Page 170 - Electromagnetics

3.3.1 The magnetic vector potential
Although the magnetic scalar potential is useful for describing fields of permanent
magnets and for solving certain boundary value problems, it does not include the effects of
source current. A second type of potential function, called the magnetic vector potential,
can be used with complete generality to describe the magnetostatic field. Because ∇·B =
0, we can write by (B.49)

B(r) =∇ × A(r) (3.127)
where A is the vector potential. Now A is not determined by (3.127)alone, since the
gradient of any scalar field can be added to A without changing the value of ∇× A.
Such “gauge transformations” are discussed in Chapter 5, where we find that ∇· A must
also be specified for uniqueness of A.
The vector potential can be used to develop a simple formula for the magnetic flux
passing through an open surface S:

m = B · dS = (∇× A) · dS = A · dl, (3.128)
S S
where is the contour bounding S.
In the linear isotropic case where B = µH we can find a partial differential equation
for A by substituting (3.127)into (3.120). Using (B.43)we have
1

∇× ∇× A(r) = J(r),
µ(r)
hence
1 1
∇× [∇× A(r)] − [∇× A(r)] ×∇ = J(r).
µ(r) µ(r)
In a homogeneous region we have
∇× (∇× A) = µJ (3.129)

or
2
∇(∇· A) −∇ A = µJ (3.130)
by (B.47). As mentioned above we must eventually specify ∇· A. Although the choice is
arbitrary, certain selections make the computation of A both mathematically tractable
and physically meaningful. The “Coulomb gauge condition” ∇· A = 0 reduces (3.130)
to
2
∇ A =−µJ. (3.131)
The vector potential concept can also be applied to the Maxwell–Boffi magnetostatic
equations

∇× B = µ 0 (J +∇ × M), (3.132)
∇· B = 0. (3.133)
By (3.133)we may still define A through (3.127). Substituting this into (3.132) we have,
under the Coulomb gauge,
2
∇ A =−µ 0 [J + J M ] (3.134)
where J M =∇ × M is the magnetization current density.

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