Page 87 - Electromagnetics Handbook

∂
∇× H e = ˆ zI 0 δ(r) +
0 E e , (2.247)
∂t
∇·
0 E e = ρ, (2.248)
∇· H e = 0, (2.249)

while those produced by the magnetic source must obey
∂
∇× E m =−ˆ zI m0 δ(r) − µ 0 H m , (2.250)
∂t
∂
∇× H m =
0 E m , (2.251)
∂t
∇· E m = 0, (2.252)
∇· µ 0 H m = ρ m . (2.253)
We see immediately that the second set of equations is the dual of the first, as long
as we scale the sources appropriately. Multiplying (2.250) by −I 0 /I m0 and (2.251) by
2
I 0 η /I m0 , we have the curl equations
0

I 0 ∂ I 0
∇× − E m = ˆ zI 0 δ(r) + µ 0 H m , (2.254)
I m0 ∂t I m0
2 2
I 0 η 0 ∂ I 0 η 0
∇× H m =− −
0 E m . (2.255)
I m0 ∂t I m0
Comparing (2.255) with (2.246) and (2.254) with (2.247) we see that
I m0 I m0 E e
E m =− H e , H m = 2 .
I 0 I 0 η
0
We note that it is impossible to have a point current source without accompanying
point charge sources terminating each end of the dipole current. The point charges are
required to satisfy the continuity equation, and vary in time as the moving charge that
establishes the current accumulates at the ends of the dipole. From (2.247) we see that
the magnetic field curls around the combination of the electric field and electric current
source, while from (2.246) the electric field curls around the magnetic field, and from
(2.248) diverges from the charges located at the ends of the dipole. From (2.250) we
see that the electric field must curl around the combination of the magnetic field and
magnetic current source, while (2.251) and (2.253) show that the magnetic field curls
around the electric field and diverges from the magnetic charge.

Dualityin a source-free region. Consider a closed surface S enclosing a source-free
region of space. For simplicity, assume that the medium within S is linear, isotropic, and
homogeneous. The fields within S are described by Maxwell’s equations
∂
∇× E 1 =− µH 1 , (2.256)
∂t
∂
∇× ηH 1 =
ηE 1 , (2.257)
∂t
∇·
E 1 = 0, (2.258)
∇· µH 1 = 0. (2.259)

Under these conditions the concept of duality takes on a different face. The symmetry
of the equations is such that the mathematical form of the solution for E is the same as

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