Tensors for Physics

10.3 Multipole Expansion and Multipole Moments in Electrostatics 171

and are proportional tor−^2 ,r−^3 andr−^4 , respectively. At large distances from the
field-generating charges, the contribution of the lowest order multipole moment will
be most important for the electric field.

10.3.4 Multipole Moments for Discrete Charge Distributions

The multipole expansion can also be applied to the electrostatic potential (10.21)of
Nchargesqi, located at positionsri. The formula for the evaluation of the multipole
moment tensors, corresponding to (10.26), is

Qμ 1 μ 2 ···μ(r)=

∑N

i= 1

qi( 2 − 1 )!!riμ 1 riμ 2 ···riμ. (10.34)

Obviously, a single charge can just have a monopole. Two charges with opposite
sign possess a dipole moment. For example, consider two charges separated by the
distancea, the unit vector parallel the vector joining them is denoted byu,viz.
q 1 =q,q 2 =−qandr^1 =(a/ 2 )u,r^2 =−(a/ 2 )u. Clearly one hasQ=0 and
all other even multipole moments vanish due to thedipolar symmetry. The dipole
moment is equal to

Qμ=pelμ=qauμ=peluμ,

wherepel=qais the magnitude of the electric dipole moment. All higher order odd
multipole moments, with= 3 , 5 ,..., are also non-zero. In particular, the octupole
moment is

Qμνλ=qa^3

15

4

uμuνuλ =a^2 pel

15

4

uμuνuλ.

At the distancerfrom the center of the charge distribution, the contribution from the
octupole moment in the electrostatic potential has an extra factor(a/r)^2 , compared
with that from the dipole moment. In most applications this factor is exceedingly
small. Hence, in this case, the octupole and higher moments can be disregarded.
Four charges, two positive and two negative ones, can be arranged such that they
constitute a quadrupole. For exampleq 1 =q,q 2 =q,q 3 =−q,q 4 =−qand
r^1 =r^2 =0,r^3 =(a/ 2 )u,r^4 =−(a/ 2 )u. In this case, one hasQ=0 and the
dipole moment as well as all other odd multipole moments are zero. The quadrupole
moment is

Qμν=

− 3

2

qa^2 uμuν.