13.8.1 Expansion of the potential

In the following we shall use a notation with greek subscripts ±, β, . . . for

cartesian components of vectors and tensors. The components of a ra-

diusvector r are indicated by x± , etc. We use the Einstein summation con-

vention: summation over repeated indices is assumed. Thus μ± E± means

3

±=1 μ± E± , which is equivalent to the inner vector product μ · E.

We concentrate on the distribution A. The sources of the potential are

external to A and we drop the superscript B for the potential to simplify

the notation. Also, we take the center of the coordinate system in r A . Now

we expand φ(r) in a three-dimensional Taylor series around the coordinate

center, assuming that all derivatives of φ exist:

‚2φ

‚φ 1

φ(r) = φ(0) + x± (0) + x± xβ (0)

‚x± 2! ‚x± ‚xβ

356 Electromagnetism

‚3φ

1

(0) + · · · .

+ x± xβ xγ (13.112)

3! ‚x± ‚xβ ‚xγ

Inserting this expansion into (13.110), we ¬nd

‚2φ

‚φ 1 (2)

(0) (1)

UAB = M φ(0) + M± (0) + M ±β (0)

‚x± 2! ‚x± ‚xβ

‚3φ

1 (3)

(0) + · · · ,

+ M ±βγ (13.113)

3! ‚x± ‚xβ ‚xγ

(n)

where M is a form of the n-th multipole of the distribution, which is a

symmetric tensor of rank n:7

(0)

M = qi (monopole), (13.114)

i

(1)

M± = qi xi± = μ (dipole), (13.115)

i

(2)

M ±β = qi xi± xiβ = Q (quadrupole), (13.116)

i

(3)

M ±βγ = qi xi± xiβ xiγ = O (octupole), (13.117)

i

etc. (hexadecapole,8 . . . ). We use an overline to denote this form of the

multipole moments, as they are not the de¬nitions we shall ¬nally adopt.

The quadrupole moments and higher multipoles, as de¬ned above, contain

parts that do not transform as a tensor of rank n. They are therefore

reducible. From the quadrupole we can separate the trace tr Q = Q±± =

Qxx + Qyy + Qzz ; this part is a scalar as it transforms as a tensor of rank 0.

De¬ning Q as the traceless tensor

Q = 3Q ’ ( tr Q) 1, (13.118)

qj (3xj± xjβ ’ rj δ±β ),

2

Q±β = (13.119)

j

the quadrupolar term in the energy expression (13.113) becomes

‚2φ ‚2φ

1 1

+ ( tr Q)∇2 φ.

Q±β = Q±β (13.120)

2 ‚x± ‚xβ 6 ‚x± ‚xβ

A real tensor in 3D space is de¬ned by its transformation property under a rotation R of the

7

(cartesian) coordinate system, such that tensorial relations are invariant for rotation. For a

rank-0 tensor t (a scalar) the transformation is t = t; for a rank-1 tensor v (a vector) the

transformation is v = Rv or v± = R±β vβ . For a rank-2 tensor T the transformation is

T = RTRT or T±β = R±γ Rβ δ Tγ δ , etc.

8 The names di-, quadru-, octu- and hexadecapole stem from the minimum number of charges

needed to represent the pure general n-th multipole.

13.8 Multipole expansion 357

Since the potential has no sources in domain A, the Laplacian of φ is zero,

and the second term in this equation vanishes. Thus the energy can just

as well be expressed in terms of the traceless quadrupole moment Q. The

latter is a pure rank-2 tensor and is de¬ned by only ¬ve elements since it is

symmetric and traceless. It can always be transformed to a diagonal tensor

with two elements by rotation (which itself is de¬ned by three independent

elements as Eulerian angles).

The octupole case is similar, but more complex. The tensor has 27 el-

ements, but symmetry requires that any permutation of indices yields the

same tensor, leaving ten di¬erent elements. Partial sums that can be elim-

inated because they multiply with the vanishing Laplacian of the potential

are of the form

Oxxx + Oyyx + Ozzx = 0. (13.121)

There are thee such equations that are not related by symmetry, thus leaving

only seven independent elements. These relations are ful¬lled if the octupole

as de¬ned in (13.135) is corrected as follows:

qj [5xj± xjβ xjγ ’ rj (xj± δβγ + xjβ δγ± + xjγ δ±β )].

2

O±βγ = (13.122)

j

This is the rank-3 equivalent of the traceless rank-2 tensor.

The energy expression (13.113) can also be written in tensor notation as

UAB = qφ(0) ’ μ · E ’ 1 Q:∇E ’ + ···,

1

(13.123)

30 O:∇∇E

6

where the semicolon denotes the scalar product de¬ned by summation over

all corresponding indices. E is the electric ¬eld vector, ∇E its gradient and

∇∇E the gradient of its gradient.9

13.8.2 Expansion of the source terms

Now consider the source charge distribution qj (r j ). The potential at an

arbitrary point r (such as rA) is given by

qj

1

φ(r) = . (13.124)

|r ’ r j |

4πµ0

j

When the point r is outside the charge distribution, i.e., |r| > |r j | for any j,

the right-hand side can be expanded with the Taylor expansion of |r ’ r j |’1

Note that we do not write a dot, as ∇ · E means its divergence or scalar product; we write

9

∇∇ and not ∇2 , as the latter would indicate the Laplacian, which is again a scalar product.

358 Electromagnetism

in terms of powers of rj /r. The general Taylor expansion in three dimensions

is

‚2

‚f 1

(a) + · · · . (13.125)

f (a + x) = f (a) + x± (a) + x± xβ

‚x± 2! ‚x± ‚xβ

So we shall need derivatives of 1/r. These are not only needed for expansions

of this type, but also come in handy for ¬elds, ¬eld gradients, etc. It is

therefore useful to list a few of these derivatives:

‚1 x±

= ’ 3, (13.126)

‚x± r r

‚2 1 x± xβ 1

= 3 5 ’ 3 δ±β , (13.127)