Page 173 - Electromagnetics

We shall find, upon investigating the general multipole expansion of A below, that this
holds for any planar loop.
The magnetic field of the loop can be found by direct application of (3.127). For the
2
2
case r a we take the curl of (3.139)and find that
µ m
ˆ
B(r) = (ˆ r 2 cos θ + θ sin θ). (3.141)
4π r 3
Comparison with (3.93)shows why we often refer to a small loop as a magnetic dipole.
But (3.141)is approximate, and since there are no magnetic monopoles we cannot con-
struct an exact magnetic analogue to the electric dipole. On the other hand, we shall
find below that the multipole expansion of a finite-extent steady current begins with the
dipole term (since the current must form closed loops). We may regard small loops as
the elemental units of steady current from which all other currents may be constructed.

3.3.2 Multipole expansion

It is possible to derive a general multipole expansion for A analogous to (3.94). But
the vector nature of A requires that we use vector spherical harmonics, hence the result
is far more complicated than (3.94). A simpler approach yields the first few terms and
requires only the Taylor expansion of 1/R. Consider a steady current localized near the
origin and contained within a sphere of radius r m . We substitute the expansion (3.89)
into (3.135)to obtain
µ 1 1 2 1
1

A(r) = J(r ) + (r ·∇ ) + (r ·∇ ) +· · · dV , (3.142)
4π V R r =0 R r =0 2 R r =0

which we view as
(0)
(2)
(1)
A(r) = A (r) + A (r) + A (r) + ··· .
The first term is merely
3
µ µ
(0)

A (r) = J(r ) dV = ˆ x i J i (r ) dV
4πr V 4πr i=1 V
where (x, y, z) = (x 1 , x 2 , x 3 ). However, by (3.26)each of the integrals is zero and we have
(0)
A (r) = 0;
the leading term in the multipole expansion of A for a general steady current distribution
vanishes.
Using (3.91)we can write the second term as
3 3 3

µ
µ
(1)
A (r) = J(r ) x i x dV = ˆ x j x i x J j (r ) dV . (3.143)
i
i
4πr 3 V i=1 4πr 3 j=1 i=1 V
By adding the null relation (3.28)we can write

x J j dV = x J j dV + [x J j + x J i ] dV = 2 x J j dV + x J i dV
j
i
i
i
j
i
V V V V V
or

1

x J j dV = [x J j − x J i ] dV . (3.144)
i i j
V 2 V
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