Chapter 14 MAXWELL’S EQUATIONS
• Introduction
I
→
− −
→
E · l = −

• Electrodynamics before Maxwell


• Maxwell’s displacement current
• Maxwell’s equations in vacuum


• The mathematics of waves
• Summary
INTRODUCTION
The contributions of James Clerk Maxwell to science
are remarkable, they are of comparable importance
to those of Newton. Maxwell summarized the laws
of electricity and magnetism, developed by Coulomb,
Faraday, and Ampère, and expressed them in the following concise mathematical form in terms of flux and
circulation, and most importantly added a missing term
that is of crucial importance. The four Maxwell Equations summarize the fundamental laws that underlie
all of electricity and magnetism.
Flux: Gauss’s Law:
Coulomb’s Law implies that the electric flux out of
a closed surface is:
→
− −
→
1
E · S =

0
 
Z



The Biot Savart Law implies that the magnetic flux
out of a closed surface is:
I
→
→ −
−
B · S = 0
Φ =

 
Although these two flux equations were derived only
for statics, they also are obeyed for electrodynamics.
Circulation:
Faraday’s Law implies that the circulation of the
electric field is coupled to the rate of change of magnetic flux.


Charge conservation is an important law of nature
that must be obeyed by the laws of electrodynamics.
Charge conservation implies that the net current flowing out of a closed surface must equal the rate of loss
of charge from the enclosed volume.
I
I
→ −
−
→

j · S +
 = 0

 
 

Maxwell realized that the above four equations for electrodynamics violate charge conservation. He showed
that an additional term, called the displacement current, that depends on the rate of change of electric
field, missing from Ampère’s Law; is needed to satisfy
charge conservation. This lecture will first discuss the
need for the introduction of a new displacement current term to satisfy charge conservation. This will be
followed by a summary of the fundamental Maxwell
Equations.
ELECTRODYNAMICS BEFORE
MAXWELL
I
 



−→
→
B −
· S

Ampère’s law gives the circulation of the magnetic
field:
Z
I
→ −
−
→
→
→ −
−
B
·
dl
=

0  j · S

• Maxwell’s equations: General
Φ =
Z
MAXWELL’S DISPLACEMENT
CURRENT
Maxwell showed that Ampère’s law could be generalized and made to satisfy charge conservation, by including an additional term to the current density called
−
→
the displacement current density, j , where:
→
−
−
→
E
j ≡ 0

Addition of this displacement current density to the
→
−
real current density j gives the Maxwell-Ampère law:
I

→
− −
→
B · l = 0
Z
− −
→
→
  j · S + 0 0



Z
 



→
−
→
E −
· S

R−
→ −
→
The second integral equals 0 j · S that is the
Maxwell-Ampère includes the sum of the the real current density and the displacement current.
107
Figure 1 Evaluation of Ampère’s Law when the closed
loop C→ 0
A formal proof for the need of the displacement
current term is as follows. Consider that the closed
loop C, in figure 1, shrinks to zero, that is  → 0 then
the surface integral becomes a closed surface. That is:
I
=0
→
− −
→
B · l = 0 = 0
Gauss’ law gives:
I
I
− −
→
→
j · S + 0 0
→
− −
→
1
E · S =

0
 
Z


I
→
−
→
E −
· S


therefore, the time derivative of Gauss’s law;
I

 
→
−
Z
→
E −

1
· S =


0  

Inserting this into the above line integral for a closed
loop C gives
I
I
→ −
−
→

j · S +
 = 0

 
 

