49,133 --> 00:29:53,366 which is having this radius greater than the uh 612 00:29:53,366 --> 00:29:55,133 square cervical charge distribution 613 00:29:55,133 --> 00:29:59,066 square radius uh which is the a is the same 614 00:29:59,666 --> 00:30:04,433 is the same as the uh as the radius of this 615 00:30:05,066 --> 00:30:07,266 complete vertical charge distribution 616 00:30:07,266 --> 00:30:09,366 so that is that is the difference in this case 617 00:30:09,366 --> 00:30:10,166 so here 618 00:30:10,166 --> 00:30:13,100 in order to determine the charge enclosed by the 619 00:30:13,300 --> 00:30:14,133 causing surface 620 00:30:14,133 --> 00:30:16,500 which is having the radius greater than that 621 00:30:16,500 --> 00:30:18,266 uh charge distribution 622 00:30:18,500 --> 00:30:22,033 uh will be having the radius equal to that 623 00:30:22,300 --> 00:30:24,600 uh that offer charge distributions here 624 00:30:24,600 --> 00:30:26,566 so in this case it is equal to a 625 00:30:26,800 --> 00:30:28,533 now going to solve the integration 626 00:30:28,533 --> 00:30:30,500 under the change is your a 627 00:30:30,966 --> 00:30:33,800 that the radius is equal to the radius of your 628 00:30:33,800 --> 00:30:35,066 charge distribution 629 00:30:35,700 --> 00:30:37,033 radical object 630 00:30:38,500 --> 00:30:40,766 now for the uh side side 631 00:30:40,766 --> 00:30:43,700 you just again need to solve this integration 632 00:30:43,700 --> 00:30:45,266 here there's no change 633 00:30:45,700 --> 00:30:48,900 because the golden circle is having the radius of r 634 00:30:49,000 --> 00:30:50,900 you again simplify this equation 635 00:30:50,933 --> 00:30:53,466 now you equate these two equation 636 00:30:53,533 --> 00:30:54,700 you can find out that 637 00:30:54,700 --> 00:30:58,366 your d is equal to a cube divided by your squared 638 00:30:58,366 --> 00:30:59,600 or not the r 639 00:30:59,700 --> 00:31:03,266 and here is the radius of your critical job 640 00:31:03,266 --> 00:31:07,800 distribution and RD Radio Secure goes in circles 641 00:31:09,000 --> 00:31:11,500 now let's summarize this history question 642 00:31:11,500 --> 00:31:13,400 in the graphical presentation 643 00:31:13,400 --> 00:31:17,600 in your if you plug this to these two equations 644 00:31:17,700 --> 00:31:21,200 now you can see that if you plot either R is equal to 645 00:31:21,200 --> 00:31:23,600 so boundary condition is R is equal to a 646 00:31:23,933 --> 00:31:26,566 apply this R is equal to a at this to the 647 00:31:26,566 --> 00:31:28,100 at this boundary condition 648 00:31:28,100 --> 00:31:32,300 your uh d is equal to a divided by C are not also 649 00:31:32,300 --> 00:31:36,100 you can apply this put this article to error here 650 00:31:36,133 --> 00:31:37,366 and you can find out this 651 00:31:37,366 --> 00:31:42,133 that is pure so this will come out to be the same 652 00:31:42,133 --> 00:31:45,333 divide by cheat or not so if you observe 653 00:31:45,333 --> 00:31:46,100 overheard that 654 00:31:46,100 --> 00:31:50,100 if you are measuring the electric field in an 655 00:31:50,100 --> 00:31:50,966 in a 656 00:31:50,966 --> 00:31:55,066 at a point inside this radical charge distribution 657 00:31:55,166 --> 00:31:56,166 with a radius 658 00:31:56,166 --> 00:31:58,766 less than the radius of this vertical chart 659 00:31:58,766 --> 00:32:00,000 distribution then 