this is election No. 14 of a V232 electromagnetic fifty 2
today we'll study electric flux lines 3
equi potential services 4
and energy density in electro static feed 5
this will cover your section 4 6
point 10 and 4 point 11 of your textbook 7
the learning objectives of today 8
uh 9
today's lecture will be to understand what is the uh 10
this electric fox lines and the equiferential services 11
and to determine the energy density in the presence of 12
and electrostatic field 13
first of all electric flux lines 14
the idea of electric flux lines 15
was first introduced by Michael Ferrari 16
in his experimental investigation 17
as a way of visualizing the electric field 18
so as you can see 19
it is some hypothetical or imaginary part 20
so an electro electric uh 21
flux line is an imaginary part 22
or hypothetical part of line or surface 23
and is a direction at any point 24
indicates the direction of electric feed 25
at that particular point so basically it is uh 26
it is an imagery part that is uh utilized uh 27
to uh represent uh the direction of electric field 28
at some particular point 29
and what is equal potential services 30
so any surface on which deep potentially seem true 31
so the potential is same 32
short on this equal potential surface 33
or this equal potential line 34
uh in case if it is a equal potential uh line 35
then the potential will be same on that line 36
and we know that 37
the intersection of equal potential surface 38
with a plane reserves 39
in a pot 40
so that is a line 41
so that is a one dimensional pot that is a 9 42
so if your equal potential surface 43
it is being cut by that one plane 44
then we know that the intersection of two planes 45
or two surfaces is a line 46
and in this case 47
that line will be known as equal potential line 48
also we know that uh 49
no work is done 50
in moving a charge from one point to another 51
along equal potential line or surface 52
why because two of the surface is potentially same 53
this v a is equal to v b 54
okay and that is equal to V 55
so there no work will be done in this case if we move 56
uh from one point to another point 57
on this equiprentious surface 58
or a line so 59
this implies that electric plus nines 60
or the direction of E 61
are always normal to electric surface 62
or e equiponential surface 63
all nines so what 64
how can we say that it is normal to this uh 65
these flat lines are normal 66
to the equiperential surface 67
9 surfaces 68
because this start product is going to be the 0 69
and this start product is only equal to 0 70
if equal is equal to 71
90 72
right 73
I repeat that electric Fox lines or in others detail 74
the direction of electric fuel 75
are always normal to equal potential services or 76
or lines because this difference 77
which is equal to the line integral of e dot TL 78
is equal to 0 so that's why 79
and this dot product is only equal to 0 80
if and only if 81
the angular displacement between this equal 82
potential line and the electric feed is equal to Z 83
is equal to uh zero 84
and that is only once 85
their angular displacement is equal to ninety 86
okay in the following uh diagram 87
uh we can see that these green lines 88
the dash lines 89
they are going to represent the ECU potential lines 90
and the story lines 91
they represent the electric free line 92
so for a positively charged 93
for a positively charged point charge 94
we know that the electric free lines 95
they are moving away from this source 96
and if we see that these green lines 97
they will represent these circular lines 98
in this case it is a circular line 99
so this circle will represent r 100
will represent n equal potential line 101
or equal potential circle in this case 102
and as you can assume like that 103
the closer to the point source 104
the for example 105
the potential is 30 volt than away from that 106
so the potential is decreasing 107
so it is 20 volt and again away it is 10 volt 108
so it is again 109
as you can see here that is equal to minus 110
that'll be so again the potential is increasing 111
in the opposite direction of electric field 112
so if we are having the two parallel plate conductors 113
that is your uh uh capacitor 114
so two parallel plate uh conductors of uh 115
oppositely charged uh uh polarity 116
we know that again the electric field uh 117
start from positive uh terminate on the inactive plate 118
and closer to this policy 119
the potential is later and then it is going to reduce 120
and uh okay this and this is this is going to be it 121
this is going to be 122