that is, the charge conservation relation is obtained.
Inclusion of the displacement current term leads to the
second integral which is essential to this proof.
Charging capacitor
An understanding of Maxwell’s displacement current
density can be obtained by considering the case of a
capacitor charging with current I as illustrated in figure 2. The current in the wire produces a measurable
magnetic field circling around the wire carrying the
current I.
For surface S1 , bounded by the closed loop C, there
is a real current I passing inwards through this surface
which is related to the non-zero circulation of B around
the closed loop C given by Ampère’s law. However, as
shown in figure 2, surface S2  which passes between the
capacitor plates, has no real current flowing through
its surface, and thus Ampère’s law implies that there is
108
Figure 2 Two surfaces S1 and S2 bounded by the same
curve C. The current passes through the surface S1
but not S2 . However, there is a changing electric flux
through S2 
no magnetic field around C. However, there remains a
measurable magnetic field around the wire carrying the
current I. Clearly something is wrong with Ampère’s
Law.
Maxwell’s displacement current corrects this flaw
in Ampère’s Law, and satisfies charge conservation.
Inclusion of the displacement current does not change
the integral of current though surface S1 since the
displacement current is zero for this surface because
→
− −
R
→
0 1 E · S = 0 whereas the real current is . However, for 2 , the real current density is zero but the
displacement current is non-zero since the electric field
between the capacitor plates in increasing. It is shown
below that the net current through both 1 and 2 are
the same if both the real and displacement currents are
summed.
Integrating the displacement current over the surface S2 gives:
Z
Z
→
−
−
→ −
→
→
E −
j · S = 0
· S
2
2 
Consider Gauss’ law for the closed surface defined by
−1 + 2 
I
Z
→
→ −
−
1
E · S =

0 
−1 +2

where −1 is used to reflect the fact that the righthand rule connecting C and 1 was taken such that 1
is taken pointing into the volume. The time derivative
of this gives;
I
Z
→
−
→
E −

1
· S =





0
−1 +2

The left-hand integral has a zero contribution for surface 1  Thus the net displacement current is given
by:
Z
Z
Z
Z
→
−
−
→−
−
→
→
→
E −

j ·S = 0
0S+
·S =




−1
2
2

Inserting this in the charge conservation relation
I
I
→ −
−
→

 = 0
j · S +

 
 

and using the fact that the real current density is nonzero only for 1  and displacement current is non-zero
only for 2 gives
Z
Z
→ −
−
→
−
→ −
→
j · S +
j · S = 0
−1
or
Z
1
2
− −
→
→
j · S =
Z
2
−
→ −
→
j · S = 
where the direction of the first integral has been reversed. That is, the total displacement current flowing through 2 equals the total real current flowing
through 1 
Thus the Maxwell-Ampère’s law can be written:
I

→
− −
→
B · l = 0
Z
1
− −
→
→
j · S + 0 0
Z
2
→
−
→
E −
· S

where the first integral equals 0  for surface S1 and
zero for surface S2 , while the second integral equals 0 
for surface S2 and is zero for surface S1  That is, the
circulation of B around the loop C is the same whether
surface 1 or surface 2 are used. Thus Maxwell’s corrected version of Ampères Law satisfies charge conservation, that is, the circulation of magnetic field is independent on whether a current is flowing or the electric
flux is changing, that is whether surface S1 or surface
S2 are used for the closed loop C.
Addition of the displacement current implies that
the circulation of the B field around a closed loop depends on both the current and the rate of change of
electric flux through the loop. This is analogous to
Faraday’s Law which relates the circulation of the E
field around a closed loop to the rate of change of the
magnetic flux through the loop. The enormous significance of the displacement current will become more
apparent after we summarize Maxwell’s Equations.
Maxwell’s displacement current states that a changing electric flux through a closed curve induces a circulation of the magnetic field around that circuit. This
is a direct analogy to Faraday’s Law where a changing
magnetic flux though a circuit induces a circulation
of the electric field around the circuit. This analogy
can be illustrated by considering the induced magnetic
field between the plates of a parallel-plate capacitor
with circular plates, see figure 3. Consider a concentric circular path of radius  in the plane of the radius  circular capacitor plates. By symmetry using
Maxwell-Ampère’s law, plus the fact that the current
→
−
density j between the plates is zero, gives
I

→
− −
→
B · l = 0
Z
− −
→
→
j · S + 0 0
Z
→
−
→
E −
· S

Figure 3 Parallel-plate capacitor having circular plates
of radius  Current of  charges the plates. Calculate
the magnetic field around the concentric circular path of
radius .