660 00:32:00,000 --> 00:32:01,766 these are direct relationship 661 00:32:01,766 --> 00:32:03,700 between the electric field 662 00:32:03,700 --> 00:32:06,866 and the radius of that Guardian Star 663 00:32:06,866 --> 00:32:10,200 Guardian surface so um 664 00:32:10,333 --> 00:32:13,733 as bigger as the radius of that garden surface 665 00:32:13,733 --> 00:32:14,466 would be 666 00:32:14,466 --> 00:32:18,900 so the electricals and the density would be greater 667 00:32:18,900 --> 00:32:19,866 because here 668 00:32:19,866 --> 00:32:22,466 you can see that it will come closer to that surface 669 00:32:22,466 --> 00:32:24,500 of that giant distribution 670 00:32:25,333 --> 00:32:28,400 so it is that in your relationship for the second one 671 00:32:28,666 --> 00:32:32,366 if you move this is your inverse relation 672 00:32:32,366 --> 00:32:35,900 that is you're moving away from that surface 673 00:32:36,133 --> 00:32:38,233 that's vertical chart distribution 674 00:32:38,733 --> 00:32:41,966 and this relation is inverse square 675 00:32:42,166 --> 00:32:42,733 that these 676 00:32:42,733 --> 00:32:45,166 inversely proportional to the square of this 677 00:32:45,166 --> 00:32:46,500 are squared and this you 678 00:32:46,500 --> 00:32:49,133 you know that the investigation of this are squared 679 00:32:49,133 --> 00:32:54,600 it is it exhibits some sort of here quadratic equation 680 00:32:54,866 --> 00:32:58,566 so that is your parabolic decaying in this case 681 00:32:58,666 --> 00:33:00,733 until parabolic quadratic equation 682 00:33:00,733 --> 00:33:02,733 so that's why it is decaying in a 683 00:33:02,733 --> 00:33:06,366 in this parabolic quadratic equation manner 684 00:33:06,966 --> 00:33:09,600 and again since you are moving away from this surface 685 00:33:09,600 --> 00:33:10,600 of charge distribution 686 00:33:10,600 --> 00:33:12,733 that's why your deeds going to decrease 687 00:33:12,733 --> 00:33:16,100 as you move away from this charge distribution 688 00:33:18,600 --> 00:33:21,800 let's solve these two examples of your book 689 00:33:21,800 --> 00:33:24,933 so in the first case your DS given 690 00:33:24,933 --> 00:33:29,600 it is having dependency on the Z direction 691 00:33:29,600 --> 00:33:32,366 and we have to calculate the chart density 692 00:33:32,700 --> 00:33:35,666 at this reduation point and we have 693 00:33:35,666 --> 00:33:37,900 we have to also can be the total charge 694 00:33:37,900 --> 00:33:42,666 and caused by the this cylinder of radius uh uh 695 00:33:42,666 --> 00:33:44,200 1 meter and uh 696 00:33:44,200 --> 00:33:49,533 the uh that uh z direction uh dimensions minus the 2 2 697 00:33:49,533 --> 00:33:50,000 2 698 00:33:50,000 --> 00:33:52,966 so that is your guardian surface you can see over here 699 00:33:53,266 --> 00:33:53,700 okay 700 00:33:53,700 --> 00:33:57,800 so first of all your Roby that is your child Dansky 701 00:33:57,800 --> 00:34:00,166 the one in Child Dansky it is equal to that 702 00:34:00,166 --> 00:34:01,133 don't be so 703 00:34:01,133 --> 00:34:01,333 only 704 00:34:01,333 --> 00:34:03,900 you have to simply take the divergence of this thing 705 00:34:03,900 --> 00:34:07,166 and it is only having the dependency on this easy 706 00:34:07,166 --> 00:34:07,266 so 707 00:34:07,266 --> 00:34:11,100 only we have to take the Basha derivative of this DZ 708 00:34:11,100 --> 00:34:14,466 