Tensors for Physics

10.4 Further Applications in Electrodynamics 175

An expansion of the potential in powers ofr′yields

φ(r+r′)=φ(r)+rμ′∇μφ(r)+

1

2

rμ′rν′∇ν∇μφ(r)+...,

or, equivalently

φ(r+r′)=φ(r)−rμ′Eμ(r)−

1

2

rμ′rν′∇νEμ(r)−....

Notice thatδμν∇ν∇μφ(r)=−δμν∇νEμ=0, for electrostatic fields,r′μrν′in these
equations can be replaced byrμ′r′ν−( 1 / 3 )r′κrκ′δμν. Thus, the expansion of the energy
(10.44) reads

W=φ(r)

∫

ρ(r′)d^3 r′−Eμ

∫

rμ′ρ(r′)d^3 r′

−

1

2

∇νEμ

∫ (

rμ′r′ν−

1

3

rκ′rκ′δμν

)

ρ(r′)d^3 r′−...

The integrals over the charge density can be expressed in terms of the multipole
moments. This leads to

W=Qφ(r)−pelμEμ−

1

6

Qμν∇νEμ−..., (10.45)

whereQis recalled as the total charge,pμelis the electric dipole moment, andQμν
is the quadrupole moment tensor.

10.4.4 Force and Torque on Multipole Moments in an

External Field

The force on a cloud of particles with chargesqjlocated at the positionsr+rj,in
the presence of an external electric fieldEisFμ=−

∑

jqjEμ(r+r

j), or, in terms

of the charge density

Fμ=−

∫

ρ(r′)Eμ(r+r′)d^3 r′. (10.46)

A power series expansion with respect tor′yields, by analogy to (10.45),

Fμ=QEμ(r)+pelν∇νEμ+

1

6

Qνκ∇κ∇νEμ−... (10.47)