Review of Electromagnetic Theory
I. Introduction:
In these notes, I will write down Maxwell’s equations and from them derive the wave
equation for the electric field and develop expressions for the Poynting vector associated with the
intensity of an electromagnetic field. With this formalism in hand, I will discuss the issue of the
coherence of an electromagnetic field. I will try to match the book’s notation as much as
possible.
II. Maxwell’s Equations
The basic equations of electromagnetism are the four Maxwell Equations and the Lorentz
force law. In principle these, together with Newton’s second law of motion are enough to
completely determine the motion of an assembly of charges given the initial positions and
velocities of all the charges. Maxwell’s equations are,
  e   / 0
(1)
 b  0
b
e  
t
(2)
(3)
e
(4)
t
Here, e and b are the electric and magnetic fields respectively. The sources for the fields are the
volume charge density  and the current density j . The two parameters in these equations are
the permittivity of free space  0  8.85 1012 F/m and the permeability of free space
 0  1.26 10 6 H/m . The vector, differential-operator in these equations is defined as



  xˆ
 yˆ
 zˆ
x
y
z
These equations can be used to determine the fields if the charge and current densities are known.
Once the fields are known, the force felt by a given charge q moving with velocity v is given by
the Lorentz force law,
(5)
F  q e  v b
  b   0 j   0 0


In principle, Equations (1)-(5) constitute all of electromagnetics. However, in practice, the charge
and current densities associated with matter are too complicated to specify and so a
phenomenological means of dealing with matter is often introduced.
III. Maxwell’s Equations in Matter
To deal with matter the relations
d   0e  p
(6)
(7)
b  0 h  m
are usually introduced. Here, p is the polarization of the medium and characterizes the field
induced in matter by the electric field. The field m is called the magnetization and is related to
the induced magnetic field in matter. We will ignore the magnetization in what follows. The
polarization of the medium satisfies the relations,
  p  b
jb 
p
t
(9)
where b is the volume charge density bound to the material medium and jb is the current due
to the charges bound to the material medium. The total charge and current densities can then
written as,
   f  b   f    p
(10)
p
j  j f  jb  j f 
t
where  f and j f are the “free” charge and current densities, respectively. Using (10) to mimic
the effects of matter lead to the versions of Maxwell’s equations written below.
  e   f / 0    p / 0
(11)
 b  0
b
e  
t
e
p
  b   0 j f   0 0
 0
t
t
(12)
(13)
(14)
We will typically be interested in situations for which the free charge and current densities
vanish. That is, situations appropriate for propagation of electromagnetic fields through net
neutral matter.
IV. Solutions to the Wave Equation in Free Space:
In free space there is no matter and hence p  0 and charge and current densities also
vanish. Maxwell’s equations then take the particularly simple form,
(15)
 e  0
(16)
 b  0
b
e  
(17)
t
e
  b   0 0
(18)
t
Then, Eqs. (17) and (18) are easily decoupled and solved as I show below. To decouple the
equations, consider the curl of (17),
b 
 2e
 e   
   0 0 2
t
 t 
where I have used (18) to form the final expression. A useful vector identity is,
 e  2e    e 
In free space the divergence of e vanishes (Eq. (15)). So, (19) can be written as,
(19)
(20)
 2 e   0 0
 2e
0
 t2
(21)
Recall that the Laplacian operator is defined as,
 2
2
2 
(22)
2e   2  2  2  e
 x  y  z 
I will consider monochromatic, plane wave solutions for fields propagating in the z-direction and
polarized in the x-direction. That is, let I’ll try a solution to (22) of the form,
(23)
e  E0 cos(t  kz  0 ) xˆ
with E0 , k , 0 , and  constant. Substitution of (23) into (21) indicates that (23) is indeed a
solution provided that,

