UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
LECTURE NOTES 7
EM WAVE PROPAGATION IN CONDUCTORS
Inside a conductor, free charges can move/migrate around in response to EM fields contained

therein, as we saw for the case of the longitudinal E -field inside a current-carrying wire that had a
static potential difference V across its ends. Even in the static case of electric charge residing on


the surface of a conductor, we saw that Einside  r   0 , but recall that this actually means (as we
 NET 
showed last semester) that the NET electric field inside the conductor is zero, i.e. Einside
r   0 .
n.b. here, we assume {for simplicity’s sake} that the conductor is linear/homogeneous/isotropic
– i.e. no crystalline structure/no anisotropies/no inhomogenities/voids/defects…


From Ohm’s Law, we know that the free current density J free  r , t  is proportional to the

 

(ambient) electric field inside the conductor: J free  r , t    C E  r , t  where:
 C = conductivity of the metal conductor  Siemens m = Ohm 1 m  and  C  1 C
C = resistivity of the metal conductor  Ohm-m  .
Thus inside such a conductor, we can assume that the linear/homogeneous/isotropic
conducting medium has electric permittivity  and magnetic permeability  . Maxwell’s


equations inside such a conductor {with J free  r , t   0 } are thus:
Using Ohm’s Law:

 

J free  r , t    C E  r , t 
 
 
  

 
E  r , t 
E  r , t 

  C E  r , t   
4)   B  r , t    J free  r , t   
t
t
  
  

1)  E  r , t    free  r , t   2)  B  r , t   0
 
  
B  r , t 
3)   E  r , t   
t
Electric charge is (always) conserved, thus the continuity equation inside the conductor is:

 

 
 free  r , t 


 J free  r , t   
but: J free  r , t    C E  r , t 
t

  
  
 free  r , t 
1

  C  E  r , t   
but:  E  r , t    free  r , t 

t





 C  free  r , t 
  r , t 
 free  r , t    C 

  free

thus:
or:
  free  r , t   0 

t
t
  
1st order linear,
homogeneous
differential equation
The general solution of this differential equation for the free charge density is of the form:



 free  r , t    free  r , t  0  e  C t    free  r , t  0  e t  relax i.e. a damped exponential !!!
Characteristic damping time:  relax    C = charge relaxation time {aka time constant}.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 


Thus, the continuity equation  J free  r , t     free  r , t  t inside a conductor tells us that any

free charge density  free  r , t  0  initially present at time t = 0 is exponentially damped /
dissipated in a characteristic time  relax    C = charge relaxation time {aka time constant},



such that at when: t   relax    C :  free  r , t   relax    free  r , t  0  e 1  0.369   free  r , t  0 
Calculation of the Charge Relaxation Time for Pure Copper:
Cu  1  Cu  1.68 108 -m   Cu  1 Cu  5.95 107 Siemens m
If we assume  Cu  3 o  3  8.85 108 F m for copper metal, then:
relax
 Cu
  Cu  Cu  Cu  Cu  4.5 1019 sec !!!
However, the characteristic (aka mean} collision time of free electrons in pure copper is
coll
Cu
coll
  Cu
vthermal
where Cu
 3.9 108 m = mean free path (between successive collisions) in
coll
Cu
Cu
Cu
 3k BT me  12 105 m sec and thus we obtain:  coll
pure copper, and vthermal
 3.2 1013 sec .
relax
Hence we see that the calculated charge relaxation time in pure copper,  Cu
 4.5 1019 sec
Cu
is  than the calculated collision time in pure copper,  coll
 3.2 1013 sec .
Furthermore, the experimentally measured charge relaxation time in pure copper is
relax
 Cu
 exp't   4.0 1014 sec , which is  5 orders of magnitude larger than the calculated charge
relax
relaxation time  Cu
 4.5 1019 sec . The problem here is that {the macroscopic} Ohm’s Law
is simply out of its range of validity on such short time scales! Two additional facts here are that
both  and  C are frequency-dependent quantities { i.e.      and  C   C   }, which
becomes increasingly important at the higher frequencies  f  2  ~ 1  relax  associated with
short time-scale, transient-type phenomena!
So in reality, if we are willing to wait a short time (e.g. t ~ 1ps  1012 sec ) then, any initial

free charge density  free  r , t  0  accumulated inside a good conductor at t = 0 will have

dissipated away/damped out, and from that time onwards,  free  r , t   0 can be safely assumed.
Note: For a poor conductor  C  0  , then:  relax    C   !!! Please keep this in mind…
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
After many charge relaxation time constants, e.g. 20 relax  t  1 ps  1012 sec , Maxwell’s

steady-state equations for a good conductor become {with  free  r , t  t   0 from then onwards}:
  
1)  E  r , t   0
  
2)  B  r , t   0
 
  
B  r , t 
3)   E  r , t   
t
 
 
  
 
  
E  r , t 
E  r , t  
   C E  r , t   
4)   B  r , t    C E  r , t   


t
t 

Maxwell’s equations for a
charge-equilibrated conductor
These equations are different from the previous derivation(s) of monochromatic plane EM
waves propagating in free space/vacuum and/or in linear/homogeneous/isotropic non-conducting
materials {n.b. only equation 4) has changed}, hence we re-derive {steady-state} wave equations

 
for E & B from scratch. As before, we apply     to equations 3) and 4):
  
  
  E  
 B
t



  

E 
2
=   E   E     C E  
=

t 
t 



2 E
E
2
=  E   2   C
=
t
t
 
 
 
2 E  r , t 
E  r , t 
2
  C
Again:  E  r , t   
and:
t 2
t
  
 
  
  B   C  E  
 E
t



  
B
2 B
2
  B   B    C
  2
t
t



2 B
B
 2 B   2   C
t
t
 
 
 
2 B  r , t 
B  r , t 
2
 B  r , t   
  C
t 2
t


Note that the {steady-state} 3-D wave equations for E and B in a conductor have an
additional term that has a single time derivative – which is analogous e.g. to a velocity-dependent
damping term associated with the motion of a 1-D mechanical harmonic oscillator.









