this is lecture No. 12 of a V222 2
electromagnetic field theory 3
today will study electro potential 4
this will cover Section 4.7 of your tax full 5
the learning objectives of today's electric would be 6
understand what is electric potential 7
and determine the elect 8
the potential difference between two points 9
what is electric potential 10
if we take the help of this analogy 11
with respect to this gravitational field 12
so in the gravitational field we know that 13
moving an object upward against the gravitational field 14
increases is gravitational potential energy 15
so by default 16
we know that due to gravity the force is a downward 17
so if we move an object upward 18
against this gravitational force 19
that in fact we are going to increase 20
the potential energy of that object 21
because we are acting against him default 22
and gravitational feed of the earth 23
so an object moving downward within the 24
within the gravitational feed 25
would lose gravitational potential merging 26
so it is the otherwise that if it did it did object 27
it is by default moving down 28
downward due to the gravity 29
or gravitational force of that earth 30
then it is going to lose its potential energy 31
and if would be the help of some external 32
as source extender force 33
we are going to move that object 34
against this gravitational force 35
then we are going to enhance the potential energy 36
of that particular object 37
so now let's relate this video back to electrostatics 38
so what happens that in this particular case 39
once you move 40
a charge in an electric fees is potential interchanges 41
and this is light out and light 42
moving a mass object in a adaptation feed 43
so in this particular object 44
if there's a negative charge created feed 45
which is invert and if there's a positive charge 46
so by default 47
it is going to attract towards the negative chart 48
however 49
if we are going to move this charge away from this 50
negative charge field 51
which is opposing field then we are go 52
we are going to add add on the uh 53
potential energy of this particular field 54
so that is the analogy between these two things 55
so again repeat that 56
it will move this positive charge away from this 57
negatively creative charge 58
inactive field which is in by default or by default 59
an attractive field we are going to 60
we're going to work against this inactively charged 61
generally electric field and 62
we are going to add in the potential energy of this 63
our electric potential of this vertical charge Q 64
so building on this in order to move 65
in order to bring two life charges near each other 66
two life charges so by default they dribble each other 67
we have to we have to work 68
so work must be done in order to separate two charges 69
again by default they are going to attract each other 70
so by we have to separate these two charges 71
so we have to again work for work is again to be done 72
so just repeat if there are two like charges 73
and they are repelling each other 74
and we want to bring them a closer to each other 75
then we'll have to do the work 76
and if they are they are to oppositely charge opposite 77
opposite quality charges 78
they are b by default attracting each other 79
and we want to sack them off 80
then we'll have to apply the work against that 81
attraction force so extended force 82
F against the electric field 83
increases the potential energy 84
in these two particular cases 85
electrify intensity due to a charge distribution 86
can be obtained from Coolum's Law 87
in general or because Isla 88
when the charge distribution is symmetric 89
so this this 90
this is the take away from our previous uh 91
two collectors 92
another way of obtaining electric feed is 93
from the electric scaler 94
potentially to be defined in this particular section 95
and why it is so important 96
in a sense 97
that this way of finding electric feed is easier 98
because it is easier to handle the scaler quantities 99
than the vector feed quantity 100
and we will see that electric 101
scale of potential is a scale of quantity 102
so that let's find out what happens 103
once we move a charge in the electric field 104
of some external charge 105
so we want to we want to move a point charge Q 106
a distance DL so this is the distance DL 107
that we want a point charge Q to be moved 108
and the work done is the product of force that we 109
apply externally and the distance it is going to cover 110
so this is by default the definition of your work 111
you know that it is f dot g 112
so in this case the travel distances 113
the this a differential length segment that is GL 114
the external forces 115
f that is acting against the external field 116
and here the external electric field 117
it is it is generated by some source 118
you can see that it is the external 119
it is the 120
it is the electric field against which you are moving 121
a charge Q a distance of the L 122
so displacement of a point charge 123
Q in an electro static field 124
which may be generated by another point source 125
align source 126
or a surface identity or wallet charge identity 127
so this we will see in our coming flights as well 128
so this is the definition 129
that work done is the product of force 130
and the distance covered by that particular point 131
charge if we move that point charge 132
DL distance in the presence of electrostatic field 133
due to some other charger 134
some other charge distribution and not that negative 135
negative sign 136
indicates that work is done by an external agent 137
this means we are doing the work 138
