Plane monochromatic wave in a conducting medium
With non-zero J (=  E) in a conducting medium Maxwell’s eqns take the form
  f
i)  . E 

 
ii)  . B  0

 
B
iii)   E  

 t
 
E
iv)   B   0 J   
t
where  f is the free charge density in the conducting medium
Consider continuity eq for the free charge:
 
 f
. J f  
t

 f
t
 
   . E
 From eq (i) after substitution, we get

 f

  f
t




  t






0
e
;
 f t    f 0 e
f

Thus continuity eq. for free charge as applicable in a conducting medium is:
 f

  f
t

Solution?

Thus for a good conductor, any free charge density

 f r , t  0 present at t = 0 will decay exponentially
and hence dissipates in a characteristic time .
For Cu, resistivity is 1.68 x 10-8 .m   = 5.95 x 107 mhos/m,
Assuming   30
   4.5  10 19 sec !!
t
A measure of how
good is a conductor
Thus Maxwell’s eqns in a conducting medium can be written as
 
i)  . E  0
 
ii)  . B  0

 
B
iii)   E  


t
 

E
iv)   B    E   
t
Wave eqn for E becomes
2
2



E
E

E
E
2
2
 E   2  
  E   2  
0
t
t
t
t


E  z , t   Es  z  e  i  t
 Wave fronts parallel to xy in this case are governed by



d Es  z 
2

    Es  i    Es  0
2
dz
2

d Es  z  2 
 k Es z   0;
2
dz
2
 Rewriting eq. (1)
where
Soln. of eq. (2):
(1)
k   

d Es  z  2 
  Es  0
2
dz
2
(2)
 2     2  i 

 i z
E s  z   E0 e
a const vector
which on substitution in eq. (2) yields
  2   2   i     0
(3)
  2   2   i     0
(3)
  must be complex
    i
Assume
which on substitution in eq. (3) yields


  2   2  2 i     2   i     0
Equating real and imaginary parts in eq. (4)
 2   2   2 
(5)
 
 
2
2     
Substitute (6) in (5):
     
4
2 
2
2
  
2
2
2
0
4
 2     4 2  2   2  2  2
2
(6)
(7 )
(4)
2

1

 2   2   1  1  2 2

2






1 1 
2



       1  2 2 
2 2    

Thus the plane wave traveling along z direction in a conducting medium with e
 

E r , t   Es  z  ei   t   z  ;     i
 The wave attenuates with propagation in a conducting medium
For a good conductor,
2
i t




:
 
 i  t  z   i   i   z
 E r , t   E0 e
e
 
 i  t  z    z
 E r , t   E0 e
e ;    
2
distance
1

1

skin depth

  
 1      


 2 
1
2
1
2
(8)
 Skin depth
  
    

 2 
1
2
 2

 
  
1



1
2
For copper
 = 0 ,  = 5.8 x 107 mhos/m ,   0 = 4 x 10-7 N/amp2
At  =2 x 1010 /s

 108 1  Good conductor

 2

 
  
1
 
0.065

1
 

2
  

7
7 

  2   4  10  5.8 10 
2
1
2
 For  = 100 /s,   0.0065 m = 0.65 cm
and For  = 108 /s,   6.5 x 10 -6 m