1

(24)
k   0 0 1/ 2
Now,  / k is the phase velocity of a traveling, plane, monochromatic wave. Numerical
evaluation of the right-hand side of (24) shows that,
 / k  3 108 m/s=c
(25)
That is, Maxwell’s equations have free-space solutions corresponding to monochromatic plane
waves propagating with a phase velocity equal to the speed of light c. This was first noted in the
1880’s and was the first definitive indication that light was an electromagnetic phenomenon. The
parameters appearing in the solution to the wave equation are: the electric field amplitude E0 , the
(angular) frequency  , the spatial frequency k and the phase constant 0 .
Problem 1:
(A) By examination of (23) show that
2
(26)
T
where T is the time for the electric field to complete one cycle at a fixed z. That is, where T is the
temporal period.
(B) By examination of (23) show that
2
k 
(27)


where  is the wavelength (that is, the spatial period) of the wave.
(C) By examination of (23) show that the phase velocity of this wave is indeed  / k.
Problem 2:
Use Eq. (17) above to show that the magnetic field b corresponding to the electric field
(23) has the form,
1
b  E0 cos( t  kz  0 ) yˆ
(28)
c
To do this problem, substitute (23) into the right hand side of (17) and then integrate to find the
magnetic field. (Do not just show that (28) works upon substitution.) In doing the integration
be careful to keep track of the limits.
Problem 3:
Show that a plane, monochromatic, electromagnetic wave propagating in the z-direction
(in free space) cannot be polarized in the z-direction. Hint: Use Eq. 15.
V. Electromagnetic Energy and the Poynting Vector
Energy is stored in electromagnetic fields and electromagnetic waves carry energy and
momentum. These facts follow from Maxwell’s equations, the Lorentz force law, and the work
energy theorem. Here, I will just state the relevant results. In linear dielectrics (materials for
which d   e   0e  p   0 (1   )e with  and  constants) the energy density (energy per unit
volume) stored in an electromagnetic field is given by,

1
1
(29)
u  e e  b b 
2
0

Note thsat our book uses  for the energy density. Here, I have still assumed that the material is
non-magnetic. The energy flux density S is known as the Poynting vector and is given by,
1
S
e b
(30)
0
The direction of the Poynting vector is the direction of energy flow and the magnitude of the
Poynting vector is the energy per unit time crossing an incremental area perpendicular to the
direction of propagation of the energy flow. Most detectors of optical radiation measure the time
averaged Poynting vector which I will denote
1 T
S
Sdt
(31)
2T T
where T is (half) the time over which the detector averages the incoming energy. If the averaging
time 2T is long compared to an optical period (and it usually is) the averaging time can be taken
to be infinite. In this case,
1 T
S
Sdt
(32)
2T T
T 
For this case (averaging time long compared to period of wave) the intensity of the
electromagnetic wave can be defined as the magnitude of the time averaged Poynting vector,

Lim 
I S
(33)
Note that the intensity is the average power per unit area. Hence it carries units of Watts/meter 2 .
For the field given by equations (23) and (28) the Poynting vector and the energy density
can be shown to satisfy the relationship,
(34)
S  cuzˆ
This relation embodies the fact that for this kind of wave, the velocity of energy flow is the same
as the phase velocity of the wave and the direction of energy flow is the same as the direction of
the propagation of the plane wave.
Problem 4:

(A) Calculate u, S , and I forthe field given by Eqs. (23) and (28).
(B) Show that for this case, S  cuz . Argue that this means that the velocity of energy
propagation is c.
It is the velocity of energy propagation that can never exceed c. The phase velocity
sometimes does exceed c in material media. For plane monochromatic waves in material media
the phase velocity and the velocity of energy propagation are the same but this is not true for
general electromagnetic wave propagation.
VI. Complex Representations of the Fields
Many calculations can be simplified by dealing with complex fields that are associated
with the physical, real-valued quantities via the relation,
 
 
e (r , t )   E (r , t )
(35)
d i
Here  indicates the operation of taking the real part. Similar definitions hold for the other field
vectors. The real physical quantities are denoted by lower case symbols and the associated
complex functions by upper-case symbols. For example, if
 