and the force we apply to move the chart 139
is against the electric field 140
and this is equal and opposite to the force 141
which is being exerted by the default 142
electrostatic field 143
against which we are going to apply the force 144
and if we 145
want to move the charge in the directional field 146
then as a natural 147
where as a natural we are not going to do anything 148
we don't do any sort of the word and feel does 149
why feel does because by default 150
either it will be repelling 151
or it will be attracting that particular fee 152
the total course 153
uh total work done are potential energy required 154
uh or 155
you can see that 156
potential energy added on into that particular uh 157
charge in moving up 158
in moving that charge from point a to point B 159
against this electric field 160
is found out by intergating 161
this particular differential word 162
over that entire part that is from a to B 163
so that is just uh uh 164
from your mathematical point of view 165
if you want to find out the complete word 166
you just need to uh 167
have the integration of that differential element 168
over that complete part a to B 169
so dividing a 170
W by Q is the potential that deposit charge 171
and this quantity 172
we also define as the potential difference 173
between point a and B 174
R 175
are the work done or equivalently 176
it is the work done by the external force 177
in moving a unit point 178
charge from point a to point B in 179
in the process of external electric v 180
and mathematically we can define it from here 181
we can define it that w 182
y q is equal to minus interval of a to B E 183
don and b l and it is in the sense 184
it is a potential difference 185
it is in the universe of wars or in otherwise 186
it is Joles Barcula 187
so electric potential point of view 188
there are some notes that if v a 189
B is negative 190
if the overall integration is negative 191
then there's a loss in potential energy in moving point 192
Q from a to B 193
and in that case the work is done by defeat 194
okay it is a V 195
a V narrative 196
remember that the work will be done by the field 197
there will be no gain in the potential energy 198
rather it will be lost in the potential energy 199
of that particular charge 200
because 201
it is moving in the direction of that particular field 202
and if we be positive then there's a gain 203
important energy of that particular charge 204
because in that case we are 205
external force is acting upon 206
and it is acting against that particular 207
electrostatic field 208
and it is going to perform the work 209
so there is a net gain in the potential energy of that 210
queue because we are 211
we are we are moving that charge 212
using that external force 213
against the direction of existing electrostatic fee 214
the third thing is VAB is independent of B 215
part of integration 216
so either you move from here to here 217
you follow this part or this part 218
the neck 219
the neck gaining potential energy will be the same 220
in determining this V a 221
b 222
a is considered by 223
conventionally as a mission point by V by 224
by B we have taken as a final point 225
so this convention 226
we are going to follow through this lecture 227
so let's take this 228
that E is the E which is being generated 229
and against which we are going to do the word 230
is being generated by by another point chart Q right 231
so we know that from kulunzla 232
that electrically generated by this particular charge 233
if it is present at the origin is Q divide 234
therefore perhaps cannot rskr 235
and if we want to move 236
and if we want to move another point charge 237
another point charge from this 238
from this point a to B falling this part 239
then what is the well done 240
so what the well done is or they put change in this uh 241
or increase in this potential energy is minus integral 242
okay to be edot BL 243
and since we are playing in our sperical partner system 244
so BL in the sperical partner system is defined at this 245
at this and since the electric field which was 246
which was generated by this uh 247
point charge against which we are doing the work is uh 248
is also really outward 249
so this start product will be only with this d R 250
d R 251
hey now we have to integrate this thing 252
and if we integrate this thing 253
so you can bring this up 254
so it is r minus 2 you involve that R rule 255
so at a result it will its integration will be 1 256
0 r r okay 257
now you put in this image to rearrange this minus sign 258
and then you will find out that this V a 259
B or the potential difference between point B and a 260
while you're moving this point check Q 261
while you're moving this point check Q is uh 262
is a Q divided by 4 by astronaut and uh 263
multiply by 1 or the distance factor of this 264
uh 265
particular the distance factor of this two point 266
initial and end point points 267
within which you are moving your point 268
charge against this existing electrostatic field 269
so again 270
these are the position factors for that initial 271
end and points 272
so building on the results of previous life 273
if we choose a reference point with potential of a zero 274
for example 275
if we choose a zero reference for potential that is 276
and that means you could zero at infinity 277
so we take that v a v 278
a is equal to zero at position rector 279
r is equal to infinity 280
so this means we are bringing a charge 281
we are bringing that charge from infinity to words 282
that point b 283
like this one 284
and so you can see like this if this is your Q 285
this is your R B 286