with respect to Z and there's only Z present over here 709 00:34:14,466 --> 00:34:16,900 and whose differentiation with respect to Z 710 00:34:16,900 --> 00:34:18,233 is equal to one 711 00:34:18,333 --> 00:34:20,966 so at the end your row is equal to this one 712 00:34:20,966 --> 00:34:23,200 and simply put this row 713 00:34:25,700 --> 00:34:29,400 y and Z and this equation 714 00:34:29,400 --> 00:34:31,766 and you can simplify this chart 715 00:34:31,766 --> 00:34:34,000 and then see at this resolution point 716 00:34:34,266 --> 00:34:38,566 so remember this is your Maxwell's first equation 717 00:34:38,566 --> 00:34:39,700 differential form 718 00:34:41,000 --> 00:34:44,100 so to find out the total charge and fold by this number 719 00:34:44,100 --> 00:34:47,600 there are two ways so the first one is the basic one 720 00:34:47,600 --> 00:34:48,066 that is 721 00:34:48,066 --> 00:34:51,466 using the direct definition of your total volume 722 00:34:51,966 --> 00:34:55,500 volume that chooses your total volume chart 723 00:34:55,500 --> 00:34:59,366 against the integration and it is in this case 724 00:34:59,366 --> 00:35:00,900 you know that your row v 725 00:35:01,466 --> 00:35:04,233 your row v it is equal to draw cause 726 00:35:05,733 --> 00:35:06,600 square 5 727 00:35:06,900 --> 00:35:11,400 and your DS as your DV in this clinical corners system 728 00:35:11,400 --> 00:35:13,766 it is r o d 5 d 0 d Z 729 00:35:15,100 --> 00:35:17,666 and your limits for the integration these are from 730 00:35:17,666 --> 00:35:20,866 since it is a slender so far 5 it is from 0 2 731 00:35:20,866 --> 00:35:21,366 2 5 732 00:35:21,366 --> 00:35:23,333 4 your uh Z 733 00:35:23,333 --> 00:35:25,066 it is from minus 2 to 2 734 00:35:25,100 --> 00:35:28,100 and uh for this road we will be scared with it 735 00:35:28,100 --> 00:35:31,166 it is so scared now you just simply integrate this 736 00:35:31,166 --> 00:35:33,600 so it's integration with the Z 737 00:35:33,600 --> 00:35:36,933 so upper minus 4 to 2 minus minus 2 738 00:35:36,933 --> 00:35:38,300 so it will be a 4 739 00:35:38,333 --> 00:35:40,666 and your integration of this thing can be 740 00:35:40,666 --> 00:35:44,000 find out using this half an angle identity 741 00:35:44,000 --> 00:35:46,166 so solve this in your own time 742 00:35:46,166 --> 00:35:48,700 so the result will come out to be bye 743 00:35:49,300 --> 00:35:52,133 because this is going to be zero 744 00:35:52,133 --> 00:35:54,433 its integration would be going to zero 745 00:35:54,500 --> 00:35:56,766 and this would be equal to 2 pie 746 00:35:57,600 --> 00:36:01,133 right this would be equal cancer why this is zero 747 00:36:01,133 --> 00:36:02,500 because sign of pie 748 00:36:02,600 --> 00:36:06,900 sign sign of 2 pie and sign of 0 is equal to Z 749 00:36:08,133 --> 00:36:08,866 and for this thing 750 00:36:08,866 --> 00:36:12,500 it is the rocky divided tree and put this up 751 00:36:12,500 --> 00:36:14,600 and also it will come out to be one of a tree 752 00:36:14,600 --> 00:36:15,966 so that is your answer 753 00:36:17,133 --> 00:36:18,633 what is the second way 754 00:36:18,666 --> 00:36:21,200 second way the way the help of your boss is long 755 00:36:21,266 --> 00:36:22,133 and for that 756 00:36:22,133 --> 00:36:24,533 your cues equal to size with the colonies to close 757 00:36:24,533 --> 00:36:26,900 and take a lot of