Tensors for Physics

176 10 Multipole Potentials

As expected, this force is also obtained as a spatial derivative of the energyW,as
given by (10.45), viz.

Fμ=−∇μW(r). (10.48)

Notice, there is no force acting on an electric dipole moment subjected to a homo-
geneous electric field with∇μEν=0.
A torque, however is exerted on an electric dipole in a spatially constant electric

field. For discrete charges, the torque isTμ=εμνλ

∑

jr

j
νEλ(r+rj). The correspond-
ing expression for a continuous charge density is

Tμ=εμνλ

∫

r′νEλ(r+r′)d^3 r′. (10.49)

The torque can be expanded by analogy to (10.47). The leading term is

Tμ=εμνλpνelEλ(r). (10.50)

10.4.5 Multipole–Multipole Interaction

The multipole–multipole interaction of a cloud 1 of charges, located around the
positionr, in a potential and electric field generated by a group of charges 2, centered
around the positionr=0, can be inferred from (10.31) with (10.33) and (10.45). The
corresponding total charges are byQ 1 ,Q 2. The electric dipole moments and electric
quadrupole moment tensors are denoted byp(μ^1 ),pμ(^2 ), andQ(μν^1 ),Q(μν^2 ), respectively.
Thepole–pole,thepole–dipoleand thepole–quadrupoleinteraction energies are

Wpole−pole=

1

4 πε 0

Q 1 r−^1 Q 2 ,

Wpole−dip=

1

4 πε 0

Q 1 Xμpμ(^2 )=

1

4 πε 0

Q 1 r−^3 rμp(μ^2 ), (10.51)

Wpole−quad=

1

4 πε 0

1

6

Q 1 XμνQ(μν^2 )=

1

4 πε 0

1

2

Q 1 r−^5 rμrνQ(μν^2 ). (10.52)

Just as the pole–quadrupole interaction, the dipole–dipole interaction is governed by
the second multipole potential tensor, which decreases, with increasing distancer,
liker−^3 ,

Wdip−dip=−

1

4 πε 0

1

2

p(μ^1 )Xμνp(ν^2 )=−

1

4 πε 0

3

2

r−^5 pμ(^1 )rμrν p(ν^2 ). (10.53)