this is some that you are e and that is very 287
very far from this 288
first point charge 289
which is generated in this electric field 290
and you're moving another point 291
chart toward this point b 292
in the presence of this electrical 293
so what you can assume that at infinity 294
the potential is zero 295
so if you put that uh potential zero over here 296
you put that uh distance infinity nah 297
the uh r a to go infinity into this equation 298
and for that on what you can do 299
that you just use this mathematical thing 300
that one divider information equal to 0 301
so you can simplify the situation into this form 302
so gently the potential at any point 303
distance are from a point charge queue 304
which is placed at origin 305
and it is generated 306
some sort of electrostatic field is 307
v is equal to q d a before by ABS or not into r 308
and we are assuming 309
under this Assumption that v is equal to zero 310
at infinity 311
is the reference from where you are moving that point 312
charge towards that point b 313
the point at so 314
the potential at any point is 315
the potential difference between that point right 316
and a chosen point which is 317
which can be a reference point 318
and in this case it is with a potential equal to zero 319
so if they are 320
if they are in number of charges or in the 321
in this case your point charge 322
which is generating the electrostatic field 323
it is not located at the origin 324
it is located at some position Rector Albert 325
then in order to find 326
in order to find out the distance 327
the distance between this source charge 328
and the destination point b 329
we just need to use involve this distance formula 330
and the distance will be equal to B 331
and the magnitude of this uh 332
uh distance vector between these two position vectors 333
so if there are there are a number of charges 334
you want to Q 335
and which are located at some position records R1R2 336
2 r 337
N and we have to find out the potential at some point 338
which is located for example 339
that point is B and that is located at distance Rector 340
Acquisition Rector R 341
then we can simply enroll the superposition principle 342
and we can sum up the potentials which are 343
which are being generated 344
which are 345
which are being generated in the presence of each 346
of individual point source 347
and then we can add them up 348
we can simplify this into 349
what's mission formation as well 350
so if we are having uh instead of a point charge 351
the electr static 352
fee is being generated by some other type of 353
charge distribution for example 354
the continent's charge distributions and those we can 355
we will know that they can be offline 356
circus and warning charge distributions 357
what we need to do do that 358
we need to simply replace this 359
cue that we had in our previous slide 360
with this integral 361
so it is a integral of uh line charger 362
line charger uh charge distribution 363
it is a integral of second charge distribution 364
it is a integral of volume charge distribution 365
and integrated over entire volume of taxing 366
and uh note that here 367
the primed prime position rectors 368
they are going to represent the position of the 4.4 369
source chart distribution 370
this is another intuition of electric potential 371
this diagram shows some values of electric potential 372
at point in the electric field of a positively charged 373
here so for example 374
this is a positively charged here it is generating 375
generating some electrostatic feel and 376
if you are moving you're moving one column of charge 377
from point a to point B 378
since it is another positively charged positive charge 379
so you are 380
you're acting against that positively charged 381
created electrostatic field 382
so in this case 383
you are going to gain the potential energy 384
and if we if somehow the other 385
we divide these gains in the uh 386
in this potential energy over some distances 387
then we just 388
just for the understanding point point of view 389
you can see that with a gain of 15 volt 390
uh 15 uh world or jewels 391
but uh jewels of potential energy in the States 392
not for E due to a point charge 393
we have assumed that um 394
the zero potential it is at some infinity distance 395
so in case if it is not an infinity distance 396
the reference point of the award 397
it is not at some infinity distance 398
then obviously it will have some some this 399
position rector 400
so in this case what we can do that we can um 401
we can denote things like this that if another 402
if any point other than infinity is chosen 403
as a reference with zero potential 404
then what we can do 405
that we can represent that potential 406
we can represent this other term 407
in this different thing with some question 408
right with some question and further 409
we can generalize this complete thing 410
for some other points 411
that the potential at any point are from a point 412
check Q which is placed that origin is 413
and and remember that here 414
this reference point of zero 415
potential is taken at some distance r e 416
and why we have taken it as a constant because it is a 417
it is that fixed it is that fixed position 418
that's why we have taken it as a constant 419
eposition factor for this one is fixed 420
however 421
since we want to make the evaluations of this potential 422
towards the destination point 423
so this destination point 424
which is at this position director R B 425
it is varying in this case 426
however 427
this part is staying as a constant because we are 428
