Tensors for Physics

10.3 Multipole Expansion and Multipole Moments in Electrostatics 169

|r−r′|−^1 =

∑∞

= 0

1

!

(− 1 )∂r−^1

∂rμ 1 ∂rμ 2 ···∂rμ

rμ′ 1 rμ′ 2 ···rμ′. (10.23)

The spatial derivatives ofr−^1 are the descending multipole potential tensors. Since
the tensorsrμ′ 1 rμ′ 2 ···rμ′,in(10.23), are contracted with irreducible tensorsX...,

the irreducible partrμ′ 1 rμ′ 2 ···rμ′ only contributes in the product. Thus (10.23)is
equivalent to

|r−r′|−^1 =

∑∞

= 0

1

!

Xμ 1 μ 2 ···μ(r)rμ′ 1 rμ′ 2 ···rμ′. (10.24)

Insertion of this expansion into (10.22) leads to

φ=

1

4 πε 0

∑∞

= 0

1

!( 2 − 1 )!!

Xμ 1 μ 2 ···μ(r)Qμ 1 μ 2 ···μ. (10.25)

Here

Qμ 1 μ 2 ···μ=

∫

ρ(r′)( 2 − 1 )!!rμ′ 1 rμ′ 2 ···rμ′d^3 r′=

∫

ρ(r)X ̃μ 1 μ 2 ···μd^3 r,

(10.26)

is the 2-pole moment of the charge distribution. The quantityX ̃μ 1 μ 2 ···μis the
ascending multipole defined in (10.16).
Due to (10.9), the expansion (10.25) is equivalent to

φ=

1

4 πε 0

∑∞

= 0

1

!

r−(^2 +^1 )rμ 1 rμ 2 ···rμ Qμ 1 μ 2 ···μ. (10.27)

Withtheintegrationvariabledenotedbyrinsteadofr′,thefirstfourofthesemultipole
moments are the

total chargeormonopole moment

Q=

∫

ρ(r)d^3 r,

thedipole moment

Qμ≡pelμ=

∫

ρ(r)rμd^3 r, (10.28)

thequadrupole moment

Qμν=

∫

ρ(r) 3 rμrνd^3 r, (10.29)