Page 373 - Electromagnetics

We can use (5.39) to perform one final simplification of the EM field representation.
2
By the vector identity ∇(∇· Π) =∇ × (∇× Π) +∇ Π we get
1 i ∂ 2
∇ (∇· Π e ) =∇ × (∇× Π e ) − P + µ Π e .
∂t 2
Substituting this into (5.36) we obtain
P i
E =∇ × (∇× Π e ) − , (5.40)

∂Π e
B = µ ∇× . (5.41)
∂t
Let us examine these closely. We knowthat B is solenoidal since it is written as the curl
of another vector (this is also clear from the auxiliary Maxwell equation ∇· B = 0). The
first term in the expression for E is also solenoidal. So the lamellar part of E must be
i
i
contained within the source term P . If we write P in terms of its lamellar and solenoidal
components by using
∂P i ∂P i
i s i l
J = , J = ,
l
s
∂t ∂t
then (5.40) becomes
i i
P s P l
E = ∇× (∇× Π e ) − − . (5.42)

So we have again succeeded in dividing E into lamellar and solenoidal components.

Potential functions for magnetic current. We can proceed as above to derive the
i
i
field–potential relationships when J = 0 but J
= 0. We assume a homogeneous, loss-
m
less, isotropic medium with permeability µ and permittivity , and begin with Faraday’s
and Ampere’s laws
∂B
i
∇× E =−J − , (5.43)
m
∂t
∂D
∇× H = . (5.44)
∂t
We write H and D in terms of two potential functions A h and φ h as
∂A h
H =− −∇φ h ,
∂t
D =−∇ × A h ,

and the differential equation for the potentials is found by substitution into (5.43):
2 ∂
i
∇× (∇× A h ) = J − µ ∂ A h − µ ∇φ h . (5.45)
m 2
∂t ∂t
Taking the divergence of this equation and substituting from the magnetic continuity
equation we obtain
∂ 2 ∂ 2 ∂ρ m i
µ ∇· A h + µ ∇ φ h =− .
∂t 2 ∂t ∂t

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