we are moving from this point towards the destination 429
so generally what we can we can write this integration 430
this integration as this 431
as a solution as a 432
as a reverse 433
as a reverse of this thing as an integration 434
we can write that this 435
this thing that it is potential 436
generally for any electric field digital point source 437
the potential is equal to minus in Tigorov e dot b L+ t 438
and here we are assuming that we are only consideringly 439
this position rector 440
and this position 441
rector is going to recover in this constructor 442
let's solve this example of your book 443
to further clarify 444
the concepts of this electric potential 445
so in this case we are having two point charges 446
minus 4 microcoolum and 5 microcoolums 447
and they are located at these lines 448
at this position factors and what we need to find out 449
we need to find out the potential at this point 450
assuming that zero potential at infinity 451
okay so in this case 452
it is a superposition problem and Q1 is this 453
Q2 is this 454
we just need to plug in the values in this case 455
and utilize the formula which is for infinity 456
for infinity 457
we know that this constant is going to be zero 458
because we are bringing a charge from infinity 459
towards this point right towards 460
toward this point B so in this case it is 1 461
0 n one 462
and here is Q1 for example is Q2 463
they are generated some sort of 464
electric field so Q1 you can still like attractive one 465
it is the deposit one like this one 466
and then you can just 467
you can just plug in these values of Q1 and Q2 468
you can find out this uh uh 469
distance factor between these 470
uh between this elevation point and this source point 471
this source point 472
and just use the magnitude of this thing and what else 473
and plug in this magnitude over here 474
and you can find out d potential at this point B 475
the potential energy or the electric potential 476
which is attained by uh 477
once we are moving this point 478
unit point charge q from 479
from infinity distance towards this point b 480
in the presence of this Q and N Q2 481
let's solve another example of your book 482
uh and it is a complex one 483
uh in this case 484
applying charge 5 nanoculum is located at this point 485
while another lines 486
uh uniform uh distribution applying charge 487
it is uh uh 488
it is uh located at y equal to 1 and z equal to 1 489
so it is you can see that it is a x axis 490
aligned line current distribution 491
so remember this thing 492
so the first thing we need to find out the uh 493
potential at this particular point 494
if we are moving uh moving 1.1 495
uh one that uh 496
uh 497
point charge from a point which is at zero potential 498
and this point is located at this zero 499
so you can assume like this 500
this is your unit charge right 501
you're moving this charge to another point a 502
in the presence of this Q1 and this 503
what you say 504
you remember we donate this line chat by this Royale 505
something like this okay 506
so the electricity being generated by this thing 507
and electrocuted engineered by this 9 source 508
and you are moving from this zero potential 509
reference point at this origin 510
and toward this another point r B 511
you can take it at this point r 512
B and you can take it as a 513
your prime you can take in this case okay 514
so uh since we we can see over here 515
the potential is being created by uh 516
two different uh sources the first one is the point 517
so the second one is the line distribution 518
and we know that the 519
you know to find out the potential 520
we can take the help of this standard equation 521
that is minus in 2 block e dot DL 522
and we know that the electric will be tested 523
you to a point charge 524
it is Q divider for perhaps or not ask here 525
and we know we 526
since it is a sperical corner system problem 527
so so this dldr yeah 528
you just uh integrate this thing right 529
integrate this thing 530
so minus minus will be cancelled out 531
so in the integration 532
you will see that it is minus r 2 533
you just involve the positive integration 534
so it will come come out to be A1 or R and minus 535
minus will be cancelled out so this is your VQ 536
right but is why this Constance coming out 537
because at reference point of zero voltage 538
it is not at some infinity distance 539
okay for the second case 540
we know that 541
electrically generated by an infinite line charge 542
it is row L divided by 2 by ABS 543
no not row and it is in this slantrical partner system 544
remember here and here 545
if we integrate this one with this back to this row 546
we know that integral of this one of our row is 547
which looks like the row is Eleanor Row 548
so recall your integration concepts over here and okay 549
so that's it 550
and here C2 is another integration constant 551
and we can combine the C 1 552
c two in the form of another constant that is c 553
and this is our equation that we will utilize 554
in the next slide 555
okay in the first case 556
we are moving from this origin point at the zero board 557
to destination point a and uh uh here 558
uh you can see that what we are going to do 559
that we are going to simply take the defense 560
we are going to imply this equation over here 561
we're going to imply this equation over here 562
but we are going to put the uh 563
this uh perpendicular distance uh 564
uh evaluations and this uh 565
position after evaluation with respect to this 566
origin and this destination point 567
so in this case in this particular case 568