Using Gauss’s Law at the surface of the capacitor plates
gives that


=
=

0 2
2 = 0 0 + 0 0 2
These give that

=
=
0  
22 
0 

22
This result is the same as for the magnetic field inside a
cylindrical conductor of radius  having a uniform cur
rent density 
2  However, this result was obtained by
calculating the induced magnetic field due to a changing flux of electric field.
MAXWELL’S EQUATIONS: GENERAL
After much hard work we finally have arrived at the
basic laws of electromagnetism as first developed by
Maxwell. These are derived from the experimental
laws of Coulomb, Faraday and Ampère plus superposition. Maxwell expressed these laws in terms of the
flux and circulation of the electric and magnetic vector
fields, as well as adding the missing term, the displacement current. The most general form of the Maxwell’s
Equations is given below.
Maxwell Equations, General form.
Flux
Gauss’s Law for electric field:
Φ =
I
→
− −
→
1
E · S =

0
 
Z



109
=
1
(Enclosed charge)
0
Gauss’s law for magnetism
Φ =
I

 
→
− −
→
B · S = 0
Circulation:
Faraday’s Law
I
→
− −
→
 E · l = −


Z
 



−→
→
B −
· S

Ampère-Maxwell law:
I
→
− −
→
 B · dl = 0


Z
→
−
−
→
→
E −
) · S
 ( j + 0



−
→
→ → −
−
→
F = ( E + −
v × B)
Maxwell used his four equations to predict that
electromagnetic waves can occur and these will travel
at a velocity in vacuum of  = √10   He was aston0
ished to discover that this equalled the measured velocity of light convincing him that light was an electromagnetic wave. It was not until after Maxwell’s death
that Hertz demonstrated that electromagnetic waves
are created via oscillating electric fields.
MAXWELL’S EQUATIONS IN
VACUUM
It helps to consider the special case of Maxwell’s Equations in vacuum, that is, where current and charge den→
−
sities are zero, j =  = 0 Then Maxwell’s equations
in vacuum simplify to the following.
Maxwell Equations for Vacuum
Flux
= 0 (Net real and displacement currents flowing
through the loop)
Remember that the total flux of the electric field
out of a closed surface is independent of the size or
shape of the Gaussian surface because the electric field
for a point charge has a 12 dependence. Gauss’s law
for the electric field gives the strength of the enclosed
charge. The total flux for the magnetic field also is independent of the size and shape of the surface because
it also has a 12 dependence. However, the net magnetic flux is always zero because the magnetic field
for an current element is tangential not radial. The
fact that the net flux is zero is equivalent to the statement that magnetic monopoles do not exist. That is,
north and south poles only occur in pairs in contrast
to charge which comes in two flavors, positive or negative.
The circulation of the electric field is a statement
of Faraday’s law of induction. Also it includes the
fact that for statics, the circulation is zero implying
that the electric field from a point charge is radial, not
tangential. A consequence for statics is that the line
R
→
→ −
−
integral → E · l is path independent allowing the
use of the concept of electric potential.
The circulation of the magnetic field is a statement
of the Ampère-Maxwell law. Remember to use a righthanded definition relating the line integral and the direction of the flux through the loop when using either
of the circulation relations.
The complete description of electromagnetism requires the four Maxwell equations plus the Lorentz
force relation that defines the electric and magnetic
fields. That is:
110
I
→
− −
→
E · S = 0
I
→
− −
→
B · S = 0

 

 
Circulation
I
→
− −
→
 E · l = −


I
Z
 



→
− −
→
 B · dl = 0 0


Z
−→
→
B −
· S

→
−
→
E −
· S




Note the symmetry between these pairs of equations both for the flux and circulation. The right-hand
side of the last equation comes from Maxwell’s displacement current. The only non-symmetric aspects
are the negative sign in Faraday’s law, and the product of the constants, 0 0  This product equals the
square of the velocity of light in vacuum,  as will be
shown next lecture. That is:
0 0 =
1
2
where the phase  is fixed with respect to the waveform
shape. It can correspond to motion in either direction.
General wave motion can be described by solutions
of a wave equation. The wave equation can be written in terms of the spatial and temporal derivatives
of the wave function Ψ() Consider the first partial
derivatives of Ψ() =  ( ∓ ) =  ()
Figure 4 Travelling wave, moving at velocity v in the 
0
direction, at times t=0 and t= −
 
