Page 85 - Electromagnetics

curl equations requires

∂B 2
∇× E 2 = J − , (2.232)
∂t
∂D 2
∇× H 2 = J m + . (2.233)
∂t
As long as we can resolve the question of how the constitutive parameters must be altered
to reflect these replacements, we can conclude by comparing (2.232) with (2.229) and
(2.233) with (2.228) that the solution to these equations is merely

E 2 = H 1 ,
B 2 =−D 1 ,
D 2 = B 1 ,
H 2 =−E 1 .

That is, if we have solved the original problem, we can use those solutions to find the
new ones. This is an application of the general principle of duality.
Unfortunately, this approach is a little awkward since the units of the sources and
fields in the two problems are different. We can make the procedure more convenient by
multiplying Ampere’s law by η 0 = (µ 0 /
0 ) 1/2 . Then we have
∂B
∇× E =−J m − , (2.234)
∂t
∂(η 0 D)
∇× (η 0 H) = (η 0 J) + . (2.235)
∂t
Thus if the original problem has solution (E 1 ,η 0 D 1 , B 1 ,η 0 H 1 ), then the dual problem
with J replaced by J m /η 0 and J m replaced by −η 0 J has solution
E 2 = η 0 H 1 , (2.236)
B 2 =−η 0 D 1 , (2.237)
η 0 D 2 = B 1 , (2.238)
η 0 H 2 =−E 1 . (2.239)
The units on the quantities in the two problems are now identical.
Of course, the constitutive parameters for the dual problem must be altered from
those of the original problem to reflect the change in field quantities. From (2.19) and
(2.20) we know that the most general forms of the constitutive relations (those for linear,
bianisotropic media) are
¯
D 1 = ξ 1 · H 1 + ¯ 1 · E 1 , (2.240)
¯
B 1 = ¯µ 1 · H 1 + ζ 1 · E 1 , (2.241)
for the original problem, and
¯
D 2 = ξ 2 · H 2 + ¯ 2 · E 2 , (2.242)
¯
B 2 = ¯µ 2 · H 2 + ζ 2 · E 2 , (2.243)
for the dual problem. Substitution of (2.236)–(2.239) into (2.240) and (2.241) gives

¯ ¯ µ 1
D 2 = (−ζ 1 ) · H 2 + · E 2 , (2.244)
η 2
0
2 ¯
B 2 = η ¯ 1 · H 2 + (−ξ 1 ) · E 2 . (2.245)
0
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