we remember that this row is the perpendical distance 569
from x y z to this case any point x y 570
z to the line 571
which is the y equal to 1 and the equal to 1 572
this is x X is a parallel point and this case 573
if we want to find out the rule 574
what we need to do that is if this is our uh uh 575
uh 576
some uh destination point and this is our source point 577
so we just need to keep this 578
take 1 coordinate seem as of your evolution point 579
and we just need 580
then need to take the distance between 581
distance factor between these two points only 582
so if we generalize this equation 583
in order to find out 584
the distance factor between two points 585
is perpendicular to ax thus 586
this will be eliminated 587
because this is a x axis parallel line 588
so as we remembered we need to keep this coordinate 589
same as the coordinate of this revolution point 590
to find out the perpendical distance between this line 591
and the evolution point now 592
simply this will be a standardized equation 593
for this particular case 594
only for this where distance factor 595
let's find out this row knot and row a 596
with respect to this line 597
and this R knot and r a with respect to this source 598
point source 599
so uh in the first case your uh 600
in the first case that is the doughnut 601
your knot is your uh uh 602
your uh your or initiating initialization point 603
remember it is uh your uh 604
you can say that your initial point 605
and here 606
you see that you're keeping this x coordinates same 607
and 608
to find out the potential distance between the line 609
and this point of origin 610
and for the second case at this point a 611
the destination point 612
we're keeping this coordinates in 613
to find out the perpetual distance between this line 614
source and this point a 615
okay now that 616
simply solve the distance factor how to find out 617
find out using this 618
so the second case is 619
you need to find out the position vector 620
or the distance vector between two points 621
and the first point is your origin 622
the second point is your destination point 623
and the source is your unit point source 624
that is minus 3 4 625
0 in this vertical example 626
so that is your uh 5 nanoculums 627
uh source point location 628
no uh 629
you 630
you know that we don't have the integration over here 631
integration constant over here why 632
because we are having minus over here 633
so the constant will be subtracted in this case 634
once we uh 635
will be eliminated once we subtract the uh both uh 636
both evaluations using the same formulation 637
may you put in these values and you can uh 638
you can simplify this thing by putting this value 639
so not here 640
the head we have Eleanor run out divide by row a why 641
why this one 642
because if you plug in this value over here 643
you take the difference of this 644
we know that Eleanor 645
something like Eleanor a divided by B is equal to 646
Eleanor 647
e minus Eleanor B 648
this is this is your love written property 649
so you can uh you can write the subtraction of uh 650
log written term as a division 651
as a log rhythmic of division of those two terms 652
your planning value 653
you are going to find out this value 654
so the second case in this case 655
you are having the reference point of 100 word 656
and it is a place that this point 657
no you just need to 658
you don't just need to plug in this value 659
you just need to really this valuation point 660
so in this case this is going to stay same 661
this is going to stay same 662
this is the distance factor 663
between your destination point 664
and your source point right 665
and what is your 666
and here your destination point is this one 667
your source point is initiating point is this one 668
and you can just plug in these values to find out 669
this resultant thing 670
and only differences instead of zero reference board 671
you are having under board reference board 672
the third thing is 673
we want to find out the difference between this 674
B and C this is simply V 675
C minus VB 676
that we have found out in this Part a and Part B 677
well here we don't need a potential reference 678
because it will use a common reference 679
to find out the potential 680
what is the VC and VB 681
then this will be eliminated by default 682
this is your assignment for your week No. 6 683
and here again note that you have to solve this 684
practice exercises from 2nd edition 685
if you're using the 3rd edition 686
kindly match those 687
equate those questions with your class 688
class medals who are on campus 689
some here today's lectures 690
so we understood what is the electric potential 691
how to find out the electric potential 692
and how to gain the electric potential 693
how to uh 694
increase the potential in the presence of some uh 695
external electric well 696
electric fields for that particular uh general charge 697
and then how to uh 698
find out the potential difference between two points 699
in the presence of uh 700
some external fields and this extended field 701
they can be due to some point source line sources 702
the charge distribution are such as 703
all while in charge distribution 704
so next time we are going to find out 705
the relationship between electric and 706
electric field and the 707
this electric potential and also 708
we are going to drive the Maxwell's second equation 709
and we will also study what is an electric and typo 710
here I thank you all 711
if you have any questions 712
they will be entertained to your emails or online 713
Eastern Corona session 714
that will have arranged at the department