 




The general solution(s) to the above {steady-state} wave equations are usually in the form of
an oscillatory function × a damping term (i.e. a decaying exponential) – in the direction of the


propagation of the EM wave, complex plane-wave type solutions for E and B associated with
the above wave equation(s) are of the general form:
 
 i kz t  1
 
 
 k 
 kˆ  E  r , t     kˆ  E  r , t 
and: B  r , t   Bo e
v
 
 
 i kz t 
E  r , t   Eo e
n.b. with {frequency-dependent} complex wave number: k    k    i  




where: k    e k   and     m k   and corresponding complex wave vector

k    k   kˆ  k   zˆ (for EM wave propagating in the kˆ   zˆ direction, here).




Physically, k    e k   is associated with wave propagation, and     m k  
is associated with wave attenuation (i.e. dissipation).
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
 i kz t 
 
 i kz t 
We plug E  r , t   Eo e
and B  r , t   Bo e
into their respective wave equations
above, and obtain from each wave equation the same/identical characteristic equation –
{aka a dispersion relation} between complex k   and  {please work this out yourselves!}:
k 2     2  i  C
Thus, since k    k    i   , then:
2
k 2     k    i     k 2     2    2ik        2  i  C
If we {temporarily} suppress the  -dependence of complex k   , this relation becomes:
2
k 2   k  i   k 2   2  2ik   2  i  C
2
2
2
We can re-write this expression as:  k        i  2k   C   0 , which must be true
for any/all values of {any of} the parameters involved. The only in-general way that this relation
can hold is if both  k 2   2    2   0 .and.  2k   C   0 . Then:
k 2   2   2 and: 2k   C
Thus, we have two separate/independent equations: k 2   2   2 and: 2k   C . We have
two unknowns: k and  . Hence, we solve these equations simultaneously to determine k and  !
From the latter relation, we see that:   12  C k . Plug this result into the other relation:
k 2   2  k 2   12  C k   k 2 
2
1 1
2
 C    2
2 2
k
Then multiply by k 2 and rearrange the terms to obtain the following relation:
k 4    2  k 2   12  C   0
2
This may look like a scary equation to try to solve (i.e. a quartic equation - eeekkk!), but it’s
actually just a quadratic equation! {So, it’s really just a leprechaun, masquerading as a unicorn!}
Define: x  k 2 , a  1 , b     2  and c    12  C  , this equation then becomes
2
“the usual” quadratic equation, of the form: ax 2  bx  c  0 , with solution(s)/root(s):
x
4
1
b  b 2  4ac
or: k 2      2  
2
2a
  
2 2
2
 4  12  C  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
This relation can be re-written as:



 2  C2  2
1
2 
k     1  1  4
2
4  2  2 4 2


2



2


 C2   1


 1


2 
2
C

  2    1  1   2 2   2    1  1     

 







On physical grounds ( k 2  0 ), we must select the + sign, hence:
12
2
2

1
 
  C  
  C  
2
2


and thus: k  k  
k     1  1  
1 1 
2
2 
  
  







2

12
 
2


 1   C   1
2 

  


Having thus solved for k (or equivalently, k 2 ), we can use either of our original two relations to
solve for  , e.g. k 2   2   2 , thus:
 2  k 2   2 
2
2



1
1
  C  
 C 
2
2
2








1
1
1
1



 
      

2
2
  





Hence {finally}, we obtain:

 


 1   C   1
2 

  

k    e k    
2

1
2


and:     m k    

 


 1   C   1
2 

  
2


The above two relations clearly show the frequency dependence of both the real and imaginary
components of the complex wavenumber k    k    i   . This physically means that EM
wave propagation in a conductor is dispersive (i.e. EM wave propagation is frequency dependent).


Note also that the imaginary part of k   ,     m k   results in an exponential
attenuation/damping of the monochromatic plane EM wave with increasing z:
 
 i kz t    z i kz t 
E  r , t   Eo e
 Eo e e
and:
k    k    i  
where:
 
 i kz t    z i kz t  k


k
B  r , t   Bo e
 Bo e e
 kˆ  E  z , t   kˆ  Eo e  z ei kz t 




The characteristic distance z over which E and B are attenuated/reduced to 1 e  e1  0.368
of their initial values (at z = 0) is known as the skin depth,  sc    1    (SI units: meters).
i.e.  sc   
1
  
1


 
2


 1   C   1
2 

  


1
2



i kz t 
E  z   sc , t   Eo e1e 


B  z   sc , t   Bo e 1ei kz t 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
1
2
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015

Lect. Notes 7
Prof. Steven Errede

The real part of k   , i.e. k    e k   determines the spatial wavelength    ,
the phase speed v   and also the group speed vg   of the monochromatic EM plane wave
in the conductor:
   
v   
2
k  

2
e k  





k   e k  
v() = propagation speed
of a point on waveform
that has constant phase .
 dk   
vg    


dk   d   d  
1
1
Phase   (kz – t) = constant.
A constant phase point on the
waveform moves: z(t) = /k + v t.
vg() = propagation
speed of energy /
information.
We will discuss phase speed v   and the group speed vg   more – later…

The above plane wave solutions satisfy the above EM wave equations(s) for any choice of Eo .
As we have also seen before, it can similarly be shown here that Maxwell’s equations 1) and 2)
 
 


 E  0 and  B  0 rule out the presence of any {longitudinal} z-components for E and B .


 For EM waves propagating in a conductor, E and B are {still} purely transverse!


