Chapter IV
Maxwell's Equations
IV.1. Introduction
Maxwell’s equations are a set of four mathematical equations that relate precisely the electric
and magnetic fields to their sources , which are electric charges and currents .They were
established by James Clerk Maxwell (1831–1879) based on the experimental discoveries of:
André Marie Ampère (1775–1836)
Michael Faraday (1791–1867)
Carl Friedrich Gauss (1777–1855)
Heinrich Hertz (1857–1894)
Oliver Heaviside (1850–1925).
Maxwell’s equations can be expressed in both integral and differential forms. In what follows,
these equations will be investigated and theirs physical meaning will be explained.
IV.2. Faraday's Law
We begin by reexamining the important discovery made by Faraday in the last century. He
observed that it was possible to induce a current in a conducting loop by changing the flux
through it. This effect could be accomplished in one of two ways, by moving the loop itself or
by actually varying the magnetic field passing through the loop. In the first case, the cause of
the current flow is the force exerted by the magnetic field on the moving charges in the
wire. In the second case, the force is the result of a time-varying vector potential and its
associated electric field. In either case, the integrated force per unit charge around the loop is
proportional to the rate of change of magnetic flux through the loop. This result is known
as Faraday's law. The mathematical form of Faraday's law is given by
=
Where
.
.
is the surface bounded by a close path . The direction of the differential surface
(4.1)
is
de ned by the direction of contour
and the right-hand rule. When we curl the fingers of the
right hand in the direction of contour
, the thumb points in the direction of a unit normal to the
surface
.
1
If we consider the surface to be fixed in space, then the time derivative in equation (4.1) applies
only to the time-varying magnetic field
. In that case, we can express the equation as
.
.
Using Stokes’ theorem, we can transform the line integral around a closed path
integral over the surface
(4.2)
into a surface
as
.
=
×
.
×
Equating the right-sides of the equations (4.2) and (4.3) we get
.
The integration on either side of this equation is over the same surface
(4.3)
.
(4.4)
bounded by closed path
; therefore, the equation is valid if and only if the two integrands are equal. That is,
.
×
and
.
×
(4.5)
The equation (4.2) is the integral form of Faraday's law and the (4.5) is its differential form. If
is not function of time , (4.3) and (4.5) reduce to the electrostatic equations.
×
.
=0
=0.
(4.6 )
(4.6 )
Example 4.1
a simple magnetic field which increases exponentially with time within the cylindrical region
<
, is given by
Where
=
is constant. Calculate the generated electric field
Solution
By choosing the circular path
=
,
<
, in the
constant by symmetry, we then have from (4.2),
2
= 0 plane, along which
must be
If we now replace a with ,
<
= 2
=
, the electric field intensity at any point is
1
2
=
let us now use the differential form (4.5) to determine E
×
Hence
=
Example 4.2
=
1
1
2
=
A time varying magnetic field given by
induced
= 0 and
where
is a constant. Find the
around the rectangular loop lying in the x-z plane bounded by :
=
as shown in Figure (3.2).
= 0,
= ,
Solution
We have
=
, the magnetic flux enclosed by the loop is:
=
.
,
=
=
.
,
=
.
cos
Since the magnetic density is uniform and normal to the plane of the loop , the induced emf can
be obtained by multiplying the area of the loop
by the magnetic flux density . The induced
emf around the loop is then given by
3
.
(
.
=
.
=
cos
)
sin
IV.2. Ampere's Circuital law
Faraday’s experimental law has been used to obtain one of Maxwell’s equations in differential
and integral form. It shows that a time varying magnetic field produces an electric field. In this
section we turn our attention to the time-changing electric field. In order to understand the
contribution of Maxwell to the Ampere's Circuital law imagine a capacitor connected to a timevarying voltage source as shown in Figure 4.1. The accumulation of charge on each electrode of
the capacitor is a time-dependent process. Because the rate of change of charge constitutes a
current, there must be a time-varying current ( ) in the circuit. This current must also establish a
time-varying magnetic field in the region. Thus, if we select an open surface
bounded by a
closed contour , Ampère’s law suggests that
where
.
is the time-varying magnetic field intensity.
= ( )
( 4.7)
Figure 4.1: The displacement current in a capacitor establishes the continuity of the
conduction current in the conductor
However, if we consider another surface
contour
as the open surface bounded by the same closed
as shown in the figure, the current passing through this surface is zero. In other words,
we now have
.
4
=0
( 4.8)
Once again, (4.7) is in contradiction of (4.8). If we eliminate the uncertainty in these equations
by setting ( ) to zero, then we cannot justify the existence of either the current in the circuit or
the magnetic field associated with it. These inconsistencies led Maxwell to predict that there
must exist a current in the capacitor. As the current cannot be due to conduction, he referred to it
as displacement current. In order to account for the displacement current, Maxwell added
another term to Ampere’s law and ensured its validity for the time-varying case as well. The
additional term is, in fact, a consequence of conservation of charge. The mathematical form of
the Ampere's circuital law is given by:
×
Where
=
+
(4.9 )
is the conduction current density (A/m2),
=
is the displacement current density (measured in A/m2 ).
The modification of Ampere’s law is one of the most significant contributions of Maxwell and it
led to the development of a unified electromagnetic field theory.
For any arbitrary open surface
bounded by a closed contour
we can rewrite (4.9) in the
integral form as
.
=
.
In analogy with the name "electromotive force" for
"Magnetomotive force" abbreviated as mmf. [ ] =
+
.
.
, the quality
.
(4.9 )
.
is known as
is the current due to the flow of
free charges crossing the surface 'S'. It can be a conduction current due to the motion of charges
in a conductor or a convection current such as due to the motion of charged cloud in space.