Thus the system of units used for the velocity of light is
the only parameter in Maxwell’s equations. Remember, that the velocity of light in vacuum is taken to
be a fundamental constant. That is, the equations
are independent of the system of units used in defining 0 and 0 . The symmetry of the Maxwell equations and the direct inclusion of the velocity of light
is related to the direct connection of the electric and
magnetic fields predicted by Einstein’s Theory of Relativity. Next lecture will use Maxwell’s equations in
vacuum to demonstrate that they lead to the prediction of electromagnetic radiation. The discussion of
electromagnetic radiation requires a knowledge of wave
motion which will be reviewed next.
MATHEMATICS OF WAVES
Consider a travelling wave in one dimension. If the
wave is moving, then the wave function Ψ ( ) describing the shape of the wave, is a function of both
 and . The instantaneous amplitude of the wave
Ψ ( ) could correspond to the transverse displacement of a wave on a string, the longitudinal amplitude
of a wave on a spring, the pressure of a longitudinal
sound wave, the electric or magnetic fields in an electromagnetic wave, a matter wave, etc. If the wave train
maintains its shape as it moves, then one can describe
the wave train by the function  () where the coordinate  is measured relative to the shape of the wave,
that is, it is like a phase. Consider that  ( = 0)
corresponds to the peak of the travelling pulse shown
in figure 4. If the wave travels at velocity v in the 
direction, then the peak is at  = 0 for  = 0 and is at
 =  at time t. That is: the fixed point on the wave
profile  () moves in the following way
 =  − 
moving in +x direction
 =  + 
moving in -x direction
That is, any arbitrary shaped wave form travelling in
either  direction can be written as:
Ψ() =  ( ∓ ) =  ()
Ψ 
Ψ
Ψ
=
=

 

and
Ψ 
Ψ
Ψ
=
= ∓

 

Factoring out
Ψ

for the first derivatives gives
Ψ
Ψ
= ∓


The sign in this equation is independent of the shape
of the waveform but depends on the sign of the wave
velocity making it not a generally useful formula.
Consider the second derivatives
2Ψ
2 Ψ 
2 Ψ
=
=
2
2

 
2
and
2 Ψ 
2 Ψ
2Ψ
=
= + 2 2
2
2

 

Factoring out
2 Ψ
2
gives
1 2Ψ
2Ψ
= 2 2
2

 
This wave equation in one dimension is independent
of both the shape of the waveform and the sign of
the velocity. There are an infinite number of possible
shapes of waves travelling in one dimension, all of these
must satisfy this one-dimensional wave equation. The
converse is that any function that satisfies this one
dimensional wave equation must be a wave in this one
dimension.
One example of a solution of this one-dimensional
wave equations is the sinusoidal wave
Ψ() =  sin([ −

]) =  sin( − )

where  =   The wave number  = 2
  where  is
the wavelength of the wave, and angular frequency  =
2. Note that this satisfies the above wave equation
where the wave velocity equals  =  =  
The Wave Equation in three dimensions is
∇2 Ψ ≡
2Ψ 2Ψ 2Ψ
1 2Ψ
+
+
= 2 2
2
2
2



 
There are an infinite number of possible solutions Ψ to
this wave equation, any one of which corresponds to a
wave motion with velocity v.
111
The Wave Equation is applicable to all forms of
wave motion, both transverse and longitudinal. That
is, it applies to waves on a string, siesmic waves, water waves, sound waves, electromagnetic waves, matter waves, etc. In the subsequent discussion of electromagnetic waves, the Maxwell Equations will be used to
deduce a wave equation. The existence of a wave equation is equivalent to proving the existence of electromagnetic waves of any wave form, frequency or wavelength travelling with the velocity given by the wave
equation.
SUMMARY
The Maxwell equations, which are the fundamental
laws of electromagnetism, have been obtained. They
play a crucial role in most branches of science and engineering. The Maxwell Equations apply to all manifestions of electromagnetism and they contain Einstein’s
Theory of Special Relativity. For example Maxwell’s
equations predict that the velocity of light is independent of motion of the frame of reference. Maxwell did
not realize this important facet of his equations: it
took Einstein to discover this crucial fact which he
presented in his seminal 1905 paper entitled ”On the
electrodynamics of moving bodies”.
Reading assignment: Giancoli, Chapter 32,
Review the physics of waves, Giancoli, Chapter 15.
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