If we consider e.g. a linearly polarized monochromatic plane EM wave propagating in the
 
i kz t 
xˆ , then:
kˆ   zˆ -direction in a conducting medium, e.g. E  r , t   E e  z e 
o
 
 
 k 
 k 
 k  i
i kz  t 
B  r , t     kˆ  E  r , t     E o e  z e 
yˆ  
 
 
 
 
 
 E  r , t   B  r , t   kˆ   zˆ


   z i kz t 
yˆ
 Eo e e

( kˆ   zˆ = propagation direction of EM wave, here)
The complex wavenumber: k    k    i    k   eik  
where: k    k   k*    k 2     2   and: k    tan 1    k   
In the complex k -plane:
  m k  
 |k|
k
k


k  e k  
k  k 2   2
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
i kz t 
E  r , t   E o e  z e 
xˆ has: E o  Eo ei E
k  |k|eik
 
k
k
|k|eik
 z i  kz  t 
yˆ  E o e z ei kz t  yˆ has: Bo  Bo ei B  E o 
Eo ei E
and that: B  r , t   B0 e e
Then we see that:

i
Thus, we see that: Bo e B 
|k|eik


Eo ei B 
|k|

Eo e 
i  E k 

k2  2


Eo e 
i  E k 


i.e., inside a conductor, E and B are no longer in phase with each other!!!


 B   E  k
Phases of E and B :
With phase difference:  B  E   B   E  k  magnetic field lags behind electric field!!!
1
2
2
Bo |k| 
C  
1

   1  
We also see that:
  
Eo  
c
   



The real/physical E and B fields associated with linearly polarized monochromatic plane
EM waves propagating in a conducting medium are exponentially damped:
 
E  r , t   e
 
B  r , t   e
E  r, t   E e


B  r, t   B e
Bo

Eo

cos  kz  t   E  xˆ
 z
cos  kz  t   B  yˆ  Bo e  z cos  kz  t   E  k   yˆ
o
1
|k   |
 z
o
2 2

C  


  1  
  




 B   E  k
2

C  

2
2


where: k    k          1  
  




1
2

    
 and: k     k    i    zˆ , k    k    i  
 k   
 B   E  k , k    tan 1 
Definition of the skin depth  sc   in a conductor:
 sc   
1
  
1


 
2

C 

 1 


1
 
2 




1
2
=
Distance z over which


the E and B fields fall
to 1 e  e 1  0.368
of their initial values.
The instantaneous power per unit volume in the conductor {ultimately dissipated as heat!} is:
 
 
 
 


p  r , t   J  r , t  E  r , t    C E  r , t  E  r , t    C E 2  r , t   Eo2 e2 z cos 2  kz  t   E  Watts m3 

The time-averaged power per unit volume in the conductor is: p  r , t  t  12 Eo2 e2 z  po e z
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Phase Speed vs. Group Speed of a Wave:
The phase speed = numerical value of v     k   at a point on the  vs. k   curve,
The group speed vg    d dk   = the local slope at a point on the  vs. k   curve:
"
"
← Technically, making this plot is wrong !!!
Why is this plot technically wrong ??? Because: k    fcn  
i.e.  is the independent variable {always plotted on the axis of abscissas (i.e. the x-axis)}
k   is the dependent variable {always plotted on the axis of ordinates (i.e. the y-axis)}
Thus, the technically correct way is to plot k   vs.  : {because k depends on  , not vice-versa!}
 dk     dk   
Then: vg    1 


 d   d 
1
i.e. vg    1 slope of k   vs.  graph
See below:
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Another way to think about this issue is to remember that the angular frequency  and
wavenumber k   are Fourier transforms of time t and position z  t  , respectively. In the spacetime domain, clearly the space position z  t  is the dependent variable, time t is the independent
variable. The Fourier transform of the dependent variable z  t  is the dependent variable k   ,
the Fourier transform of the independent variable t is the independent variable  .
Thus here, for the physics associated with propagation of EM plane waves in a conductor,
with frequency-dependent real-component wavenumber k   :


k    e k  
The phase speed: v   


k  
2

C 
 

 1 



1
2 
 




1
2
1
2


 
 1   C   1
2 

  


1
2
1


2 
2


 dk   
1
 d   

 C 

1 
The group speed: vg   

 
  1 
2 
dk   d   d 
d 
 
  


 


1
1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
So let’s work out what the group speed vg   is for an EM plane wave propagating in a conductor.
Using the chain rule of differentiation:
2

dk  

 
 1   C   1

d
2 

  


1
2
2


 d 
 1   C   1

2 d 

  


1
2
2



 
2


 1   C   1
2 

  


 
2

C 

 1 

1

 
2 




 

2 

1
2


1
2

1 1
   1 
  2  C    3 
2 2
     
2


C 

 1 

1

 





1
2
 C 
  


2
 
1  C 
  
2
2
1 

1
2
2 2 
2
2

C 

 C 
 1 

1
1



 
 








2


C 

1
2
2





 1
 
  
1   C   1  1 

2
2

  2
  




 1   C   1 1   C  



  
   





The group speed of an EM plane wave propagating in a conductor is: (eeek!!!)
1
 dk   
vg     
 
 d 
1
 
2


 1   C   1
2 

  


1

2
1


2


 C 
  
 1




1


2
2

 2
C 
C  


 1 
  1 1  
 






   




The relation between phase speed vs. group speed of an EM plane wave propagating in a conductor is:
v   
10


k  
1
 
2


 1   C   1
2 

  


1
2
vg    v   
1


 1
1 
 2


2




 C 


  
2
2

 
 
1   C   1 1   C 

  
  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.







UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
EM Wave Complex Impedance in a Conductor:
The complex vector impedance associated with an EM wave propagating in a conductor is:
 
 
 
 
 
 
 1 
E  r , t;    H *  r , t;  
E  r , t ;    B*  r , t ;  
Z  r , t;    E  r , t;    H  r , t;   

2
2
 
 
H  r , t;  
B  r , t; 
If the electric and magnetic fields associated with the EM wave propagating in the conductor are:
 
 
 
 k 
 k 
i kz t 
i kz t 
yˆ
E  r , t   E o e  z e 
xˆ and: B  r , t     kˆ  E  r , t     E o e  z e 
 
