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Name : Amit Tiwari
Roll No. 13110008
Dept : Electrical Engineering
1
4- Maxwell Equations

.E 
0
B
 E  
t
.B  0
E
  B   0 J   0 0
t
B   A
A
E  V 
t
V

 V  0 0 2  
t
0
2
2
 A
 A  0 0 2   0 J
t
2
2
In Lorentz Gauge ‘V’ and ‘A’ satisfy
inhomogeneous wave equation.
Wave travels with speed ‘c’.
z
R
tr  t 
c
Retarded
position
Particle trajectory
R
W(tr)
q
LienardWiechert
Potential
Present position
P
Observer
r
V (r , t ) 
x
1
40

 (r ' , t r )
R
y
1
qc
d ' 
40 Rc  R.v
General Expression of Electric and Magnetic field :
E (r , t ) 
q
R(c  v )u
40
( R.u )3
2
2
Velocity Field
/Generalized
Coulomb Field

q
R( R  (u  a))
40
( R.u )3
Acceleration/
Radiation Field

R E (r , t )
B( r , t ) 
c

u  c R v
How to derive this long
expression ?
A
E  V 
t
General Expression of Electric and Magnetic field :
E (r , t ) 
q
R(c  v )u
40
( R.u )3
2
2
Velocity Field
/Generalized
Coulomb Field

q
R( R  (u  a))
40
( R.u )3
Acceleration/
Radiation Field

R E (r , t )
B( r , t ) 
c

u  c R v
P
 (measured from instantaneous
posn of charge)
q
Rest Frame of the charge

v


q


E
3 
 2 2
2
2
   (1   sin  ) 2 
v 4
 
c 5
A2
O’
O
A1
D1
r1  ct0
r2  c(t0   ) K
D2
O
O’
A1
A2
r1  ct0
o
A1
r2  c(t0   )
O
O’
A2
r1  ct0
o
A1
O
O’
A1 A2
Which Inner Field Line connects with which outer
field line ?
Ans : (Gauss Law )
Electric field lines do not end in charge free space.
D1
K
D2
O
O’
A1
A2
D1
K
D2
O
O’
A1
A2
1
sin d
 E.ds  2q 0  12 (1  12 sin 2 )3/ 2
2
sin d
 2q  2
2
2
3/ 2
0  2 (1   2 sin  )
tan 1
1

tan 2
2
r1  ct0
r2  c(t0   )
1
2

D2 K 
 E"rad"
E perp   lim
 0 KD
1 


D2 K   ar sin 
 lim
 
 0 KD
c
1 

2

D2 K 
 E"rad"
E perp   lim
 0 KD
1 



q


E"rad" 
3 
 2 2
2
2
2


(
1


sin

)


2


D2 K
 ar sin 
 lim
 
 0 KD
c
1 

qra sin 
E perp  2 2
2
2
3/ 2
 c (1   sin )
q
a sin 

2
3
r c (1   cos  )
D1
K
D2
O
O’
A1
A2
qra sin 
E perp  2 2
2
2
3/ 2
 c (1   sin )
q
a sin 

2
3
r c (1   cos  )
r
Ɵ
OO’
A1A2
Radiation Pulse
t = 10 seconds
Radiation Pulse
t = 20 seconds
28
Primary References :
Primary References :
Snippets of Physics - Why does an
Accelerated Charge Radiate? |
Series Article | May 2009, 14 (05)
Author : T Padmanabhan
Other References :
Introduction to Electrodynamics by
David J Griffiths
Classical Electromagnnetic Radiation
by MA Heald and JB Marion
Electricity and Magnetism by
Edward M Purcell
Motivation:
Lectures by Shri Anand
Sengupta
Feynman Lectures on Physics by
Richard Feynman