Multipole Expansions of Electric Fields

display

1





2



3



Φ

E

(ℓ)

Φ

E

()

Φ

E

(

r

0

)

phase

plot type

index ℓ

0

index 

0

radius,

r

0

10

measurement density, ϱ

100

charge density, ρ

•

-

+

#

i

j

k

charge separation, d

1

In classical electromagnetic theory, the potential of a reasonably localized charge distribution can be represented by a multipole expansion, which is based on a sum of terms of the form

-(ℓ+1)

r



Y

ℓ

(θ,ϕ)

. The contributions from

ℓ=0,1,2,3,…

represent monopole, dipole, quadrupole and octupole sources, respectively, and sums of these can generate any realistic potential. The gradient of the potential gives the electric field. Gauss's law then equates the flux of the electric field through a Gaussian surface to the source density within the boundary of that surface. Though Gauss's law can facilitate the expression of the electric field of a point charge, expansion of the electric field created by a configuration of point charges requires a sum including higher-order multipole functions. Dipoles, quadrupoles and octupoles all have a unique angular distribution of flux, all of which integrate to zero. Each of these distributions represents a solution of Poisson's equation. The monopole potential (as from a point charge) is spherically symmetrical, varying as

1/r

.