it's equal potential line in this case 123
now let's consider the case of two point charges 124
so the first case is once they are oppositely charged 125
for opposite polarity charges 126
so they are going to attract each other 127
so this is your electrify lines 128
and you can assume that these greens 129
they are going to represent the eco potential line 130
uh lines in this case 131
so the potential is going to be steam 132
the resultant potential 133
due to these two point charges is going to stay same 134
on these uh 135
hypothetical uh dotted lines green lines 136
and at the point charges they are of same quality 137
then they are going to repel each other 138
and in this case you can see that 139
this is going to be the equal potential 140
the result and equal potential lines 141
due to this oppositely charge charge quality charges 142
and uh let's see 143
what are the electric plus lines 144
and differential services 145
for the dipole 146
so this is the same case that we discussed for the uh 147
fine charge in the previous time 148
so for the dipole 149
this is uh for example it is it's positive chart right 150
and this is negative chart 151
then we know that it will start from positive charge 152
and terminate on the negative charge 153
like this way right 154
and so these are the flux lines or electrical lines 155
and what are the equal potential 156
the resulted equal potential lines 157
due to this die call will be represented like this 158
dashed circle like this one 159
know that the lines of force very very important 160
take away from these three force lines 161
that lines of force 162
electrical lines or electric front lines 163
or the direction of E field 164
is always normal to equal potential surface 165
you can see here that it is normal 166
it is normal 167
it is normal to the secret potential surface hotline 168
well again it is normal at the start 169
it also it is going to normal 170
it is also going to normal so we're gonna do that 171
we can take away from this slide 172
that electric free lines or electric press lines 173
they're always normal to the equi potential services 174
uh some applications of this uh 175
ecubential services are we know that uh 176
if we place uh if we place these conducting bodies 177
or conducting spheres in the electric field 178
then uh the surface of this conducting body 179
it is going to represent this equal potential volume 180
because for this conducting bodies 181
due to its smallest skin bag 182
the charges are going to stay on the surface of these 183
uh radical type of conducting bodies 184
due to the metallic property 185
or the higher conductivity property 186
or the least is or the minimum skin effect of the uh 187
conductors 188
these charges will stay at the surface of conductors 189
and charge will flow on this surface of conductor 190
so we can treat this conductor as a 191
as an ECU potential volume 192
in the presence of electric clip 193
the second example is very famous and practical example 194
that is your electro cardio diagram 195
electrocardio diagram 196
that is your ECG and you're a medical 197
and it is bi 198
medical application of this ecubrancial services 199
so a typical application of field mapping 200
or electric flex lines or ecubrancias surgery 201
found in the diagnosis of human heart 202
our human heart beads in response 203
in response to an electric speed potential 204
that we apply using the electrodes 205
uh Pisces electrodes across the uh heart 206
uh on the uh uh once we plant 207
plant those electrodes on the chest across that 208
across the boundaries of that heart and the then heart 209
then heart therefore the heart 210
so basically the heart will beat 211
in response to the electric fuel potential 212
which is being applied across this heart 213
and then what we are going to monitor 214
uh the heart can be characterized at the typo 215
as a typo with the field map 216
similar to that on last line 217
uh such a field map is useful 218
in detecting abnormal heart position 219
for example electro cardiogram 220
so the uh the uh we can 221
if we see some of the abnormalities in this uh 222
uh Bible pattern uh 223
which is generated by this uh 224
heart beating against the standard uh 225
uh 226
type of pattern that we started on our earliest slides 227
then 228
there will be some abnormalities in the heartbeat of 229
and the heart so first so what we do 230
we apply the electric feed potential 231
using the electrodes across this uh 232
heart and then heartbeats 233
heartbeats and generate electric feed lines right 234
