Page 363 - Electromagnetics

We assume the material constants are symmetric about some plane, say z = 0. Then
(x, y, −z) = (x, y, z),
µ(x, y, −z) = µ(x, y, z),
σ(x, y, −z) = σ(x, y, z).
That is, with respect to z the material constants are even functions. We further assume
that the boundaries and boundary conditions, which guarantee uniqueness of solution, are
also symmetric about the z = 0 plane. Then we define two cases of reflection symmetry.

Conditions for even symmetry. We claim that if the sources obey
i
i
i
i
J (x, y, z) = J (x, y, −z), J (x, y, z) =−J (x, y, −z),
x
x
mx
mx
i
i
i
i
J (x, y, z) = J (x, y, −z), J (x, y, z) =−J (x, y, −z),
y
my
my
y
i
i
i
i
J (x, y, z) =−J (x, y, −z), J (x, y, z) = J (x, y, −z),
mz
mz
z
z
then the fields obey
E x (x, y, z) = E x (x, y, −z), H x (x, y, z) =−H x (x, y, −z),
E y (x, y, z) = E y (x, y, −z), H y (x, y, z) =−H y (x, y, −z),
E z (x, y, z) =−E z (x, y, −z), H z (x, y, z) = H z (x, y, −z).
The electric field shares the symmetry of the electric source: components parallel to the
z = 0 plane are even in z, and the component perpendicular is odd. The magnetic field
shares the symmetry of the magnetic source: components parallel to the z = 0 plane are
odd in z, and the component perpendicular is even.
We can verify our claim by showing that the symmetric fields and sources obey
Maxwell’s equations. At an arbitrary point z = a > 0 equation (5.1) requires

∂E z ∂E y ∂ H x i
− =−µ| z=a − J | z=a .
mx
∂y ∂z ∂t
z=a z=a z=a
By the assumed symmetry condition on source and material constant we get

∂E z ∂E y ∂ H x i
mx
− =−µ| z=−a + J | z=−a .
∂y z=a ∂z z=a ∂t z=a
If our claim holds regarding the field behavior, then

∂E z ∂E z
=− ,
∂y ∂y
z=−a z=a

∂E y ∂E y
=− ,
∂z ∂z
z=−a z=a

∂ H x ∂ H x
=− ,
∂t ∂t
z=−a z=a
and we have

∂E z ∂E y ∂ H x i
− + = µ| z=−a + J | z=−a .
mx
∂y ∂z ∂t
z=−a z=−a z=−a
So this component of Faraday’s lawis satisfied. With similar reasoning we can showthat
the symmetric sources and fields satisfy (5.2)–(5.6) as well.
© 2001 by CRC Press LLC