C dot DS here 758 00:36:26,933 --> 00:36:29,100 since it is a sylintrical object 759 00:36:29,100 --> 00:36:29,666 so again 760 00:36:29,666 --> 00:36:33,200 there are three surfaces surfaces side top and bottom 761 00:36:33,466 --> 00:36:36,900 since your d it is in d d direction 762 00:36:37,266 --> 00:36:40,233 so for this size for this size 763 00:36:40,500 --> 00:36:43,800 there's no zero component is d so for this size 764 00:36:43,800 --> 00:36:47,900 your this integration is going to be zero right 765 00:36:47,900 --> 00:36:48,866 because again 766 00:36:48,866 --> 00:36:51,666 the angle between them will be equal to ninety 767 00:36:53,600 --> 00:36:56,200 okay and cause of 90 is equal to zero 768 00:36:57,133 --> 00:36:59,400 remember this thing that we have discussed earlier 769 00:36:59,400 --> 00:37:03,566 access as well so for this top and bottom surface 770 00:37:03,566 --> 00:37:05,666 it was yellow here for the first of all 771 00:37:05,666 --> 00:37:07,766 for the top surface your DS is equal to again 772 00:37:07,766 --> 00:37:09,300 in the slanderal quality system 773 00:37:09,300 --> 00:37:13,000 is equal to row divided row and Z for the top 774 00:37:13,000 --> 00:37:15,000 it is equal to 2 right 775 00:37:15,100 --> 00:37:18,266 and your radius it is the range from 0 to 1 776 00:37:18,266 --> 00:37:20,400 your 5 equal to 0 to 5 777 00:37:20,466 --> 00:37:22,200 and you can solve this integration 778 00:37:22,200 --> 00:37:24,400 know how to solve this integration 779 00:37:24,400 --> 00:37:27,600 your Z is constant so simply put it equal to 2 780 00:37:28,066 --> 00:37:31,000 your deal uh so it is a Roku by 3 781 00:37:31,000 --> 00:37:32,633 so solution is 1 by 3 782 00:37:32,666 --> 00:37:35,733 and again this uh uh cost square five solution 783 00:37:35,733 --> 00:37:38,233 you know it is 5 from your previous life 784 00:37:38,466 --> 00:37:41,000 likewise for the for this bottom 785 00:37:41,000 --> 00:37:42,900 the only difference is your normal 786 00:37:42,900 --> 00:37:46,033 to that surface element it is minus easy 787 00:37:46,066 --> 00:37:50,366 and also use easy for 2 - 2 at this bottom surface 788 00:37:50,700 --> 00:37:52,100 solve this integration again 789 00:37:52,100 --> 00:37:53,400 you will find out the same answer 790 00:37:53,400 --> 00:37:54,766 that is because minus 791 00:37:54,766 --> 00:37:56,133 minus been going out to be canceled 792 00:37:56,133 --> 00:38:00,266 so answer is 2 5 by 3 some all the 3 intergation 793 00:38:00,266 --> 00:38:01,933 so the answer is the same that is 4 794 00:38:01,933 --> 00:38:04,833 5 by 3 that we found out in the previous life 795 00:38:06,266 --> 00:38:08,400 let's solve this uh problem 796 00:38:08,400 --> 00:38:11,400 uh for your uh surical charge distribution 797 00:38:11,400 --> 00:38:13,300 so the case is for this 798 00:38:13,300 --> 00:38:18,400 uh uh for this charge distribution with videos r 799 00:38:18,533 --> 00:38:22,266 R and inside that your dependencies on a row 800 00:38:22,266 --> 00:38:23,966 not r divided by r 801 00:38:23,966 --> 00:38:27,366 and outside this radius of this radical chart 802 00:38:27,366 --> 00:38:28,566 distribution here 803 00:38:28,566 --> 00:38:30,033 it will be equal to zero 804 00:38:30,733 --> 00:38:33,066 so the first thing we have to find out 805 00:38:33,066 --> 00:38:35,866 the electric field inside that