Lecture 3 - Time dependent EM fields relaxation, Propagation Notes | EduRev

: Lecture 3 - Time dependent EM fields relaxation, Propagation Notes | EduRev

 Page 1


Module I: Electromagnetic waves
Lecture 3: Time dependent EM ?elds: relaxation,
propagation
Amol Dighe
TIFR, Mumbai
Page 2


Module I: Electromagnetic waves
Lecture 3: Time dependent EM ?elds: relaxation,
propagation
Amol Dighe
TIFR, Mumbai
Outline
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Page 3


Module I: Electromagnetic waves
Lecture 3: Time dependent EM ?elds: relaxation,
propagation
Amol Dighe
TIFR, Mumbai
Outline
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Coming up...
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Page 4


Module I: Electromagnetic waves
Lecture 3: Time dependent EM ?elds: relaxation,
propagation
Amol Dighe
TIFR, Mumbai
Outline
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Coming up...
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Stationary and non-stationary states
I
Stationary state, by de?nition, means that the currents are
steady and there is no net charge movement, i.e.
r
~
J
s
= 0 or
@
@t
= 0 (1)
These statements are equivalent, due to continuity.
I
If the initial distribution of charges and currents does not satisfy
the above criteria, they will redistribute themselves so that a
stationary state is reached.
I
This process of “relaxation” happens over a time scale that is
characteristic of the medium, called the relaxation time.
Page 5


Module I: Electromagnetic waves
Lecture 3: Time dependent EM ?elds: relaxation,
propagation
Amol Dighe
TIFR, Mumbai
Outline
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Coming up...
Relaxation to a stationary state
Electromagnetic waves
Propagating plane wave
Decaying plane wave
Stationary and non-stationary states
I
Stationary state, by de?nition, means that the currents are
steady and there is no net charge movement, i.e.
r
~
J
s
= 0 or
@
@t
= 0 (1)
These statements are equivalent, due to continuity.
I
If the initial distribution of charges and currents does not satisfy
the above criteria, they will redistribute themselves so that a
stationary state is reached.
I
This process of “relaxation” happens over a time scale that is
characteristic of the medium, called the relaxation time.
Relaxation time
I
The continuity equation, combining withr
~
D =, gives
r
@
~
D
@t
=r
~
J (2)
I
Using
~
D =
~
E and
~
J =
~
E,
r(1+


@
@t
)
~
J = 0 (3)
I
The solution to this differential equation is
~
J =
~
J
s
+(
~
J
0

~
J
s
)e
t=
(4)
where J
0
is the initial current distribution
I
 == is the relaxation time
I
@
@t
=r
~
J,
~
E =
~
J=, etc. relax at the same rate.
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