Spherical Multipole Moments of A Point Charge
The electric potential due to a point charge located at is given by
where is the distance between the charge position and the observation point and is the angle between the vectors and . If the radius of the observation point is greater than the radius of the charge, we may factor out 1/r and expand the square root in powers of using Legendre polynomials
This is exactly analogous to the axial multipole expansion.
We may express in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2)
Substituting this equation for into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem
where the functions are the spherical harmonics. Substitution of this formula into the potential yields
which can be written as
where the multipole moments are defined
.
As with axial multipole moments, we may also consider the case when the radius of the observation point is less than the radius of the charge. In that case, we may write
which can be written as
where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics
The two cases can be subsumed in a single expression if and are defined to be the lesser and greater, respectively, of the two radii and ; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion
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