this is election No. 17 of Amy 2 2
3 2 Electromagnetic Queen theory 3
today we will have an overview of question number one 4
linear isotropic and homogeneous dielectrics properties 5
continuation 6
and relaxation time of the dielectric materials 7
this will cover your sections of 5.7 8
and 5.8 of your book 9
and this is academic week No. 10 of your semester 10
the learning objectives of uh 11
today's lecture would be 12
to have an overview of question No. 1 13
to study linear 14
isotropic and homogeneous dialect materials 15
to understand what is continuity equation 16
and to determine relaxation time inside and uh 17
dielectric 18
let's now discuss 19
some of the properties of these dialect materials 20
and what technologies we use 21
once we define these dialectrics 22
so the first one is 23
once we see that our material is linear 24
then it means that d raise 25
meaningally with respect to e 26
remember the secretion d is equal to abslon e 27
the constitute of relationship 28
so 29
it says that materials for which action are the Sigma 30
because for the Sigma 31
you know that J is equal to Sigma E 32
right 33
uh for this uh this type of the materials 34
uh this outline 35
or this conductivity does not 36
re in the region being considered 37
and this means that they are seen at all times 38
indeed 39
3D special partner system and also there that is 40
you know that it is independent of x 41
y and z coordinate system 42
so if it is independent of your geometric 43
geometrical distribution 44
then it is going to stay the same toward the 45
this region and then this material it will 46
it will be known as the homogeneous meat material 47
okay so first of all 48
if the d 49
it is directly proportional to E inside a material 50
then if this material is known as the linear 51
and for that material if this uh 52
value of this absline dependentivity 53
and the conductivity 54
they are going to stay the same throughout the region 55
and they are independent of the special coordinates 56
than 57
their material is known as the homogeneous material 58
in case 59
if it is an 60
in homogeneous or mono homogeneous material 61
then when this means that 62
he is dependent upon the space coordinates 63
and you can take the example of the primativity 64
of the atmosphere 65
because it is changing with respect to the altitude 66
so materials uh 67
for which it be the electric 68
first density 69
and the electric field are in the same direction 70
are set to be isotropic 71
so if they are uh in the same direction means that uh 72
they are well 73
since we know that they are in their director 74
quantities and if they are in the same uh directions 75
uh this means that uh their abslon and Sigma 76
they are not going to uh raid their 77
they will stay as the constant 78
these two things 79
this thing will be stay at the constant 80
so this means they are they are in the same direction 81
and these types of material 82
they are known as the isotropic materials 83
and for an isotropic materials 84
or non isotropic materials 85
your your this of flux tensity 86
your lactropine 87
and polarization are not parallel to this permativity 88
or this acceptability and in this case 89
it is going to have the 9 components 90
because it will vary in all X 91
y and Z directions 92
and this variation is referred to as cancer 93
cancer metrics this thing you can see here 94
the variation in the x direction 95
y direction and z direction like this one 96
duration in the x y direction 97
duration in x z direction 98
duration in y z direction 99
so this combination is going able 100
reserved in a 9 element metrics 101
and such a metrics 102
you can also see that these metrics exhibit 103
for the crystalline materials 104
and the magnetized plasma 105
these these are the two examples 106
for d and asotropic materials 107
and here you can see that these components of the d 108
they are they are 109
they are not directly proportional to this uh E x 110
but they are involving this uh 111
multiplied of this uh radiation terms of the abstract 112
now summarizing this a dialectic material is linear 113
if for this d is equal to absent E 114
the E does not change 115
once the electric field is applied on that material 116
so this means that under the influence of external 117
lactropine the permativity of that material 118
it is going to steady constant 119
for dialecting materials 120
to be qualified as homogeneous materials 121
uh for this given d 0 absnar not e 122
e does not change from one point to another point 123
so this means that your action 124
it is going to stay constant through this given region 125
and a dialectromic material is set to be isotropic 126
if E does not change in different directions 127
the same idea hose for a connecting materials 128
for which the home law is applied 129
that is Jay Z go to absolutely not Jay Z go to Sigma e 130
and in case uh it is uh linear then uh 131
your stigma does not uh 132
change once the electricity supply 133
and if it is homogeneous then 134
stigma is going to stay constant 135
at all the points in the region 136
and if it is isotropic then stigma 137
the conductivity 138
does not vary with respect to direction in the 139
and different directions 140
so for linear 141
isotropic and homogeneous media are formulas 142
dried for free space 143
can be applied for these types of demetrial 144
which are linear isotropic 145
and homogeneous 146