generate electric feed lines that we study 235
that we basically read using the sensors out 236
using the electrodes and then we plot that on the Ecgs 237
now or in this case 238
it is going to give you a pattern of like a typo 239
national study how to determine this energy density 240
in the presence of electrostatic feeds 241
so first of all 242
we consider our region free of elect electric field 243
so there's no point chart presence in this region 244
for example this region 245
there's no point chart present in this region 246
and now we are going to move three point charges Q1 247
Q2 Q3 from infinity in this charge free region 248
to determine the energy present 249
in the assembly of charges 250
we have to determine the amount of work necessary 251
to assemble them so we basically 252
we are going to calculate the amount of work 253
to determine this energy 254
required to assemble these three charges 255
first of all 256
no work is required to transfer Q1 from infinity to P1 257
why 258
because we are not working against any electric field 259
which is present in this sorcery or charge free region 260
and uh 261
so because the space is initially charged sorcery 262
so there's no electricity 263
we are not we are not going to perform any of work 264
and in the second case of work is being done 265
once we transfer Q2 from infinity to point P2 266
and in this case 267
we are going to work against the electric field 268
which has been created due to the presence of this Q1 269
so 270
we are working against electric field due to this Q1 271
and we know that uh uh we know that that 272
that work is equal to q b 273
that we have already drive earlier 274
so that work is equal to Q2 275
since we are moving this uh Q2 276
and then potential uh 277
uh difference between this point P 2 and uh uh 278
this P1 is going to be v 2 uh v 2 1 279
in the third case 280
uh now we are uh moving the uh point P3 281
point er point chart Q2 Q3 er at point P3 282
and we are placing this point charge to 3 283
from infinity from infinity to point P3 284
in the presence of electric feed 285
which is now being created by the presence of Q 286
2 and Q1 so we are working against two 287
against an electric feed 288
which is the resultant of electric feed 289
due to these two point charges 290
now and now the potential difference will be 291
the potential difference between this point P3 and P2 292
and the potential difference 293
between this point B3 and B1 294
now the total work done 295
while positioning the three charges can be summed up 296
like this one so far 297
uh first Q1 charge it is equal to zero 298
for the Q2 charge it is q 2 times v 2 1 299
for Q3 charge q q q uh w 3 300
that is q 301
q 3 charge it is uh q 3 times uh v 3 1 plus v 3 2 302
and these are the definitions of the 303
which 1 is the potential at point 2 due to Q1 304
potential at point 3 due to point a Q 305
point charge Q1 306
and potential at point 3 due to point charge Q2 307
now if we place these charges in a reverse order 308
that we if we place this charge Q3 fault 309
then again 310
since no charge is present in that free charge 311
free zone so the well done will be zero 312
and now we move this point 313
charge Q 314
in the presence of point charge 3 in that region 315
so this will be equal to Q2 316
Q2 times 3 to 3 317
and then likewise 40 Q1 in the presence of charge q 3 318
n Q2 so it could be q 1 times q 1 v 1 2+ v 1 3 319
uh now we add up these two equations 320
and we take these Q ones common 321
then we can simplify this equation like this 322
that's v 1 what is v 1 so v 1 is the potential 323
potential difference or the potential at 324
at this uh point uh 325
B1 due to this point charge Q2 1 326
Q3 and so Jordan potential at point B at point B1 327
P2 and P 3 r V1 V2 and V3 328
so V1 is the total potential at point V1 329
due to point charges Q2 and Q3 330
and likewise 331
this V2 is the total potential at point P2 332
where we are we are bringing this Q2 333
uh and that is differentially to point uh 334
point charge Q1 and Q3 335
and likewise that point Q3 may be a point P3 336
the protector 337
prevention is due to the point charge at Q1 and Q2 338
and then we can set bring this to on this side 339
so this is the total well done total well done 340
once we are moving three charges 341
three charges to a free to a source predision 342
and then we can simplify this equation 343
in a smashing promise 344
in case we are having the continuous chart distribution 345
then 346
we simply replace this summation with this integral 347
