Page 156 - Electromagnetics

(3.61)and writing the derivatives in Cartesian coordinates we obtain

1 1 1 2 1
1

(r) = ρ(r ) + (r ·∇ ) + (r ·∇ ) +· · · dV . (3.90)
4π V R r =0 R r =0 2 R r =0

For the second term we can use (3.57)to write

ˆ
1 R ˆ r
1

(r ·∇ ) = r · ∇ = r · = r · . (3.91)
R r =0 R r =0 R 2 r =0 r 2

The third term is complicated. Let us denote (x, y, z) by (x 1 , x 2 , x 3 ) and perform an
expansion in rectangular coordinates:
3
3

∂ 2
2
(r ·∇ ) 1 = x x j 1 .

i
R ∂x ∂x R

r =0 i=1 j=1 i j r =0

It turns out [172] that this can be written as

1 2 ¯
2
1

(r ·∇ ) = ˆ r · (3r r − r I) · ˆ r.

R r 3
r =0

Substitution into (3.90)gives
¯
Q ˆ r · p 1 ˆ r · Q · ˆ r
(r) = + + +· · · , (3.92)
4π r 4π r 2 2 4π r 3
which is the multipole expansion for (r). It converges for all r > r m where r m is the
radius of the smallest sphere completely containing the charge centered at r = 0 (Figure

¯
3.11). In (3.92) the terms Q, p, Q, and so on are called the multipole moments of ρ(r).
The first moment is merely the total charge

Q = ρ(r ) dV .
V
The second moment is the electric dipole moment vector

p = r ρ(r ) dV .
V
The third moment is the electric quadrupole moment dyadic

¯ 2 ¯
Q = (3r r − r I)ρ(r ) dV .
V
The expansion (3.92)allows us to identify the dominant power of r for r r m .
The first nonzero term in (3.92)dominates the potential at points far from the source.
Interestingly, the first nonvanishing moment is independent of the location of the origin
of r , while all subsequent higher moments depend on the location of the origin [91]. We

can see this most easily through a few simple examples.
For a single point charge q located at r 0 we can write ρ(r) = qδ(r − r 0 ). The first
moment of ρ is

Q = qδ(r − r 0 ) dV = q.
V
© 2001 by CRC Press LLC