sperical charges 806 00:38:35,866 --> 00:38:37,500 solution so for this case 807 00:38:37,500 --> 00:38:39,300 your radius of integration 808 00:38:39,300 --> 00:38:42,866 to determine this Rovine integration 809 00:38:42,866 --> 00:38:45,966 would be the same as the radius of your garden surface 810 00:38:45,966 --> 00:38:49,300 that is your arm now you solve this integration 811 00:38:49,300 --> 00:38:53,066 you can take this doughnut outside the integration 812 00:38:53,300 --> 00:38:58,166 this R is also constant this R is going to be Q right 813 00:38:58,533 --> 00:39:00,666 and the integration of the science data 814 00:39:00,666 --> 00:39:02,700 respect to data is minus false data 815 00:39:02,700 --> 00:39:06,466 put this uh uh put this uh uh 816 00:39:06,466 --> 00:39:08,733 that uh integration limits are 817 00:39:08,733 --> 00:39:10,200 in this vertical coordinate system 818 00:39:10,200 --> 00:39:12,300 of the data that we know that it is from 0 819 00:39:12,300 --> 00:39:13,100 2 5 820 00:39:13,100 --> 00:39:17,833 4 d complete square and for 5 it is from 0 to 2 pi 821 00:39:17,966 --> 00:39:19,200 now solve this integration 822 00:39:19,200 --> 00:39:20,200 it is very simple 823 00:39:20,200 --> 00:39:22,966 so sign the integration is minus fast data 824 00:39:23,000 --> 00:39:27,166 then this R Q intervision is R 4 / 4 825 00:39:27,166 --> 00:39:30,800 put this limit inside this and you can 826 00:39:30,800 --> 00:39:33,033 for the left hand side for this 827 00:39:33,933 --> 00:39:38,100 for this close integral side that is your flag 828 00:39:38,200 --> 00:39:40,166 you can use the definition of that 829 00:39:41,733 --> 00:39:45,266 secretary of that sphere so that is equal to 4 bioscare 830 00:39:45,300 --> 00:39:49,000 you know that these equal to astronaut e 831 00:39:49,000 --> 00:39:50,600 electrical intensity 832 00:39:50,766 --> 00:39:52,866 and then you can simply simplify this equation 833 00:39:52,866 --> 00:39:54,033 to find out the e 834 00:39:54,066 --> 00:39:57,666 inside that um that's radical object 835 00:39:58,900 --> 00:40:01,666 so for that outside of that circle object 836 00:40:01,700 --> 00:40:04,700 so the only difference is as I mentioned earlier 837 00:40:04,700 --> 00:40:06,933 that your Gaussian surface 838 00:40:06,933 --> 00:40:09,533 it will be a combination of two Gaussian surface 839 00:40:09,533 --> 00:40:12,100 two spherical limits intergation limits 840 00:40:12,100 --> 00:40:15,600 so the first one is approaching till the radius of that 841 00:40:16,066 --> 00:40:19,400 till the radius of that cerical charge distribution 842 00:40:19,400 --> 00:40:21,200 and the second one is outside of that 843 00:40:21,200 --> 00:40:22,266 cerical charge distribution 844 00:40:22,266 --> 00:40:26,666 because if this is your are out of the 845 00:40:26,666 --> 00:40:29,133 all of your uh cerical charge distribution 846 00:40:29,133 --> 00:40:30,966 this is your causing circle 847 00:40:30,966 --> 00:40:32,700 so obviously from here to here 848 00:40:32,700 --> 00:40:35,866 you are going to have this volume chart dancing 849 00:40:35,866 --> 00:40:36,733 and also this 850 00:40:36,733 --> 00:40:39,700 this volume chart dancing is going to be zero 851 00:40:39,900 --> 00:40:43,533 so you don't only simply have to fall the integration 852 00:40:43,533 --> 00:40:45,800 which is up to there up to this 853 