and this cue by this 348
uh this densities so in this first case 349
so it is uh line chart density 350
in the second case it is surface chart density 351
and in the third case it is the walling chart density 352
and this mission is being replaced by this line 353
in trigger surface in trigger line 354
this volume in trigger to find out the total charge 355
to find out the total charge present in that volume 356
surface or line 357
no uh 358
we uh further uh take the last uh this volume charger 359
uh continuous uh uh country chart dance to example 360
uh that total uh uh energy dance report this case 361
the total work done in this case 362
and we uh we take the help of this uh 363
first equation of Maxwells that is Robizi 364
put that dot e we just plug in this thing and we flip 365
apply some of the vector identities 366
the divergence theorem and some simplifications 367
that you can consult in the book 368
then it can simplify 369
it can be simplified to this uh uh dropper.com 370
so the energy density are total work done for this 371
walling chart density 372
so is the volume check it out d dot e 373
but in other words 374
we know that these equal to astronaut E 375
and we can also simplify this dot term 376
as the astronaut E square 377
so 378
this is the energy store that we work on in this case 379
now we define another termology 380
that is the electrostatic energy density 381
so this is the well done or energy 382
or energy per unit volume 383
you can see here jewels uh 384
your uh this uh jewels per uh cubic meter 385
and also you can see it uh as a rate of change of uh 386
this uh 387
energy density with respect to volume 388
differential volume 389
so what we need to do 390
we only we cut out this in children in this case 391
simply cut out this in children in this case 392
because differential is going to cut this 393
or eliminate this in children 394
and this is going to stay like half of d dot e 395
or half of astronaut E squared or is equal to d squared 396
divided to ABS cannot but in other words 397
we can see that the energy store 398
or energy density is also the volume 399
in trigger of electrostatic energy text 400
let's solve this example of your book that is 4.17 401
two point charges 402
okay the 3 point charge 403
3 point charges are located at origin 404
at this point and this point 405
and we have to find out the energy in the system 406
now we first ways do is to utilize this mission formula 407
and what we can do here so we want 408
we want when we know that it is equal to the potential 409
due to charge Q2 410
and Q3 411
at this point P1 412
so we take this s P1 we take this s P2 413
we take this s P3 right 414
and Q2 it is at the distance of 1 415
we can see here this point is at the distance of 416
0 0 1 right 0 417
0 418
1 419
so p is P1 is actually at the origin 420
and this P3 is at the distance of 1 0 421
0 like this way 422
so the distance between 423
that is using the distance formula 424
it is only simply the z axis that is the 1 425
and for this Q3 it is also the quatu one right 426
because only axis coming into play in this case 427
therefore this V2 428
V2 is the potential at this point V1 due to discharges 429
Q1 a Q3 430
no uh 431
this is your yeah 432
this is your elevation point right 433
and from here to here again 434
the distance is 1 however 435
from here to here the distance can be found out 436
using your Pytaborus theorem right 437
uh so that is equal to 438
that is very diagonal in this case 439
right and we know that is 440
it is it is the 441
some of the scales of the two sides scheduled 442
scheduled of the some of the scales of the two sides 443
and that will be at route 2 square root 2 444
and for the third case uh 445
for this case again 446
uh for this Q1 447
it is distances 1 and for Q2 448
the distance is again the diagonal that is Q2 449
so it is at the potential at point P3 450
due to this point chart Q1 and Q2 451
then you can simply plug in these value 2 1 452
2 2 1 453
2 3 454
take care of these polarities 455
and here you can see the polarities as well 456
and you can simplify that it is 13.37 nanojoles 457
no alternatively 458
what is the second way 459
so the second way is instead of the submission formula 460
you involve your basic uh 461
definition of finding all this electrostatic energy 462
so what was that you bringing 463
bringing this charge Q1 from infinity till this point 464
right so for this case no work will be done 465
secondly you bring this charge 466
Q2 in the presence of the electric field of 467
due to this Q1 and then you bring this charge Q2 468
