424 Chapter 15Orthogonal expansions. Fourier analysis
This analysis of the potential is important for the description of the electrostatic
interaction of charge distributions, for example in the theory of intermolecular forces.
The quantities
(15.29)
in (15.28) are called the electric multipole momentsof order lof the charge
distribution (strictly,Q
lis the component of the multipole in the direction OP, see
Chapter 16). The expansion (15.28) is called the multipole expansionof the potential.
The first few moments are
total charge
dipole moment (15.30)
quadrupole moment
EXAMPLE 15.4The field of a dipole
For the system shown in Figure 15.3 of two unlike
charges+qand−qseparated by distance r:
1.Q
01 = 1 +q 1 − 1 q 1 = 10
and the total charge is zero.
whereμ 1 = 1 qris the (scalar) dipole moment of the pair of charges, andμ 1 cos 1 θis
the component of the dipole moment in the direction OP.
and, becauseP
l(cos 1 θ)is an even function ofcos 1 θ, every even-order multipole
moment is zero.
r
q
r
22222
1
2
31
2
1
2
= 3
⋅−−
(cos )θ ⋅ (coos ( ) )
2π+−=θ 10
Qqr qr qr
11
2
1
2
=− +==cosθθθθcos(π ) cos μcos
Qqr
iN2 ii i1221
2
=− 31
=∑
(cos )θ
Qqr
iN1 ii i1=
=∑
cosθ
iN0 i1=
=∑
QqrP
liiilli=
∑
(cos )θ
.................................................................................................................................................................................................................................................................................................................................................................................................o
+q
−q
p
R
r/ 2
r/ 2
.......................θ
Figure 15.3