 
Then:
 k *    *  z  i kz t 
i kz t 
xˆ  
E e e
E o e  z e 
yˆ
   o
 


Z  r , t;    
2
 k   
i
kz


t




z
yˆ

 E o e e



 k *   
  
 |k   |2 


E o
2
e 2 z
E o
2
e 2 z
 k*   
zˆ   
zˆ Ohms 
 |k   |2  


 
 k*   
zˆ
Note that {again}: Z  r , t ;      
2 
 has no explicit time dependence
 |k    | 
 
 k*   
 k    i   
Z  r ,     
zˆ    2

 zˆ  Ohms  .
2
2
 |k   | 



k









Complex
impedance is
manifestly a
complex
frequency-domain
quantity
 i 
 i 
Note that since: k *    k    i    k*   e k    k   e k  
and:
    
   B     E  
 k   
k    tan 1 
We can also equivalently write this expression as:
 k   e  ik   
 
 k*   
 zˆ
Z  r ,     
zˆ   
   |2 
 |k   |2 

k
|




   i B   E   
    i E   B  
 
e
zˆ   
e
zˆ

 |k   | 
 |k   | 




© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
EM Wave Propagation in a Conductor – Special/Limiting Cases:
a) Good conductors:  C   Conductivity of a good conductor:  C   i.e.C  1  C  0  .


and:  C   , i.e.  C  1 for a good conductor. Then:
 

Since: k  k  ik
1
2

 
 C 


1
k 
1 
 
2 




2
 
  C  


1 
  
2 



2
1
2
   C 



2   
1
2

  C
 C

2
2

  C
 C

2
2
and:
2

 
 C 


1
 
1 
 
2 




1
2
 
  C  


1 
  
2 



2
1
2
   C 



2   
1
2


 In a good conductor  C  1 :
 

k       
 C
and skin depth:  sc   
2
1
  

2
 C
.


FORMULAS FOR EM WAVE PROPAGATION IN A GOOD CONDUCTOR  C  1
 

 C
k       
and:
2
Wavenumber, k   
2
    
  
n.b. in a perfect conductor:  C  
 k       

   
 sc   
1
2
k  
  

 C
2

 C
2
k  

2
  
1
  
0

2
 C
 2 sc    2
2
 C
    
1
  tan 1
 k   
k     B   E   tan 1 
1

But: tan 1  45 

4

    B   E  45 
0
2
 sc    skin depth 

4


 B lags E by  45 in a good conductor.

n.b. In a perfect conductor:  C  ,   45  4
In a typical good conductor (e.g. gold/silver/copper/…):
12
 C    1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
For optical frequencies/visible light region:   1016 radians sec . A good conductor typically has
 C  107 Siemens m and   3 o , and at optical frequencies:  C    37.7  1 is satisfied.
If the conductor is  non-magnetic (e.g. copper, aluminum, gold, silver, platinum… etc.)
   o  4 107 Henrys /m .
 C
o C
1
1016  4 107 107  2





k


 2.51108 radians /m
Then:  
 


2
2
2


And:     2 k   = wavelength in good conductor  2.51108 m  25.1nm
cf w/ vacuum wavelength: o 
2 2 c c 2  3 108

 
 1.885 107 m  188.5 nm
16

ko
f
10
vacuum
     25.1nm  good
conductor   o  188.5 nm  wavelength 
   188.5 nm
 7.52 at optical frequencies,   1016 rad/sec.
Vacuum/conductor -ratio:  o  
nm


25.1




  
1

 4.0 109 m  4.0 nm !!!
Skin depth:  sc   
  
2
 This explains why metals are opaque at optical frequencies,   1016 radians/sec
{and also e.g. explains why/how silvered sunglasses work!}
Compare these results for EM waves propagating in conductors at optical frequencies to those
for EM waves propagating in conductors, but instead at very low frequencies – e.g. the AC line
frequency, f AC  60 Hz   AC  2 f AC  120 rad/sec , where the criterion for a good
conductor,  C    1015  1 is certainly well-satisfied:

 C 120  4 107 107 

kAC   AC 
  48.7 radians /m
2
2




2
 0.129 m  12.9 cm
AC 
k


At f = 60 Hz: oAC  5  106 m !!

6
 oAC  5 10 m  3.87  107 !!
 AC 0.129 m

60 Hz AC skin depth:  AC  AC  2.05  102 m  2.05 cm !!
sc

2
 Need at least 3-4   sc  several  10 cm to screen out unwanted 60 Hz AC signals !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Phase speed vs. group speed in a Good conductor:
 
in a good conductor, where:  C   1
  
 C
Given that: k       
2
The phase speed and group speed in a good conductor are respectively:
1
 dk   
and: vg    
 
 d 


v   

k  
 C
2
1
1

 C 1  1 



2

2
2
 C
 2v   !!!
2
Complex impedance in a Good conductor:
 |k   | e  ik   
 
 k*   
   ik  
ˆ
Z  r ,     
z
zˆ   
e
zˆ  Ohms 





2
2
 |k   | 
 |k   | 
 |k   |







Since: k       
And:
 C
2
Then: |k   | = k 2     2    2k     C


 B lags E by  45
in a good conductor.
    
1
  tan 1  45   B     E  
 k   
k    tan 1 
Hence in a good conductor:
 
  i B   E  
 good 
  i E   B  
   i E   B  
Z cond
zˆ 
e
zˆ 
e
zˆ
e
 r ,     

C
 C
  C 
   i E   B    
 C   i E   B  
e
zˆ  
zˆ  Ohms 

e


 C


lin

Define {real scalar}: Z med
since


good

Then {real scalar} Z cond



C

Z
lin
med
 
C

1
2
lin
 Z med
   1 in a good conductor!
C

 
 
 
i     B   
i 
zˆ
Since {in general} the complex impedance is: Z  r ,    Z  r ,   e Z   zˆ  Z  r ,   e E
We see the phase of the complex wave impedance for a good conductor is:
    
1
  tan  1  45   E     B      B     E     k  
 k   
 Z    tan 1 
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
 C 

 1
  
k   B   E  
Instantaneous EM Wave Energy Densities in a Good Conductor:
uEM  u
EM
E
u
1
  1 2 1    1  
B     EE   
B B 
   E2   
2
  2   2
  2

EM
M
Prof. Steven Errede

4
 45
{in a good conductor}
 
 
E  r , t   Eo e  z cos  kz  t   E  xˆ and B  r , t   Bo e  z cos  kz  t   E    yˆ
Where: Bo 
|k   |

