Tensors for Physics

174 10 Multipole Potentials

cf. Sect.7.5. Thus the electric polarizationP=D−ε 0 E,cf.(7.58)ofdielectric
material is determined by

ρi=−∇νPν. (10.41)

The volume integral overρivanishes. Multiplication ofρibyrμand subsequent
integration over a volumeVwhich totally encloses the internal charges, yields the
macroscopic electric dipole momentpelμof this charge distribution, see (10.28). From
(10.41) follows

pμel=

∫

V

ρ(r)irμd^3 r=−

∫

V

rμ∇νPνd^3 r. (10.42)

Due to

rμ∇νPν=∇ν(rμPν)−Pν∇νrμ=∇ν(rμPν)−Pμ,

and the application of the Gauss theorem, one finds

pelμ=−

∫

∂V

nνPνrμd^2 s+

∫

V

Pμd^3 r.

The surface integral, taken over a surface outside the internal charge distribution,
yields zero. Thus the macroscopic electric dipole moment is the volume integral
over the electric polarization:

pelμ=

∫

Pμd^3 r. (10.43)

This means, the electric polarizationPis the density of electric dipole moments.
These dipole moments can be permanent dipoles of molecules or dipoles induced by
an electric field.

10.4.3 Energy of Multipole Moments in an External Field

The electrostatic energy of a cloud of particles with chargesqjlocated at the positions
r+rj, in the presence of an electrostatic potentialφisW =

∑

jqjφ(r+r

j).

It is assumed that the positionris in the center of the charge cloud and thatφis
generated by other charges, which are far away. Alternatively, when the charges are
characterized by a charge densityρ, the energy is

W=

∫

ρ(r′)φ(r+r′). (10.44)