Tensors for Physics

10.2 Ascending Multipoles 167

are also solutions of the Laplace equation. These are theascending multipole
potentials. They are proportional torand thus 0 forr=0. Apart from the fac-
tor( 2 − 1 )!!, the ascending multipoles are just the irreducible tensors constructed
from the components ofr,viz.

X ̃μ 1 μ 2 ···μ=( 2 − 1 )!!rμ 1 rμ 2 ···rμ. (10.16)

Examples for= 0 , 1 , 2 ,3areX ̃=1,X ̃μ=rμand

X ̃μν= 3 rμrν = 3 rμrν−r^2 δμν, (10.17)

X ̃μνλ= 15 rμrνrλ = 3

(

5 rμrνrλ−r^2 (rμδνλ+rνδμλ+rλδμν)

)

. (10.18)

By analogy to (10.12), the multiplication of the symmetric traceless tensor of rank,
constructed from the components of the vectorr, by a component of this vector and
subsequent contraction, yields a corresponding tensor of rank−1. In particular,
(10.12) implies

rμrμ 1 rμ 2 ···rμ− 1 rμ =



2 − 1

r^2 rμ 1 rμ 2 ···rμ− 1. (10.19)

Similarly, from∇μXμ 1 μ 2 ···μ− 1 μ=0 and (10.9) follows

∇μrμ 1 rμ 2 ···rμ− 1 rμ =

2 + 1

2 − 1

rμ 1 rμ 2 ···rμ− 1. (10.20)

For=1, the relationsrμrμ=r^2 and∇μrμ=3 are recovered. The nontrivial case
=2 can be used to verify the factors occurring on the right hand side of (10.19)
and (10.20).

in Electrostatics 10.3 Multipole Expansion and Multipole Moments

in Electrostatics

10.3.1 Coulomb Force and Electrostatic Potential

The Coulomb force exerted on a chargeqat the positionr, by a chargeQ, located
at the positionr′=0, is

F=

qQ

4 πε 0

1

r^3

r,