2

  C  

Eo   1  
  




k       
And:
v   
 C
2
1
2
Eo 
  C
 C
Eo 
E for a good conductor,
 o

,


2


for a good conductor.
 C
k  
 C
2
Then:
1
1   1

uEEM  r , t    E 2   E  E   Eo2 e 2 z cos 2  kz  t   E  and:
2
2
2



1 2
1
1 2 2 z

uMEM  r , t  
B 
B B 
Bo e
cos 2  kz  t   E  k   C Eo2 e2 z cos 2  kz  t   E  k 
2
2
2
2
Time-averaging these quantities over one complete cycle:
1  

u  r , t    u  r , t  dt

0
1
1
1 

u EEM  r , t    Eo2 e 2 z  cos 2  kz  t   E  d   Eo2 e 2 z
0
 4
2
 12

1 
1  

  1
uMEM  r , t   C Eo2 e 2 z  cos 2  kz  t   E  k  d   C  Eo2 e2 z   C    Eo2 e2 z
0
 4   
2
   4
 12


1


EM 
uTot
 r , t   uEEM  r , t   uMEM  r , t    1  C  Eo2e2 z n.b. Exponentially attenuated in z !!!
4   
1     1 2 2 z 
 
EM 
But:  C   1 for a good conductor,  uTot  r , t    C    Eo e 
2     2

  
i.e. the ratio:

uMEM  r , t    C 


 1 or: uMEM  r , t   uEEM  r , t  for a good conductor.


EM 
uE  r , t    
 Vast majority of EM wave energy is carried by the magnetic field in a good conductor !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15
UIUC Physics 436 EM Fields & Sources II
 1  
Poynting’s Vector: S  E  B 

Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
1  
1
S r ,t  
EB 
Eo Bo e 2 z cos k  zˆ

2
k 

4
 

1
1 2 2 z  |k|

Eo Bo e 2 z cos k 
Eo e
EM wave intensity (aka irradiance): I  r   S  r , t  
 cos k 
2
2


But:
|k|cos k


k

 C

2

 
 C
1  k  2 2 z 1  C 2 2 z


Eo e
 I r   S r ,t  
  Eo e
2   
2 2
2

 
b.) Special/Limiting Case of a Fair Conductor:  C   1  Must use exact formulae!
  
c.) Special/Limiting Case of a Poor Conductor: (i.e.  an insulator):
 
Here:  C   1 . Conductivity of poor conductor:  C  0 i.e.C  1  C    .
  
Complex wavenumber: k  k  i , with: k  k    e k   and:       m k   .




Noting that to 1st order in the Taylor series expansion: 1   1  12  for  1 , thus:
2

C 
 

 1 

k    
1

2 
 




2

 
 C 


    
1 
1

2 
 




 k     
1
1
2

2

 
2

1  C 
1

1  


2  2   

1
 
2
2

1  C 
1

1  


2 
2   


1
2
 
1  C 
2  

2 
2   
2



1
2
  
  C2 1


 C
2 2
2

4 
1

and:      C
for a poor conductor.
2


1
     2  C 
1  
 C 
  C   1 i.e.     k   .
In a poor conductor 
  1 , the ratio:  k   
2   
  
     
 The complex wavenumber k  k  i is primarily real, because   k in a poor conductor.
    
1  C 
1  1  C  
Phase angle in a poor conductor: k   B   E  tan 1 
  tan  
  
 ~ 0 1
 2     2   
 k   


  B   E  k   E , i.e. B and E are nearly in phase with each other in a poor conductor
(i.e. dissipation/losses very small in a poor conductor).
16
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
In a typical poor conductor, e.g. pure water:
Water has a huge static electric permittivity (due to the permanent electric dipole moment of
water molecule):  H 2O  81 o  @ f  0 Hz   at P  1 ATM and T  20 C  , however, at optical
frequencies   1016 rad sec  :  H 2O    1.777 o , where  o  8.85 1012 Farads m .
Since water is  non-magnetic:  H 2O  o  4 107 Henrys m
 index of refraction: nH 2O     H 2O    H 2O  o o  1.333 at optical frequencies.
The conductivity of pure water is:  CH 2O  1 CH 2O  1 2.5 105 -m  4.0 106 Siemens m
 at P  1 ATM
and T  20 C  . Thus, the criteria for a poor conductor  C    2.54 1011  1
is certainly satisfied at optical frequencies.
The wavenumber in pure H 2O at optical frequencies is:
k H 2O        o  1016 1.777  8.85  4 107  4.45 107 radians m
The wavelength in pure H 2O is: H 2O  2 k H 2O  1.413  107 m  141.3 nm at optical frequencies.
cf w/ the vacuum wavelength: o  c f  2 c   1.885 107 m  188.5 nm
 
Note that the optical wavelength ratio:  o
 H O
 2
Skin depth:  sc   
1
  

1
1
2
C
 188.5 nm
 1.333  nH 2O for a poor conductor.
 
nm
141.3

 
for a poor conductor  C   1 .
 
  
For pure H2O at optical frequencies:
o 1 
1
1
4  107
 1

 H 2O      C
 C
 
 5.65 104 rad m
5 
12
2
 2
 2  2.5 10  1.777  8.85 10
 scH O   
2
1
H O
2
 1.7688 103 m  1.77 km
n.b. neglects/ignores Rayleigh scattering process – visible light
photons elastically scattering off of H2O molecules. atten  10m
vis
1

  H 2O     2  C 
1   1 
1
1


  C  
 1.27 1011  1
Ratio: 


5
12
16
 kH O   
2    2  2.5 10  1.777  8.85 10 10
 
 2

 H O 
Phase difference: k   B   E  tan 1  2   1.27  1011 radians   1 i.e.  B   E  k   E
 kH O 
 2 