Example 4.2
The magnetic field intensity in free space is given as
A/m, where
=
(
t
z)
is a constant quantity. Determine (a) the displacement current density and (b) the
electric field intensity.
Solution
In free space the conduction current density
(4.9a)
×
=
=
is zero. Thus from the Ampere's circuital law
That is:
5
×
=
(
0
cos(
t
z)
z)
, ( /
)
z) ,
( /
)
t
0
By integrating the displacement current density with respect to time, we obtain the electric flux
density as
sin(
=
t
Finally, the electric field intensity in free space is
=
IV.3. Gauss’s laws
=
sin(
z) ,
t
( / )
In our study of electrostatics, we obtained a mathematical statement for Gauss’s law in the point
(differential) form as
The volume charge density
.
=
(4.10 )
is zero within a dielectric medium because the effect of polarized
charges is already included in the definition of relative permittivity (or dielectric constant ). The
arguments used to derive this equation are equally applicable in the time-varying case. The only
difference is that both
and
are now time-dependent field quantities. Equation (4.10) is one
of the four Maxwell equations. It can also be expressed in integral form as
.
=
where ( ) is the total charge present at any time
surface .
= ( )
within the volume
(4.10 )
bounded by a closed
Because the magnetic flux is always continuous, Gauss’s law derived earlier for
magnetostatic fields is also valid when the fields vary with time. Thus,
where the magnetic flux density
.
= 0
(4.11 )
is now a time-varying field. This equation completes the set
of four Maxwell’s equations. It can also be written in integral form as
.
=0
Equations (4.11a) and (4.11b) signify that the magnetic charges do not exist.
6
(4.11 )
IV.3. Conservation of charge
We consider a conducting region bounded by a closed surface
We assume that the volume charge density in the region is
is shown in Figure 4.2.
, and the current leaving the surface
can be described in terms of the volume current density . The total current crossing the closed
surface
in the outward direction is
( )=
.
(4.12)
Figure 4.2: A conducting region bounded by surface s with an outward charge flow
Since current is simply a flow of charge per second, an outward flow of charge must decrease the
charge concentration by the same amount within the region bounded by s. Thus, the rate at which
the charge is leaving the surface must be equal to the rate at which the charge is diminishing in
the bounded region. Therefore, we can also express the current as
where
( )
is the total charge enclosed by the surface at any time t. We can write
volume charge density
(4.13)
in terms of the
as
=
(4.14)
where the integral is taken throughout the region enclosed by s. Combining (4.12), (4.13), and
(4.14), we obtain
.
(4.15 )
Equation (4.15) is called the integral form of the equation of continuity and is a mathematical
expression of the principle of conservation of charge. It states that any change of charge in a
region must be accompanied by a flow of charge across the surface bounding the region. In other
7
words, the charge can be neither created nor destroyed, but merely transported. The closed
surface integral on the left-hand side of (4.15) can be transformed into a volume integral by
applying the divergence theorem. Since the volume under consideration is stationary, the
differential with respect to time can be replaced by a partial derivative of volume charge density.
We can now rewrite (4.15) as
.
or
. +
=0
Since the volume under consideration is arbitrary, the only way for the preceding equation to be
true in general is for the integrand to vanish at each point. Hence,
. +
=0
(4.15 )
This is the differential (point) form of the equation of continuity.
For static fields where
= 0, Maxwell's equations in integral form become:
.
.
=0
.
=0
=
=0
.
Whereas the low of conservation of charge becomes
.
Example 4.3
Given
=
(
law in differential form.
Solution
)
=0
(Wb/m2) and it is known that
= 6 × 10 rad/s,
= 2 rad/m
Using Faraday's law in differential form, we have
8
= 0,find
by using Faraday's
×
×
=
×
And,
=
=
=
0
.
.
sin (
)
cos (
.
=
Numerical application
0
1
)
cos (
2 × 10 × 36
3 × 6 × 10 × 10 × 4 × 10
)
cos (6 × 10
= 10 cos (6 × 10
2
)
2
)
( / )
( / )
Example 4.4
The magnetic field intensity of an electromagnetic wave propagating in a lossless medium
with
=9
and
=
Find E (z, t) ?
is
( , ) = 0.3
(10
+ /4 )
(A/m)
Solution
We have
×
and
Hence
( , )=
=
( , )
=
×
×
0.3
×
37.7
0.3
(10
0.3
9
(10
(10
(10
+ /4 )
+ /4 )
+ /4 )
+ /4 )
(V/m)
IV.4. Laplacian operator
An operator that occurs frequently in the study of field theory, which will be used in the
following chapter, is called the Laplacian operator, symbolically written as
the divergence of a gradient of a scalar function. Simply put, if
scalar function, the Laplacian of
is a continuously differentiable
is
.
1. In rectangular coordinates
+
=
which yields
. It is defined as
+
=
.
+
+
+
+
(4.16 )
From (4.16) it is evident that the Laplacian of a scalar function is a scalar and involves
second-order partial differentiation of the function. By simple transformations, we can
express the Laplacian of a scalar function in cylindrical and spherical coordinates as
2. In cylindrical coordinates
3. In spherical coordinates
=
1
=
1
+
1
1
+
+
+
(4.16 )
1
(4.16 )
In our discussion of electromagnetic fields we will also encounter expressions of the form
where
is a vector field. Such an expression is known as the Laplacian of a vector field
,
and
defined as:
.
×
×
In Cartesian coordinates, the expression (4.17) becomes:
where:
=
2
=
+
2
2
+
10
2
2
2
+
+
2
2
2
(4.17)
(4.18)
(4.19)