Page 368 - Electromagnetics

To obtain a formula for V l we take the divergence of (5.9) and use (5.11) to get
2
∇· V =∇ · V l =∇ · ∇φ =∇ φ.
The result,
2
∇ φ =∇ · V,
may be regarded as Poisson’s equation for the unknown φ. This equation is solved in
Chapter 3. By (3.61) we have
∇ · V(r )

φ(r) =− dV ,
V 4π R

where R =|r − r |, and we have
∇ · V(r )

V l (r) =−∇ dV . (5.12)
V 4π R
Similarly, a formula for V s can be obtained by taking the curl of (5.9) to get
∇× V =∇ × V s .
Substituting (5.10) we have
2
∇× V =∇ × (∇× A) =∇(∇· A) −∇ A.
We may choose any value we wish for ∇· A, since this does not alter V s =∇ × A.
(We discuss such “gauge transformations” in greater detail later in this chapter.) With
∇· A = 0 we obtain
2
−∇ × V =∇ A.
This is Poisson’s equation for each rectangular component of A; therefore

∇ × V(r )

A(r) = dV ,
V 4π R
and we have
∇ × V(r )

V s (r) =∇ × dV .
V 4π R
Summing the results we obtain the Helmholtz decomposition

∇ · V(r ) ∇ × V(r )

V = V l + V s =−∇ dV +∇ × dV . (5.13)
V 4π R V 4π R
Identification of the electromagnetic potentials. Let us write the electromagnetic
fields as a general superposition of solenoidal and lamellar components:
E =∇ × A E +∇φ E , (5.14)
B =∇ × A B +∇φ B . (5.15)
One possible form of the potentials A E , A B , φ E , and φ B appears in (5.13). However,
because E and B are related by Maxwell’s equations, the potentials should be related to
the sources. We can determine the explicit relationship by substituting (5.14) and (5.15)

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