Tensors for Physics

Chapter 10

Multipole Potentials

Abstract In this chapter descending and ascending multipole potentials are
introduced, their properties are discussed and the dipole, quadrupole and octupole
potentials are considered in more detail. An application is the multipole expansion
of electrostatics, the multipole moments, like electric dipole, quadrupole, octupole
moments are defined. Further applications in electrodynamics are the calculation of
the induced dipole moment of a metal sphere, the electric polarization expressed as
dipole density, the determination of the energy of multipole moments in an external
field, as well as the multipole-multipole interaction. An application of the multipole
expansion for the pressure and velocity in hydrodynamics yields the Stokes force
acting on sphere.

Multipole potentialsare tensorial solutions of the Laplace equationΔφ... = 0.
Depending on their behavior forr→∞and atr=0,descendingandascend-
ingmultipole potentials are distinguished. More specifically,

descending multipole potentialsapproach 0 forr→∞, and diverge forr→0,
ascending multipole potentialsare0atr=0 and diverge forr→∞.

10.1 Descending Multipoles

10.1.1 Definition of the Multipole Potential Functions.

As noticed before, cf. Sect.7.6.3, the spherical symmetric solution of the Laplace
equation, which vanishes forr→∞,is

X 0 =r−^1. (10.1)

It is understood thatr>0. The spatial derivative∇μr−^1 ≡∂∂rμr−^1 is also a solution
of the Laplace equation. This follows from

Δ

(

∂

∂rμ

r−^1

)

=

∂^2

∂rλ∂rλ

(

∂

∂rμ

r−^1

)

=

∂

∂rμ

Δr−^1 = 0.

© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_10

163