 B and E are nearly in phase with each other in pure H2O at optical frequencies.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
17
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
For pure H2O at low frequencies – e.g. 60 Hz AC line frequency  AC  2 f AC  120 rad sec  :
The electric permittivity at f = 60 Hz is  HAC2O  f  60 Hz   80 o  80  8.85  1012 Farads m
and  HAC2O  o  4  107 Henrys m . Conductivity of pure H 2O :  CH 2O  4.0 106 Siemens m
 
Note that the criteria for a poor conductor:  AC C
  H O AC
 2

4.0  106

 15  1

12
80
8.85
10
120





is not satisfied at the 60 Hz AC line frequency – i.e. at low enough frequencies,
even poor conductors such as pure water are actually quite good conductors !!!
Thus, for the following, we must use the good conductor approximations:
H 2O
H 2O
k AC
    AC
  
H O
AC
  
2
2
k
H 2O
AC
 
H O
 AC  AC
C
2
2

 AC o C
2
120  4  107  4 106
 3.08 105 rads m
2

 2.04  105 m cf w/ vacuum wavelength: o  c f AC 
2 c
 AC
 5.00 106 m
 o  5.00  106 m
 24.495
Vacuum/good conductor wavelength ratio:  H 2O  
5
 AC  2.04  10 m
Skin depth for pure H2O at 60 Hz AC line frequency:  HAC2O  1  HAC2O  3.25 104 m  32.5km
This may seem like a large distance scale associated with the attenuation of the 60 Hz EM
waves propagating in pure water, however compare the skin depth to the wavelength at this
H 2O
 1.77 106 m , i.e. we see that  HAC2O  HAC2O , as we expect
frequency:  HAC2O  32.5km vs. AC
for the case of a good conductor !!!


The ratio  HAC2O k HAC2O  1 for pure H2O at 60 Hz AC line frequency, which is what we expect for
a good conductor {this ratio should be  1 for a poor conductor!}.



Thus, the phase difference is: k   B   E  tan 1  HAC2O k HAC2O  tan 1 1   45o
4


o
which again is what we expect for a good conductor, i.e. B lags E by 45 !
18
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Phase speed vs. group speed in a Poor conductor:
1

 
Given that: k      and:      C
in a poor conductor, where:  C   1
2

  
 The complex wavenumber k  k  i is primarily real, because   k in a poor conductor.
The phase speed in a poor conductor is:
v   
1


1


compare to vacuum/free space: c 
k    

 o o
The group speed in a poor conductor is:
1
 dk   
1
 v   !!!
vg     
 
d




Complex impedance in a Poor conductor:
 k   e  ik   
 
 k*   


 zˆ      e  ik   zˆ  Ohms 
Z  r ,     
zˆ   

2
2
 |k   | 
 |k   | 
 |k   |







But:
k      and:    
And:
k     B   E   tan 1 
1
 
 C in a poor conductor, where:  C   1
2
  
    
1  C 
1  1  C  
  tan  
  
 ~ 0 1
 2     2   
 k   
 poor 
Hence in a poor conductor: Z cond
r ,  
E and B are  inphase with each other
for a poor conductor.


poor
lin
zˆ  Z cond
zˆ . Define {real scalar}: Z med

.


Then {real scalar} characteristic longitudinal EM wave impedance for a poor conductor is:

 Ohms 

poor
lin
Z cond
 Z med

n.b. Compare to free space: Z o 
o
 376.8  Ohms 
o
 
 
 
i     B   
i  
zˆ .
Since {in general} the complex impedance is: Z  r ,    Z  r ,   e Z zˆ  Z  r ,   e E
The phase of the complex wave impedance for a poor conductor is:
    
1  C 
1 
  tan   
   0   E     B  
k
2









 Z    tan 1 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
19
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
Instantaneous EM energy densities in a poor conductor:  C   1
  



1
  1 2  1    1  
uEM  r , t   uEEM  r , t   uMEM  r , t     E 2   
B     EE   
B B 
2
  2   2
  2



The physical/instantaneous purely real time-domain E and B fields are:
 
 
E  r , t   Eo e  z cos  kz  t   E  xˆ and: B  r , t   Bo e  z cos  kz  t   E  k  yˆ
2

C  


where: Bo  Eo   1  

  




|k|
k    

v
where: v 
1

 k    ,
and:    c
2

1

then: uEEM  r , t    E 2 
2
1 2

uMEM  r , t  
B 
2
1
2
 
Eo   Eo for a poor conductor,  C   1 .
  

1

k  

for a poor conductor.
|k|   
for a poor conductor.
1   1 2 2 z
cos 2  kz  t   E  and:
cE  E   Eo e
2
2


1
1 2 2 z
cos 2  kz  t   E  k 
B B 
Bo e
2
2
Time-averaging these quantities:


1
1 2 2 z
1
1



uEEM  r , t    Eo2 e 2 z and: uMEM  r , t  
Bo e
  Eo2 e 2 z   Eo2 e2 z
4
4
4
4
1
1
1


EM 
uTot
 r , t   uEEM  r , t   uME  r , t    Eo2e2 z   Eo2e2 z   Eo2e2 z
4
4
2

1
EM 
Thus: uTot
 r , t    Eo2e2 z for a poor conductor,  C   1 .
2
  

The ratio of {time-averaged} electric/magnetic energy densities for a poor conductor:
1 2 2 z

 Eo e
  H 2O 
uEEM  r , t 
o
1
1
4
1




tan
 k H 2O   o





  1  H 2O   c

k
B
E
EM
1 2 2 z

2
uM  r , t 
 k H 2O 
 Eo e
4


 EM wave energy is shared  equally by the E and B fields in a poor conductor!
Instantaneous Poynting’s Vector for EM waves propagating in a poor conductor:
 
 
1  
S r,t   E r ,t  B r ,t  

20
 
 
1  
1  2 2 z
cos k zˆ
S r ,t  
E r ,t  B r ,t  
Eo e


2 o
1
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
1  2 2 z

Eo e
Intensity of EM waves propagating in a poor conductor: I  r   S  r , t  
2 o
Reflection of EM Waves at Normal Incidence from a Conducting Surface:

In the presence of free surface charges  free and/or free surface currents, K free the boundary
conditions obtained from (the integral forms of) Maxwell’s equations for reflection and
refraction at e.g. a dielectric-conductor interface become:

1 E1   2 E2   free
BC 1): (normal D at interface):
 = normal to plane of interface
 = parallel to plane of interface





BC 2): (tangential E at interface): E1  E2  0  E1  E2

BC 3): (normal B at interface):
B1  B2  0  B1  B2

1 || 1 || 
B1 
B2  K free  nˆ 
BC 4): (tangential H at interface):
1
2
21
where n̂  is a unit vector  to the interface, pointing from medium (2) into medium (1).
21
{n.b. do not confuse n̂  with the EM wave polarization vector n̂ !!!}
21


Note: For Ohmic conductors (i.e. “normal” conductors obeying Ohm’s law J free   C E )


there can be no free surface currents - i.e. K free  0 , because K free  0 would require an infinite


E -field at the boundary/interface! ( J free  0 inside the conductor is fine/OK…)
Suppose  a boundary/interface (located in the x-y plane at z = 0) between a non-conducting
linear/homogeneous/isotropic medium (1) and a conductor (2). A monochromatic plane EM
wave is incident on the interface, linearly polarized in  x̂ direction, traveling in the  ẑ direction,
approaches the interface/boundary from the left {in medium (1)} as shown in the figure below:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
21
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 
 1

B  kˆ  E
v
 
1
i k z  t
and: Binc  r , t   E oinc e  1  yˆ
v1


1


i  k z t
i  k z t
Reflected EM wave {medium (1)}: Erefl  r , t   E orefl e  1  xˆ and: Brefl  r , t    E orefl e  1  yˆ
v1


k


i  k z  t 
i  k z  t 
xˆ and: Btrans  r , t   2 E otrans e 2
yˆ
Transmitted EM wave {medium (2)}: Etrans  r , t   E otrans e 2
 
Einc  r , t   E oinc ei k1z t  xˆ
Incident EM wave {medium (1)}:

n.b. complex wavenumber in {conducting} medium (2): k2  k2  i 2

 



In medium (1) EM fields are: ETot1  r , t   Einc  r , t   Erefl  r , t 

 



and: BTot1  r , t   Binc  r , t   Brefl  r , t 




In medium (2) EM fields are: ETot2  r , t   Etrans  r , t 




and: BTot2  r , t   Btrans  r , t 
Apply BC’s at the z = 0 interface in the x-y plane:




BC 1): 1 E1   2 E2   free but: E1  E1z  0 and: E2  E 2 z  0  0  0   free   free  0
BC 2): E1  E2  E oinc  E orefl  E otrans


BC 3): B1  B2 but: B1  B1z  0 and: B2  B2 z  0  0  0
BC 4):
1
1
B1|| 
1
2

B2||  K free  nˆ 
21
or: E oinc  E orefl   E otrans

but: K free  0 
k
1 
Eoinc  E orefl  2 E otrans  0
1v1
2


  v k    v 
with:    1 1 2    1 1  k2
 2   2 
 1    
2
E oinc
Thus we obtain: E orefl  
 Eoinc and: E otrans 



1



1



or:
 E orefl

 E o
 inc
  1   

  1   

 E otrans
and:  
 Eoinc


2

 1  



  v k    v 
with:    1 1 2    1 1  k2
 2   2 
Note that these “Fresnel” relations for reflection/transmission of EM waves at normal incidence
on a non-conductor/conductor boundary/interface are identical to those obtained for reflection /
transmission of EM waves at normal incidence on a boundary/interface between two linear nonconductors, except for the replacement of  with a now complex  for the present situation.
  v k    v 
Note also that here,  is frequency-dependent, i.e.        1 1 2    1 1  k2   .
 2   2 
22
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
For the case of a perfect conductor, the conductivity  C   {thus resistivity, C  1  C  0 }
2 C
 both k2   2 
2
then: k2    i   1  i 
  and since: k2  k2  i 2
  v k    v 
and since:    1 1 2    1 1  k2    
 2   2 
Thus, for a perfect conductor, we see that: E orefl   E oinc and: E trans  0
Thus, for a perfect conductor, the reflection and transmission coefficients are:
 Eo
R   refl
 Eo
 inc
2
E orefl

  

Eoinc

2
 E orefl

 E o
 inc
  E orefl

  E o
  inc
*

  1 and: T  1  R  0


We also see that for a perfect conductor, for normal incidence, the reflected wave undergoes
a 180o phase shift with respect to the incident wave at the interface/boundary at z = 0 in the x-y
plane. A perfect conductor screens out all EM waves from propagating in its interior.
For the case of a good conductor, the conductivity  C is finite-large, but not infinite.
The reflection coefficient R for monochromatic plane EM waves at normal incidence on a good
conductor is not unity, but very close to it. {This is why good conductors make good mirrors!}
 Eorefl
For a good conductor: R  

 Eoinc
2
E orefl

  

Eoinc

2
 E o
  refl
 E
 oinc
  E orefl

  E
  oinc
*
2
*

 1     1   
1  
 

 

 1  
 1  1  

  v k    v 
Where:    1 1 2    1 1  k2 and: k2  k2  i 2 . For a good conductor: k2   2 
 2   2 
Thus:
2 C
2
 1v1    1v1  2 C
C
1  i   1v1
1  i 
 k2  

2
2 2
 2 
 2 
  
Define:   1v1
C
Then:    1  i 
2 2
Thus, the reflection coefficient R for monochromatic plane EM waves at normal incidence on
a good conductor is {n.b. frequency-dependent!}:
E o
R  refl
E
oinc
with:
  1v1
2
2
*
 1    1     1    i   1    i
1  




 
 
1  
 1    1     1    i   1    i
2
2
  1      




2
2
  1      
C
   
2 2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
23
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Obviously, only a {very} small fraction of the normally-incident monochromatic plane EM
wave is transmitted into the good conductor, since R  1 and T  1  R , i.e.:

 1   2   2 
T  1 R  1 

2
2
 1      
  1
with:   1v1
C
   
2 2


Note that the transmitted wave is exponentially attenuated in the z-direction; the E and B
fields in the good conductor fall to 1/e of their initial {z = 0} values (at/on the interface) after the
monochromatic plane EM wave propagates a distance of one skin depth in z into the conductor:
 sc   
1
 2  

2
2 C
Note also that the energy associated with the transmitted monochromatic plane EM wave is
ultimately dissipated in the conducting medium as heat.
In {bulk} metals, since the transmitted wave is {rapidly} absorbed/attenuated in the metal,
experimentally we are only able to study/measure the reflection coefficient R. A full/detailed
mathematical description of the physics of reflection from the surface of a metal conductor as a
function of angle of incidence i.e. R  , inc  and also requires the use of a complex dispersion
relation k     v     c  n   with complex k    k    i   and a complex
propagation speed v    v    iv    c n   with accompanying complex index of refraction
n    n    i   , and is hence commensurately more mathematically complicated….
So-called ellipsometry measurements of the EM radiation reflected from the surface of the metal
as a function of angle of incidence yields information on the real and imaginary parts of the
complex index of refraction of the metal n    n    i   , and thus the real and imaginary
parts of the complex dielectric constant and/or the complex electric susceptibility of the metal,
since n        o  1   e   or n 2        o  1   e   .
If interested in learning more about this, e.g. please see/read Physics 436 Lect. Notes 8, and e.g.
please see/read Optics, M.V. Klein, p. 588-592, Wiley, 1970 {P436 reference book on reserve in the
Physics library}. Please also tsee/read he UIUC P402 Optics/Light Lab Ellipsometry Lab Handout C4
and especially the references at the end. Available at: http://online.physics.uiuc.edu/courses/phys402/exp/C4/C4.pdf
We will discuss the dispersive nature of dielectric, non-conducting materials in the next lecture…
But first, we need to remind the reader of the full Maxwell’s equations in matter…
24
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
Full Maxwell Equations in Matter:

The electromagnetic state of matter at a given observation point r at a given time t is described
by four macroscopic quantities:

 free  r , t   aka {free} charge density
1.) The volume density of free charge:
 
 aka electric polarization
2.) The volume density of electric dipole moments: P  r , t 
 
3.) The volume density of magnetic dipole moments: M  r , t   aka magnetization


J free  r , t   aka {free} current density
4.) The free electric current/unit area:
All four of these quantities are macroscopically averaged - i.e. the microscopic fluctuations
due to atomic/molecular makeup of matter have been smoothed out.


The four above quantities are related to the macroscopic E and B fields by the four Maxwell
equations for matter (see Physics 435 Lect. Notes 24, p. 14):
1) Gauss’ Law:
Auxiliary relation:
  
 
1
 E  Tot    free  bound  , where: bound  
o
o

 


D   o E   & constitutive relation: D   E





Electric polarization       o  E   0  e E , electric susceptibility:  e    1
 o 
 
   
 D   o E     free
 
2) No magnetic charges/monopoles:  B  0


 1  
 
 
B     H  M & constitutive relation: B   H
Auxiliary relation: H 
o



 
B
H
M
 E  
  0
 0
3) Faraday’s Law:
t
t
t
  

 
 

M=   1 H   m H , magnetic susceptibility:  m    1
Magnetization:
 o 
 o 

 



E
  B  o J ToT  o J D with: J D   o
4) Ampere’s Law:
t

 mag
 
P


 mag
P
P
J bound 
Total current density: J ToT  J free  J bound  J bound J bound    M
t


 
 
P
E
  B  o J free  o  M  o
  o o
t
t

 

D
  H  o J free  o
t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
25
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede
 



We also have Ohm’s Law: J   C E and the 3 continuity equation(s):  J   
t
associated with subscript index  for   free, bound and total electric charge conservation.
For many/most (but not all!!!) physics problems, e.g. in optics/condensed matter physics,
materials of interest are frequently non-magnetic (or negligibly magnetic) and have no free

 
charge densities present, i.e.  free  0 . If   o , then: M  0 and thus: H  B o in such
non-magnetic materials.

Then Maxwell’s equations in matter, for  free  0 and M  0 reduce to:
 
 
1   
 D  0 or:  E   P  bound
1) Gauss’ Law:
o
o
 
2) No magnetic charges:  B  0

 
B
 E  
t
3) Faraday’s Law:


 

E
P
  B  o o
 o
 o J free
4) Ampere’s Law:
t
t


 
We also have Ohm’s Law J free   c E and the Continuity eqn.  J free  0 {here}.
Then applying the curl operator to Faraday’s Law:



  
 1 

J free   
  
2 E
2P
  E  
  B   o o 2  o 2  o
   E   2 E  bound   2 E
t
t
t
t
o






We obtain the inhomogeneous wave equation:



 1 2 E 1 

J free
2P
 E  2 2  bound  o 2  o
{and a similar/analogous one for B }
c t
t
t
o


2
source terms
For nonconducting/poorly-conducting media, i.e. insulators/dielectrics, the first two terms on
the RHS of the above equation are important – e.g. they explain many optical effects such as
dispersion (wavelength/frequency-dependence of the index of refraction), absorption, double –
refraction/bi-refringence, optical activity, . . . .

  

Note that the bound   P term is often zero, e.g. if the electric polarization P is uniform:


 
  P P P



where:   xˆ 
yˆ  zˆ and: P  x  y  z  0 if P is uniform
x y
z
x
y
z
 
 
 
or, if the polarization P  E , where e.g. E  r , t   Eo cos  kz  t    xˆ , then: P=0 also.
26
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 7
Prof. Steven Errede

J free

E
 o C
For good conductors (e.g. metals), the conduction term o
is the most
t
t
important, because it explains the opacity of metals (e.g. in the visible light region) and also
explains the high reflectance of metals.
All source terms on the RHS of the above inhomogeneous wave equation are of importance for
semi-conductors – however a proper/more complete physics description of EM wave propagation
in semiconductors also requires the addition of quantum theory for rigorous treatment…
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
27