With the identification in Sec. 12.2 of the fields associated
with point charge and current sources, we are ready to construct
fields produced by an arbitrary distribution of sources. Just as the
superposition integral of Sec. 4.5 was based on the linearity of
Poisson's equation, the superposition principle for the dynamic fields
hinges on the linear nature of the inhomogeneous wave equations of Sec.
The scalar potential for a point charge q at the origin, given by
(12.2.1), can be generalized to describe a point charge at an
arbitrary source position r' by replacing the distance r by
|r - r'| (see Fig. 4.5.1). Then, the point charge is replaced
by the charge density evaluated at the source position
multiplied by the incremental volume element dv'. With these
substitutions in the scalar potential of a point charge, (12.2.1),
the potential at an observer location r is the integrand of the
The integration over the source coordinates r' then superimposes
the fields at r due to all of the sources. Given the charge
density everywhere, this integral comprises the solution to the
inhomogeneous wave equation for the scalar potential, (12.1.10).
In Cartesian coordinates, any one of the components of the vector
inhomogeneous wave-equation, (12.1.8), obeys a scalar
equation. Thus, with / Ji, (1) becomes
the solution for Ai, whether i be x, y or z.
We should keep in mind that conservation of charge implies a
relationship between the current and charge densities of (1) and (2).
Given the current density, the charge density is determined to within
a time-independent distribution. An alternative, and often less
involved, approach to finding E avoids the computation of the charge
density. Given J , A is found from (2). Then, the gauge
condition, (12.1.7), is used to find . Finally, E is found
Sinusoidal Steady State Response
In many practical situations involving radio, microwave, and
optical frequency systems, the sources are essentially in the
sinusoidal steady state.
Equation (1) is evaluated by using the charge density given by (3),
with r r' and t t - |r -
where k /c. Thus, the quantity in brackets in the
second expression is the complex amplitude of at the location
r. With the understanding that the time dependence will be
recovered by multiplying this complex amplitude by exp (j t)
and taking the real part, the superposition integral for the complex
amplitude of the potential is
From (2), the same reasoning gives the superposition integral for
the complex amplitude of the vector potential.
The superposition integrals are often used to find the radiation
patterns of driven antenna arrays. In these cases, the distribution
of current, and hence charge, is independently prescribed everywhere.
Section 12.4 illustrates this application of the superposition
If fields are to be found in confined regions of space, with part
of the source distribution on boundaries, the fields given by the
superposition integrals represent particular solutions to the
inhomogeneous wave equations. Following the same approach as used in
Sec. 5.1 for solving boundary value problems involving Poisson's
equation, the boundary conditions can then be satisfied by
superimposing on the solution to the inhomogeneous wave equation
solutions satisfying the homogeneous wave equation.