Page 369 - Electromagnetics

into Ampere’s and Faraday’s laws. It is most convenient to analyze the relationships
using superposition of the cases for which J m = 0 and J = 0.
With J m = 0 Faraday’s lawis
∂B
∇× E =− . (5.16)
∂t
Since ∇× E is solenoidal, B must be solenoidal and thus ∇φ B = 0. This implies
that φ B = 0, which is equivalent to the auxiliary Maxwell equation ∇· B = 0.Now,
substitution of (5.14) and (5.15) into (5.16) gives
∂
∇× [∇× A E +∇φ E ] =− [∇× A B ] .
∂t
Using ∇× (∇φ E ) = 0 and combining the terms we get

∂A B
∇× ∇× A E + = 0,
∂t
hence

∂A B
∇× A E =− +∇ξ.
∂t
Substitution into (5.14) gives

∂A B
E =− + [∇φ E +∇ξ] .
∂t
Combining the two gradient functions together, we see that we can write both E and B
in terms of two potentials:

∂A e
E =− −∇φ e , (5.17)
∂t
B =∇ × A e , (5.18)
where the negative sign on the gradient term is introduced by convention.

Gauge transformations and the Coulomb gauge. We pay a price for the simplicity
of using only two potentials to represent E and B. While ∇× A e is definitely solenoidal,
A e itself may not be: because of this (5.17) may not be a decomposition into solenoidal
and lamellar components. However, a corollary of the Helmholtz theorem states that a
vector field is uniquely specified only when both its curl and divergence are specified. Here
there is an ambiguity in the representation of E and B; we may remove this ambiguity
and define A e uniquely by requiring that

∇· A e = 0. (5.19)
Then A e is solenoidal and the decomposition (5.17) is solenoidal–lamellar. This require-
ment on A e is called the Coulomb gauge.
The ambiguity implied by the non-uniqueness of ∇· A e can also be expressed by the
observation that a transformation of the type

A e → A e +∇ , (5.20)
∂
φ e → φ e − , (5.21)
∂t

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