by simply replacing the abslanaut with this abslanaut 147
abslanaut are abslineaut into Abslineaut I 148
and we know that this is all equal to simply be 149
abstinaut abstlont 150
so the simple thing is if it is a cooling star only 151
you have to replay 152
this actual knot is equal to absolute knot 153
absna RD 154
the product of this optional knot 155
and the relative permittivity of that medium 156
and for the similarly on the 157
all the previous formulas 158
you have to replace this abstlon 159
knot with this abstlon 160
which is equal to this abstlon knot into the 161
a relative primativity of that but the medium 162
let's say discuss what is this continuity question 163
and what is this relaxation time 164
so uh we know that 165
due to the principle of charge conservation 166
the time rate of decrease of charge 167
within a given volume 168
within a given volume so the 169
it is uh you know it is at decrease 170
so it is represented as the negative quantity 171
so it is at decrease in the rate of change of this uh 172
this charge with respect to time inside the volume 173
and it is equal to the net outward current 174
flowing to that close surface 175
so that is a definition so we know that the current 176
it is equal to date of change of the charge 177
with respect to time 178
and if it is decreasing inside the 179
if you charge it decreasing inside of warning 180
then it will be termed as a minus term 181
and from this um uh if we are given the 182
this uh 183
this electric condensity information 184
and we evaluate the closing trigger of this 185
electric condensity 186
with respect to this surface element 187
that it also gives out this uh 188
current coming out of that close surface 189
so that is the definition of this uh current 190
so where in this uh definition q in 191
Q in is the total chart 192
which was residing inside this closed surface 193
and it is decreasing respect to time 194
let's simplify the things we know 195
that q it is equal to the volume 196
tribular of the volume chart density 197
and we can change this to operators 198
the linear operators of the integration 199
and the differentiation 200
and let's apply this uh 201
divergent here on this first term right 202
uh so this uh you can uh you can solve this digit 203
it is closed 204
in circumstances that it is equal to the open volume 205
table of the divergence of this human factor quantity 206
and from this so you can wear with this volume trigger 207
and this volume trigger 208
and then these two volume triggers 209
they are going to cancel out 210
and you're going to laptop with this uh 211
divergents of this electric uh credensity 212
uh is related with 213
the rate of change of this wall in identity 214
it decrease in rate of change of this wall of identity 215
and 216
this equation is known as the continuity of current 217
equation or the continent equation 218
so it states that 219
it states that uh it is derived from the principle of 220
conservation of charge and it states that 221
there can be no accumulation of charge 222
at any point inside that given volume 223
so in this particular example 224
it is going to decrease in this case 225
because the neck current 226
it is going outside of this closed surface 227
so for steady state plans 228
we know that there's no rate of change of this uh 229
volume identity has from Continental Christian 230
this ten dot g is equal to zero 231
showing that the total charge 232
leaving a volume 233
is seen as the total charge entering it 234
so there's no divergence in this case 235
and uh also it is all the equency of your 236
uh controls uh 237
current law that at one particular node 238
the current entering is equal to the current living 239
and in this key in this also 240
uh also the case of this uh 241
close surface if it's the close surface 242
the current entry in is equal to the current out 243
then there will be no flux outside of this 244
outside of this complete close surface 245
so consider introducing a charge 246
let's let's introduce a charge inside this 247
uh at close volume 248
okay and uh 249
from this uh we know that 250
uh I think the constituted relationship 251
b is equal to ABS or not e as del dot e 252
it is equal to will be directed ABS or not and not 253
let's substitute the continuity question 254
how to substitute 255
and first of all 256
substitute this on vibration into this causes bar 257
so we have this thing and from here 258
uh from here what we can assume things 259
okay from here right 260
put this Del dot J inside this continent equation 261
the arms are inside this continent equation and then uh 262
you can uh further simplify the things 263
so okay 264
uh okay 265
so what is this uh you have to 266
uh okay 267
so their dark G it was equal to minus rate of change 268
of describing identity 269
so J's from onside is equal to Sigma E 270
and uh let's plug in other things okay 271
so Sigma is coming out as a question 272
so your dial dot e it is equal to 273
will be divided by absent from this cause as well okay 274
but if you rearrange this equation 275
you will find out that it is homogeneous linear 276
first charter differential equation 277
and all 278
herit is known as the ordinary differential equation 279
because it is a first charter differential equation 280
when I solve this equation 281
since uh it can be uh it can be swallowed 282