E (r , t )  E0ei bt  kz   gx
(36)
then,
 
(37)
e (r , t )  E0 cos t  kz   x
b
g
Any (real) equation that is linear in the field vectors holds for the complex fields if they hold for
the real fields. Maxwell’s equations (and the linear dielectric wave equations) are linear
equations and so hold with the replacements of the field vectors by their associated complex
counterparts. This replacement often introduces considerable computational simplification.
However, care must be taken when the basic equations are not linear in the field vectors. The
Poynting vector and the energy density are both quadratic functions of the field vectors and so it
is not true that they hold (in general) for the complex fields. In the discipline of nonlinear optics
the polarization has a nonlinear dependence on the electric field vector and so drags
nonlinearities into Maxwell’s equations and so one must again use care in dealing with complex
fields in that formalism. (Again, the complex notation is still useful in these cases.)
Problem 5
(A) Show that for monochromatic plane waves of the form (36) that the time averaged energy
density and Poynting vectors are correctly given by,

1  *
S
EB
2 0
(38)
1 1   * 1  *
u   E  E 
BB
2 2
0
(B) What is the significance of the extra factors of (1/2) that appear in these equations but not in
Eqs. (29) and (30)?
F
G
H
IJ
K
VII. Superpositions of Plane Waves
Discussion of the plane, monochromatic solutions to the wave equation is useful for two
different reasons. The first is that this waveform is the simplest sort of wave that models many
real features of actual electromagnetic waves. The second is that a completely general solution to
the wave equation can be built out of a superposition of plane waves of differing frequencies and
amplitudes. A general solution to the free space wave equation for a wave polarized in the xdirection and propagating in the positive z-direction could be written as,


(39)
e ( z, t )  x A() cos (t  z / c)  () d
z
0
b
g
VIII. Interference and Coherence
Consider the electromagnetic field at a detector which is the sum of two different fields
(of the same amplitude) which are combined at the detector. That is, let,
 
e r , t  A cos( 1t  k1z1  1 )  A cos( 2 t  k 2 z2   2 ) x
(40)
The corresponding magnetic field is,
 
1
b (r , t )  A cos( 1t  k1z1  1 )  A cos( 2 t  k 2 z2   2 ) y
(41)
c
Here, z1 and z2 refer to the distances the respective beams travel form their source to the detector
and 1 and  2 are the phases of these beams at t  0 and at the position of the source. The
Poynting vector for this field is,
 1  
A2

S
e b  z
{cos2 1t  k1z1  1  cos2  2 t  k2 z2   2 
0
 0c
(42)
b gb
g
b
g
b
g b
g
 2 cosb
 t k z  g
cosb
 t k z  g
}
1
1 1
1
2
2 2
2
The intensity at the detector is then,

A2
I S 
1  1  2cos ( 1   2 )t  ( k1z1  k 2 z2 )  1   2
2 0 c
b
e
A
I
e1  cosb(   )t  (k z  k z )     gj
c
I  2I e
1  cosb
(   ) t  ( k z  k z )     g
j
gj
2
1
2
1 1
2 2
1
2
(43)
0
0
1
2
1 1
2 2
1
2
Here I 0 is the intensity of either beam considered alone. The details of the expression (43) will
be discussed in class.
Problem 6:
Why won’t two independent laser beams interfere with each other even if they have the
same (nominal) frequency? Let Eq. (43) guide your reasoning.
Problem 7:
Consider the propagation of an electromagnetic wave through a linear dielectric. A linear
dielectric is one in which the polarization is directly proportional to the electric field,


p   0 e
Here, take the susceptibility  to be a scalar constant greater than one.
(A) Derive the wave equation appropriate for this medium.
(B) What is the phase velocity of monochromatic waves in this medium?
(C) What is the index of refraction of this medium?
Problem 8:
(A)
Find an expression for the intensity measured at a detector if three monochromatic beams
that are perfectly coherent with each other are combined on a detector if the intensity of each
beam considered alone is I 0 . Assume that the three beams leave their common source with the
same phase but travel different distances to the detector.
(B) What are the maximum and minimum values the intensity of part (A) can take? (Consider the
path lengths traveled to be variable).