Q3 in the presence of electric field of this Q1 469
N Q2 so uh for the second case 470
uh for first grade zero 471
for the second case it is uh Q1 472
uh Q2 okay so potential 473
we know that it is Q1 474
and what is this distance formula 475
so the distance formula is d 476
uh distance formula is D so here uh the source 477
the source source 478
okay source and destination 479
so destination is your uh destination is your 480
so your decision P1 481
P2 and P3 right 482
so this is your destination 483
P2 is your destination and this is P1 is your source 484
so this is your distance from order in this case 485
and uh for the uh for this third case 486
your uh distance formula for this one 487
uh for the first case it is uh okay 488
this is your destination in this case please 489
please 490
your destination and you have to come from P1 and P 491
P1 to P3 it is simply uh 1 1 492
0 - 1 0 493
0 - 0 0 494
0 and for this one it is destination 495
we know that it is 1 0 496
0 and the source we know that it is 0 497
0 1 and then you can simplify these two uh 498
things and put back these values and uh 499
it will come out to you 500
so take care of this more as well right 501
this transformer uh this modulation modesty 502
uh once you take the difference of these two position 503
rectors then you can take it a scale root 504
uh uh modern and the answer is same 505
another example of your book 506
so in this case 507
a chart distribution which vertical symmetry is given 508
so this vertical chart distribution given in this case 509
and you're serving the volume chart 510
dance to your pro v is equal to constant 511
per or not once you are within this sphere 512
that is your radial distance from 0 to r 513
and out of this uh sperical entity 514
your chart entity is equal to 0 515
and we need to determine the energy store 516
stored in this region 517
we know that elected 518
fever's radical chart distribution from uh 519
Gaza's law uh 520
in uh this 4 point d section 521
that it is equal to runaut r divided t F snaraut a r 522
I did magnitude of this one 523
we know that it is a simply 524
I take take away this uh 525
junitractor and magnitude scared 526
simply takers scared of this fan 527
so we need to find out this uh energy density right 528
that is equal to half up B volume trigger lock 529
astronaut is clear right 530
and what we need to do 531
we need to plug in this value right 532
now the only variable in this case is our scale 533
because there's no cheater time 534
there's no fight time 535
with respect to select the partner system 536
so these things are being that as a constant 537
uh for this vertical corner system 538
we know that DV is equal to r square sign Peter d 539
r N d t 35 and uh we can uh simplify this integration 540
these two integration that this is equal to 4 541
5 you can solve this at your own time 542
and since RSK is there so RS R it will become RSK 543
RSK will come R4 its integration will be R5 544
we have F5 and just plug in this range of 0 to R 545
and this is your energy density in this complete 546
sperical entity with charge distribution will be 547
this is your assignment for week No. 7 548
you have to follow these two weekly 549
it is two question from chapter number four 550
practice exercises 551
and please follow this 2nd edition of the textbook 552
and off campus student 553
you can concern the on campus student 554
for the exact questions 555
so I'm here today's lectures to Posttrophone 556
we understood what is uh this electric flex lines 557
and what are the equipracial services 558
and the equipracial lines 559
and then we uh uh we uh 560
we studied how to determine this electric density 561
our electrostatic energy 562
in the presence of electrostatic fees 563
once we are bringing one or more than one charges into 564
some charge fee is done 565
in the next class 566
we are going to start off with our chapter No. 5 567
that is electric fees in material space 568
instead of free space that we have been starting up 569
you know absent not 570
we will study some materials 571
which are having properties other than the free space 572
we will study in that properties of those materials 573
in that particular lecture 574
the next lecture and we will 575
we will also understand what are these two types of 576
the crimes that that are the convection 577
current and the conduction 578
here I thank you all if you have any questions 579
they will be entertained to your emails or online 580
synchron 581
a session that will have arranged at the department