Page 84 - Electromagnetics Handbook

is a solution to (2.224). This is the principle of superposition; if convenient, we can
decompose s in equation (2.224) as a sum (2.226) and solve the two resulting equations
(2.225) independently. The solution to (2.224) is then (2.227), “by superposition.” Of
course, we are free to split the right side of (2.224) into more than two terms — the
method extends directly to any finite number of terms.
Because the operators ∇·, ∇×, and ∂/∂t are all linear, Maxwell’s equations can be
treated by this method. If, for instance,

∂B 1 ∂B 2
∇× E 1 =− , ∇× E 2 =− ,
∂t ∂t
then

∂B
∇× E =−
∂t
where E = E 1 + E 2 and B = B 1 + B 2 . The motivation for decomposing terms in a
particular way is often based on physical considerations; we give one example here and
defer others to later sections of the book. We saw earlier that Maxwell’s equations can
be written in terms of both electric and (fictitious) magnetic sources as in equations
(2.169)–(2.172). Let E = E e + E m where E e is produced by electric-type sources and E m
is produced by magnetic-type sources, and decompose the other fields similarly. Then

∂B e ∂D e
∇× E e =− , ∇× H e = J + , ∇· D e = ρ, ∇· B e = 0,
∂t ∂t
with a similar equation set for the magnetic sources. We may, if desired, solve these
two equation sets independently for E e , D e , B e , H e and E m , D m , E m , H m , and then use
superposition to obtain the total fields E, D, B, H.

2.9.2 Duality
The intriguing symmetry of Maxwell’s equations leads us to an observation that can
reduce the effort required to compute solutions. Consider a closed surface S enclosing a
region of space that includes an electric source current J and a magnetic source current
J m . The fields (E 1 ,D 1 ,B 1 ,H 1 ) within the region (which may also contain arbitrary
media) are described by

∂B 1
∇× E 1 =−J m − , (2.228)
∂t
∂D 1
∇× H 1 = J + , (2.229)
∂t
∇· D 1 = ρ, (2.230)
∇· B 1 = ρ m . (2.231)

Suppose we have been given a mathematical description of the sources (J, J m ) and have
solved for the field vectors (E 1 , D 1 , B 1 , H 1 ). Of course, we must also have been supplied
with a set of boundary values and constitutive relations in order to make the solution
unique. We note that if we replace the formula for J with the formula for J m in (2.229)
(and ρ with ρ m in (2.230)) and also replace J m with −J in (2.228) (and ρ m with −ρ
in (2.231)) we get a new problem to solve, with a different solution. However, the
symmetry of the equations allows us to specify the solution immediately. The new set of

© 2001 by CRC Press LLC