Tensors for Physics

170 10 Multipole Potentials

and theoctupole moment

Qμνλ=

∫

ρ(r) 15 rμrνrλd^3 r. (10.30)

Up to=3, the expansion (10.27) of the electrostatic potential reads

φ=

1

4 πε 0

(

r−^1 Q+r−^3 rμQμ+

1

2

r−^5 rμrνQμν+

1

6

r−^7 rμrνrλQμνλ+...

)

.

(10.31)

The next higher moment, pertaining to=4, is thehexadecapole moment.The
parts of the potential associated with the monopole, dipole, quadrupole, octupole
and hexadecapole moments are proportional tor−^1 ,r−^2 ,r−^3 ,r−^4 andr−^5 , respec-
tively. Thus at large distances from the charges, the contribution of the lowest order
multipole moment will be most important for the electrostatic potential.
Just the lowest non-vanishing multipole moment is independent of the choice of
the origin chosen in the integral (10.26) for the evaluation of the multipole moments.
Notice, all multipole moments with≥1 vanish, when the charge density has
spherical symmetry. Furthermore, when the charge density is an even function ofr,
i.e. whenρ(−r)=ρ(r), all odd multipoles with= 1 , 3 ,...are zero. Similarly,
for an odd charge density whereρ(−r)=−ρ(r), all even multipoles with=
0 , 2 , 4 ,...vanish.

10.3.3 Electric Field of Multipole Moments

The electric fieldEμ, caused by the electrostatic potentialφ, is determined byEμ=
−∇μφ.Dueto(10.3), the expansion (10.27) yields

Eμ=

1

4 πε 0

∑∞

= 0

1

!( 2 − 1 )!!

Xμμ 1 μ 2 ···μ(r)Qμ 1 μ 2 ···μ. (10.32)

Notice, that in this sum, the multipole potential tensor of rank+1 is multiplied
with the-th multipole moment.
The first few terms in the expansion (10.32) for the electric field are

Eμ=

1

4 πε 0

(

r−^3 rμQ+ 3 r−^5 rμrνQν+

5

2

r−^7 rμrνrλ Qνλ+...

)

.

(10.33)

The parts of the electric field associated with the monopole, dipole and quadrupole
moments are determined by the dipole, quadrupole and octupole potential tensors,