Vector spherical harmonics and their application to ... | Exercises Physics

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Vector spherical harmonics and

their application to magnetostatics

R

G Barrera?,

G

A

EstevezS

and

J

GiraldoOlJ

f

Institute

of

Physics, University of Mexico, Mexico

20,

DF, Mexico

$

Department of Mathematics, Chemistry, Physics and Computer Science, Inter American University,

San German,

PR

00753, USA

S;

Departamento de Fisica. Universidad Uacional de Colombia, Bogota, DE, Colombia

Received 29 June 1984, in final form 27 March 1985

Abstract

An

alternative and somewhat systematic

definition of the vector spherical harmonics, in

analogy

with the commonly used scalar spherical harmonics, is

presented. The new set

of

vector spherical harmonics

satisfies the properties

of

orthogonality and complete-

ness. and is compared with other existing definitions of

vector spherical harmonics. Some applications to prob-

lems in magnetostatics are illustrated.

1.

Introduction

Vector spherical harmonics

(VSH)

have been used

in the expansion

of

plane waves to study the ab-

sorption and scattering

of

light by a sphere (see, for

example. Bohren and Huffman 1983). They have

also been widely used in nuclear and atomic physics

(see, for example, Blatt and Weisskopf 1978).

The definitions of the various existing sets of

VSH

in different fields of physics are often dictated by

convenience. For example, one method

of

defining

such sets makes use of an operator which is propor-

tional

to

the usual orbital angular momentum

operator of quantum mechanics. When this

operator acts upon the scalar spherical harmonics

(SSH)

function, it generates one (out

of

a triad of)

VSH.

The purpose of this note is to develop an

alternate set

of

VSH

which is particularly useful in

classical electrodynamics.

An

alternative, simple

treatment based

on

SSH

uses the scalar Debye

potentials (Gray 1978a, Gray and Nickel 1978).

The layout

of

this paper is as follows.

In

52

we

present a brief review

of

SSH,

63 is

devoted to the

definition and formal properties

of

VSH

and finally,

in

$4,

we illustrate the usefulness

of

the

VSH

defined in

8.3

with several examples dealing with

11

Present address: Institute of Theoretical Physics, Chal-

mers University of Technology, Goteborg, Sweden.

Zusammenfassung

Eine alternative und mehr

sys-

tematische Definition vektorieller Kugelfunktionen,

analog

zu

den gewohnlich verwendeten skalaren Kugel-

funktionen wird vorgectellt. Der neue Satz vektorieller

Kugelfunktionen

erfullt

die Orthogonalitats- und

Vollstandigkeitsbedingungen: er

wird verglichen mit

vorhandenen anderen Definitionen vektorieller Kugel-

funktionen und zur Veranschaulichung auf Probleme

der Magnetostatik angewendet.

magnetostatic multipole moments and related

electromagnetic problems.

2.

Review

of

scalar spherical harmonics

This review is intended primarily to define notation

and underscore the parallel between the use of

SSH

and the

VSH

to be introduced

in

83.

Familiarity

with the properties and uses

of

the

SSH

at the level

of

development presented in the standard elec-

tromagnetism text

of

Jackson (19751 will be as-

sumed. Where possible we will follow the notation

of Jackson.

A

crucial property

of

the

SSH.

U,,,,

(8.

C$

I.

is the

completeness or closure relation. i.e. any arbitrary

function

g

of

8.6

can be expanded as

g(&+)=

c

Ao,,I'~,,l(8.+1.

(2.1)

Further.

if

g

is a

function

of

other variables, e.g.

S

and

r.

then the expansion coefficients

A,,,,

are func-

tions

of these additional variables. The computa-

tion

of

the expansion coefficients is made relatively

simple by the orthogonality condition

f

d.RY?,,(B.

4)I',,,J&

6)=

6,d

,,,,,

(2.2)

I

=(l

P>>

=

~

I

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Vector spherical harmonics and

their application to magnetostatics

R G Barrera?, G A EstevezS and J GiraldoOlJ

1. Introduction

in $4,we illustratetheusefulness of the VSH

defined in 8.3 withseveralexamplesdealingwith

A crucial property of the SSH. U,,,, ( 8. C$ I. is the

288 R G Barrera, G A Estivezand J Giraldo

where d R = sin 8 dB d 4 andtheintegral J d R is

overthewholerange of angles 8, d. The coeffi-

The evaluation of the coefficients is often simp-

yi,(e, 4 + n ) = (-1)"'yI,,,(e. 4 )

Sincethe coefficients of theexpansionfor V'QE

For r > r' or r < r' the right-hand side of equation

f =Dr-tl*i1 r > r'. (2.11)

tinuity of f at r = r' requires that D = C(r')''+'. To

range of r from r = r'- E to r = r'+ E. This yields

(d'x = r'drdR) which arecharacteristic of the

1-90 R G Barrera. G A Esttvezand J Giraldo

erty of the VSH is the way they are related to the C

--F(rl)Yl,n. l ( 1 r’ T 1) (3.14)

harmonics Wlm^ are presented^ in table^ 1.^ From the

values of I, m, there is a trivial orthogonality which

follows from the definition of Yl,,,, W,,,, and a,,,,.

analogous to equation (2.2), and valid for all I. l ' ,

factor I ( l + 1) in equations (3.21) could have been

incorporated in the definitions of W,,, and a,,,, to

to relate M,,, to the current sources J , the reason

C X B = V X ( D X A ) = - J. (4.4)

Vector spherical harmonics and their application to magnetostatics 293

@ M = -

consisting of two D-shaped rings each of radius a ,

there will be no netdipolemoment(thedipole

butas soon as transformationsareintroducedthe