Page 362 - Electromagnetics

Chapter 5

Field decompositions and the EM
potentials

5.1 Spatial symmetry decompositions

Spatial symmetry can often be exploited to solve electromagnetics problems. For
analytic solutions, symmetry can be used to reduce the number of boundary conditions
that must be applied. For computer solutions the storage requirements can be reduced.
Typical symmetries include rotation about a point or axis, and reflection through a
plane, along an axis, or through a point. We shall consider the common case of reflection
through a plane. Reflections through the origin and through an axis will be treated in
the exercises.
Note that spatial symmetry decompositions may be applied even if the sources and
fields possess no spatial symmetry. As long as the boundaries and material media are
symmetric, the sources and fields may be decomposed into constituents that individually
mimic the symmetry of the environment.

5.1.1 Planar field symmetry

Consider a region of space consisting of linear, isotropic, time-invariant media having
material parameters (r), µ(r), and σ(r). The electromagnetic fields (E, H) within this
i
i
s
region are related to their impressed sources (J , J ) and their secondary sources J = σE
m
through Maxwell’s curl equations:
∂E z ∂E y ∂ H x i
− =−µ − J , (5.1)
mx
∂y ∂z ∂t
∂E x ∂E z ∂ H y i
− =−µ − J , (5.2)
my
∂z ∂x ∂t
∂E y ∂E x ∂ H z i
− =−µ − J , (5.3)
mz
∂x ∂y ∂t
∂ H z ∂ H y ∂E x i
− = + σ E x + J , (5.4)
x
∂y ∂z ∂t
∂ H x ∂ H z ∂E y i
− = + σ E y + J , (5.5)
y
∂z ∂x ∂t
∂ H y ∂ H x ∂E z i
− = + σ E z + J . (5.6)
z
∂x ∂y ∂t
© 2001 by CRC Press LLC