00:40:46,000 --> 00:40:47,566 uh up to this uh 854 00:40:47,733 --> 00:40:50,733 uh radius of this uh cervical charge distribution 855 00:40:50,733 --> 00:40:53,200 and here you can again relate the same 856 00:40:53,200 --> 00:40:54,466 uh case No. 4 857 00:40:54,466 --> 00:40:58,100 that the total charge enclosed by this uh 858 00:40:58,100 --> 00:41:01,033 by this causing surface is equal to D 859 00:41:01,533 --> 00:41:04,566 complete charge which is uh 860 00:41:04,733 --> 00:41:06,466 uh which is uh 861 00:41:06,733 --> 00:41:09,800 which is on this sperical charge distribution 862 00:41:09,800 --> 00:41:11,766 having the radius of this are 863 00:41:12,900 --> 00:41:13,200 now 864 00:41:13,200 --> 00:41:16,700 you again simplify this integration as of a last slide 865 00:41:16,933 --> 00:41:20,033 and what is so left hand side is going to be 866 00:41:20,733 --> 00:41:21,866 stay the same because 867 00:41:21,866 --> 00:41:24,966 it is not having the dependencies on this integration 868 00:41:25,333 --> 00:41:25,866 and only 869 00:41:25,866 --> 00:41:28,200 we have to take the integration of that surface 870 00:41:28,200 --> 00:41:30,100 at your factor gaussian surface 871 00:41:30,400 --> 00:41:30,766 however 872 00:41:30,766 --> 00:41:33,966 for the right hand side so your robot is constant 873 00:41:33,966 --> 00:41:36,366 your eyes going to be constant 874 00:41:37,666 --> 00:41:41,033 uh right and you can solve this integration 875 00:41:41,600 --> 00:41:44,266 and this integration is going to be R cube 876 00:41:44,266 --> 00:41:47,700 R cube integration is going to be R 4 by 4 877 00:41:47,700 --> 00:41:50,566 so this R will be cancelled by this R4 878 00:41:51,300 --> 00:41:53,400 and R cube will be staying over here 879 00:41:53,400 --> 00:41:55,300 and then you can simplify this integration 880 00:41:55,300 --> 00:41:56,866 to find out this user 881 00:41:58,333 --> 00:41:59,566 so I'm here today's lecture 882 00:41:59,566 --> 00:42:01,466 so we understood what is causes law 883 00:42:01,466 --> 00:42:03,700 how to find out the electric field 884 00:42:03,900 --> 00:42:04,566 intensity 885 00:42:04,566 --> 00:42:07,500 and electric first density using this causes law there 886 00:42:07,500 --> 00:42:10,700 and we divide the max size of first equation 887 00:42:10,700 --> 00:42:14,700 in the integral and in the differential dot notation 888 00:42:14,866 --> 00:42:15,500 and finally 889 00:42:15,500 --> 00:42:19,666 we found out some useful applications of this 890 00:42:19,666 --> 00:42:21,966 Gaza's law that is applicable to that 891 00:42:21,966 --> 00:42:23,766 symmetrical charge distribution 892 00:42:23,766 --> 00:42:27,200 to find out the electrical instance and intensity 893 00:42:27,200 --> 00:42:30,066 and electrical agency using this Gaza's law 894 00:42:31,300 --> 00:42:33,700 next time we are going to study 895 00:42:33,700 --> 00:42:35,900 or understand this electric potential 896 00:42:35,933 --> 00:42:38,200 and determine the potential difference 897 00:42:38,200 --> 00:42:40,466 due to this electric charge distributions 898 00:42:42,733 --> 00:42:45,066 here I thank you all if you have any questions 899 00:42:45,066 --> 00:42:48,900 that will be entertained to your emails or online 900 00:42:48,900 --> 00:42:49,333 central 901 00:42:49,333 --> 00:42:52,600 a session that will have arranged at the department