Let denote the surface of a perfectly conducting scatterer with unit
normal vector
.
may be either open or closed. The incident
electric field
is due to an impressed source in the
absence of the scatterer. The boundary condition is such that the sum
of the incident,
,
and the scattered,
, electric fields has no tangential
component on the perfectly conducting body surface, i.e.,
where the subscript ``tan" denotes the components tangential to the
surface .
is the electric current which is induced on the surface
due to the incident field. If
is open, we regard
as the
vector sum of the currents on opposite sides of
.
The scattered electric field can be represented by the so-called vector potential and the scalar potential which are produced by the surface current, as below:
The magnetic vector potential, , and the electric scalar
potential
are given by [47]:
An time dependence is assumed and is suppressed, and
, where
is the
wavelength. The permeability and permittivity of the surrounding medium are
and
, respectively, and
and
are the
arbitrarily located observation point and source point, respectively.
The surface charge density
is related to the surface divergence of
through the equation of continuity,
where is the surface divergence operator.
Substituting (
) into eq (
), an
integro-differential
equation for
is given by
With and
given by eqs (
) and
(
), (
) is the so-called electric
field integral equation (EFIE). In the next subsection, the method of
moments is applied to obtain a matrix equation for the unknown surface
current.