15
Maxwell’s Equations for the
Electromagnetic Field

Maxwell’s equations describe the basic laws of the electromagnetic field. Over the
40 years preceding Maxwell’s enunciation of his equations (1865) the four fundamental laws describing the electromagnetic field had been discovered. These are
known as Ampe`re’s law, Faraday’s law, Coulomb’s law, and the magnetic continuity
law. These four laws were cast by Maxwell, and further refined by his successors, into
four differential equations:
=ÂH¼jþ

=ÂE¼À

@D
@t

@B
@t

ð15-1aÞ

ð15-1bÞ

= ÁD ¼

ð15-1cÞ

=ÁB¼0

ð15-1dÞ

These are Maxwell’s famous equations for fields and sources in macroscopic media:
E and H are the instantaneous electric and magnetic fields, D and B are the
displacement vector and the magnetic induction vector, and j and  are the current
and the charge density, respectively. We note that (15-1a) without the term @D=@t is
Ampe`re’s law; the second term in (15-1a) was added by Maxwell and is called the
displacement current. A very thorough and elegant discussion of Maxwell’s equations is given in the text Classical Electrodynamics by J. D. Jackson, and the reader
will find the required background to Maxwell’s equations there.
When Maxwell first arrived at his equations, the term ð@D=@tÞ was not present.
He added this term because he observed that (15-1a) did not satisfy the continuity

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


equation. To see that the addition of this term leads to the continuity equation, we
take the divergence =Á, of both sides of (15-1a).
= Á ½=  HŠ ¼ ð= Á jÞ þ

@
ð= Á DÞ
@t

ð15-2aÞ

The divergence of curl is zero, so the left-hand side is zero and we have
ð= Á jÞ þ

@
ð= Á DÞ ¼ 0
@t


Next, we substitute (15-1c) into (15-2b) and find that
!
@
=Ájþ
¼0
@t

ð15-2bÞ

ð15-3aÞ

or
=Ájþ

@
¼0
@t

ð15-3bÞ

which is the continuity equation. Equation (15-3b) states that the divergence of the
current ð= Á jÞ is equal to the time rate of change of the creation of charge ðÀ@=@tÞ.
What Maxwell saw, as Jackson has pointed out, was that the continuity equation
could be converted into a vanishing divergence by using Coulomb’s law, (15-1c).
Thus, (15-1c) could only be satisfied if


@
@D
=Ájþ

¼=Á jþ
¼0
ð15-4Þ
@t
@t
Maxwell replaced j in Ampe`re’s law by its generalization, and arrived at a new type
of current for the electromagnetic field, namely,
j!jþ

@D
@t

ð15-5Þ

for time-dependent fields. The additional term @D=@t in (15-5) is called the
displacement current.
Maxwell’s equations form the basis for describing all electromagnetic
phenomena. When combined with the Lorentz force equation (which shall be discussed shortly) and Newton’s second law of motion, these equations provide a
complete description of the classical dynamics of interacting charged particles and
electromagnetic fields. For macroscopic media the dynamical response of the aggregates of atoms is summarized in the constitutive relations that connect D and j with
E, and H with B; that is, D ¼ "E, j ¼ E, and B ¼ H, respectively, for an isotropic, permeable, conducting dielectric.
We can now solve Maxwell’s equations. The result is remarkable and was the
primary reason for Maxwell’s belief in the validity of his equations. In order to do
this, we first use the constitutive relations:
D ¼ "E

ð15-6aÞ

B ¼ H


ð15-6bÞ

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Equations (15-6a) and (15-6b) are substituted into (15-1a) and (15-1b), respectively,
to obtain
=ÂH¼j þ"
= Â E ¼ À

@E
@t

ð15-7aÞ

@H
@t

ð15-7bÞ

Next, we take the curl ð=ÂÞ of both sides of (15-7b):
= Â ð= Â EÞ ¼ À

@
ð= Â HÞ
@t

ð15-8Þ

We can eliminate = Â H in (15-8) by using (15-7a), and find that



À@
@E
= Â ð= Â E Þ ¼ 
j þ"
@t
@t
so
= Â ð= Â EÞ ¼ À

@j
@2 E
À " 2
@t
@t

ð15-9Þ

The left-hand side is known from vector analysis to reduce to
= Â = Â E ¼ =ð= Á EÞ À =2 E

ð15-10Þ

Equation (15-9) then reduces to
=ð= Á EÞ À =2 E ¼ À

@j
@2 E
À " 2

@t
@t

ð15-11Þ

Finally, if there are no free charges then  ¼ 0 and (15-1c) becomes
= Á D ¼ "= Á E ¼ 0
or
=ÁE¼0

ð15-12Þ

Thus, (15-11) can be written as
=2 E À "

@2 E
@j
¼ À
@t
@t2

ð15-13Þ

Inspection of (15-13) quickly reveals the following. If there are no currents, then
j ¼ 0 and (15-13) becomes
=2 E ¼ "

@2 E
@t2


ð15-14Þ

which is the wave equation of classical optics. Thus, the electric field E propagates
exactly according to the classical wave equation. Furthermore, if we write (15-14) as
=2 E ¼

