Page 157 - Electromagnetics

Figure 3.12: A dipole distribution.

Note that this is independent of r 0 . The second moment

p = r qδ(r − r 0 ) dV = qr 0
V
depends on r 0 , as does the third moment

¯ 2 ¯ 2 ¯
Q = (3r r − r I)qδ(r − r 0 ) dV = q(3r 0 r 0 − r I).
0
V
If r 0 = 0 then only the first moment is nonzero; that this must be the case is obvious
from (3.61).
For the dipole of Figure 3.12 we can write

ρ(r) =−qδ(r − r 0 + d/2) + qδ(r − r 0 − d/2).
In this case

¯
¯
Q =−q + q = 0, p = qd, Q = q[3(r 0 d + dr 0 ) − 2(r 0 · d)I].
Only the first nonzero moment, in this case p, is independent of r 0 .For r 0 = 0 the only
nonzero multipole moment would be the dipole moment p. If the dipole is aligned along
the z-axis with d = dˆ z and r 0 = 0, then the exact potential is
1 p cos θ
(r) = .
4π r 2
By (3.30)we have

1 p
ˆ
E(r) = (ˆ r2 cos θ + θ sin θ), (3.93)
4π r 3
which is the classic result for the electric field of a dipole.
Finally, consider the quadrupole shown in Figure 3.13. The charge density is

ρ(r) =−qδ(r − r 0 ) + qδ(r − r 0 − d 1 ) + qδ(r − r 0 − d 2 ) − qδ(r − r 0 − d 1 − d 2 ).

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