using this technique of separating the variables 283
so take the variables of this uh 284
Ruby on one side and the uh 285
this uh radio belt of this uh 286
time term on the second side 287
integrate both sides 288
so uh integration of this thing right 289
integration of this thing with respect to Rovi 290
so integration of one of a Rovi with respect to Rovi 291
it is we know that Eleanor Rovi 292
because you know that integration of one of our X 293
it was equal to I don't know facts right 294
so remember these things from your future point 295
uh problem solving as well 296
integrational constant with respect to t 297
it is t and how to get this construct of integration c 298
right we know that at t is equal to 0 299
at t is equal to zero 300
your will be it is equal to will we not 301
so plug in this failure 302
you will find out that this constant 303
it is equal to Eleanor will we not 304
so let's simplify this term 305
uh okay these two are Ellen terms 306
so bring this Ellen term over here 307
so Ellen of uh will be 308
minus Eleanor will be not 309
we know that the subtraction of Ellen terms 310
it is equal to the 311
division of these two the 312
division of these two terms 313
and then you can you can take the natural lower 314
lower technique of these two terms 315
so that relationship 316
you know from your elementary maths 317
okay and how to fully simplify 318
so take the anti lower both sides 319
so once you will take the anti lower both sides 320
you will left over with this will be divided by 321
will be not 322
and on the right hand side 323
the whole thing which was optional 324
which was this exponential risk from minus t 325
divided by minus t in two row divided by this signal 326
divided by abslon 327
when this term signal divided by abslon 328
it can be defined as Dr 329
and this this is known as the time construct as well 330
and I'll try to able to see that 331
it is also relaxation time as well 332
so once you take the engine over both sides 333
you can simplify these terms 334
so this is this is a exponentially decaying expression 335
so so 336
this is your starting charge 337
inside your clothes region 338
and it is decaying exponentially 339
with respect to this factor 340
and since it is decaying so here this TR 341
it is at time constant thing 342
and it is defined in these seconds 343
and further it is equal to absline 344
divided by this Sigma of that vertical medium 345
so once we introduce that 1 346
inchar density at the interior 347
inside a material it will decay 348
resulting in the chart 349
movement from the interior point at 350
at which it was introduced 351
to the surface of the material 352
and the time constant TR or the decay 353
it is known as relaxation time 354
or the rearrangement time 355
but the charge is moving toward the surface of that 356
material that dialect material 357
and it is also defined as relaxation time 358
is the time it is for a charge please 359
in the interior of the material to draw to E 360
raise to power minus 1 and as we solve it 361
it is equal to 36 6.8% of this initial value 362
so the relaxation time is a time in which 363
so once you will put that t is equal to t 364
r in the expression 365
so basically t r 366
is a time after which 367
only 36.5% of the initial charge will be left 368
inside that material 369
if this will be not is placed 370
placed inside that material so uh 371
it is very short for uh 372
for good conductor and very long for good dialectry 373
let's see the example for a good conductor 374
the relaxation time is so short 375
that 376
most of the charge will vanish from the interior point 377
and immediately appear at the surface 378
within a very 379
short time because the conductivity of this uh 380
these conductors for example the cover 381
it is very high so once you you saw this 382
we know that developing 383
permitting of of the conductor it is equal to 1 384
so for this t r relaxation time 385
it is equal to Absline R into Absline R 386
there are the stigma 387
so it is coming out to be tenders from minus 19 388
which is very very small 389
small time 390
there is for a good dialect 391
relaxation time is very long 392
once the charge 393
it is introduced at the centre of this dialect 394
it remains wherever it is 395
pays for times maybe up to days 396
so the king is very small in this case 397
uh because the connectivity is very small 398
temperature minus 17 optional for this uh quads uh 399
dialect 400
it is five to solve this expression for this dialect 401
and you can see that it is 51 point two days 402
so after 51.2 days 403
the inserted charge inside the dialectic 404
it will decay to 36 point uh as uh that uh 8 person 405
summing up today's lecture 406
so today we had an overview of your question No. 1 407
we studied what is linear 408
isotropic and homogeneous dialectrics 409
we understood what is continental equation 410
and how it is utilized to determine the relaxation 411
timer are a dialectic 412
next time 413
we're going to study very important topic of your 414
uh electrostatic that is the electric Bondi condition 415
that what happens once an electric field 416
it moves it travels from one medium to another medium 417
with different dialectic properties 418
here I thank you all 419
if you have any questions 420
uh they will be entertained 421
uh through your emails or online 422
synchronous sessions 423
that we have arranged at the department