1 @2 E
1=" @t2

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ð15-15Þ


then we have
=2 E ¼

1 @2 E
v2 @t2

ð15-16Þ

where v2 ¼ c2. The propagation of the electromagnetic field is not only governed by
the wave equation but propagates at the speed of light. It was this result that led
Maxwell to the belief that the electromagnetic field and the optical field were one and
the same.
Maxwell’s equations showed that the wave equation for optics, if his theory was
correct, was no longer a hypothesis but rested on firm experimental and theoretical
foundations.
The association of the electromagnetic field with light was only a speculation

on Maxwell’s part. In fact, there was only a single bit of evidence for its support,
initially. We saw that in a vacuum we have
=ÁE¼0

ð15-12Þ

Now it is easy to show that the solution of Maxwell’s equation gives rise to an
electric field whose form is
E ¼ E0 eiðkÁrÀ!tÞ

ð15-17aÞ

where
E ¼ Ex ux þ Ey uy þ Ez uz

ð15-17bÞ

E0 ¼ E0x ux þ E0y uy þ E0z uz

ð15-17cÞ

k ¼ kx ux þ ky uy þ kz uz

ð15-17dÞ

r ¼ xux þ yuy þ zuz

ð15-17eÞ

k Á r ¼ kx x þ ky y þ kz z


ð15-17fÞ

Substituting (15-17a) into (15-12) quickly leads to the relation:
kÁE¼0

ð15-18Þ

where we have used the remaining equations in (15-17) to obtain (15-18). The wave
vector is k and is in the direction of propagation of the field, E. Equation (15-18)
is the condition for orthogonality between k and E. Thus, if the direction of
propagation is taken along the z axis, we can only have field components along
the x and y axes; that is, the field in free space is transverse. This is exactly what
is observed in the Fresnel–Arago interference equations. Thus, in Maxwell’s theory
this result is an immediate consequence of his equations, whereas in Fresnel’s theory
it is a defect. This fact was the only known difference between Maxwell’s theory and
Fresnel’s theory when Maxwell’s theory appeared in 1865. For most of the scientific
community and, especially, the optics community this was not a sufficient reason to
overthrow the highly successful Fresnel theory. Much more evidence would be
needed to do this.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Maxwell’s equations differ from the classical wave equation in another very
important respect, however. Returning to (15-13), Maxwell’s equations lead to
=2 E À "

@2 E
@j

¼ À
@t
@t2

ð15-13Þ

The right-hand term in (15-13) is something very new. It describes the source of the
electromagnetic field or the optical field. Maxwell’s theory now describes not only the
propagation of the field but also enables one to say something about the source of
these fields, something which no one had been able to say with certainty before
Maxwell. According to (15-13) the field E arises from a term @j/@t. More specifically
the field arises not from j, the current per se, but from the time rate of change of the
current. Now this can be interpreted, as follows, by noting that the current can be
written as
j ¼ ev

ð15-19Þ

where e is the charge and v is the velocity of the charge. Substituting (15-19) into
(15-13), we have
=2 E À "

@2 E
@v
¼ e ¼ e_v
@t
@t2

ð15-20Þ


The term @v/@t is obviously an acceleration. Thus, the field arises from accelerating
charges. In 1865 no one knew of the existence of actual charges, let alone accelerating charges, and certainly no one knew how to generate or control accelerating
charges. In other words, the term (e)@v/@t in 1865 was ‘‘superfluous,’’ and so we
are left just with the classical wave equation in optics:
=2 E À "

@2 E
¼0
@t2

ð15-21Þ

Thus, we arrive at the same result from Maxwell’s equations after a considerable
amount of effort, as we do by introducing (15-21) as an hypothesis or deriving it
from mechanics. This difference is especially sharp when we recall that it takes only a
page to obtain the identical result from classical mechanics! Aside from the existence
of the transverse waves and the source term in (15-13), there was very little motivation to replace the highly successful Fresnel theory with Maxwell’s theory. The only
difference between the two theories was that in Fresnel’s theory the wave equation
was the starting point, whereas Maxwell’s theory led up to it.
Gradually, however, the nature of the source term began to become clearer.
These investigations, e.g., Lorentz’s theory of the electron, led physicists to search
for the source of the optical field. Thus, (15-13) became a fundamental equation of
interest. Because it plays such an important role in the discussion of the optical field,
(15-13) is also known as the radiation equation, a name that will soon be justified. In
general, (15-13) has the form of the inhomogeneous wave equation.
The solution of the radiation equation can be obtained by a technique called
Green’s function method. This is a very elegant and powerful method for solving
differential equations, in general. However, it is quite involved and requires a considerable amount of mathematical background. Consequently, in order not to
detract from our discussions on polarized light, we refer the reader to its solution


Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Figure 15-1

Radiating field coordinates arising from an accelerating charge; P is the observation point (From Jackson).

by Jackson (Classical Electrodynamics). Here, we merely state the result. Using
Green’s function method, the solution of the radiation equation in the form given
by (15-20) is found to be
!
È
É
e
n
Eðr, tÞ ¼
 ðn À vÞ Â v_
4"0 c2 3 R

ð15-22aÞ

where
¼1ÀnÁv

ð15-22bÞ

and n ¼ R/R is a unit vector directed from the position of the charge to the observation. The geometry of the moving charge is shown in Fig. 15-1.
In the following chapter we determine the field components of the radiated field
for (15-22) in terms of the accelerating charges.


REFERENCES
1. Jackson, J. D., Classical Electrodynamics, Wiley, New York, 1962.
2. Sommerfeld, A., Lectures on Theoretical Physics, Vols. I–V, Academic Press, New York,
1952.
3. Born, M. and Wolf, E., Principles of Optics, 3rd ed., Pergamon Press, New York, 1965.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.