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Axisymmetric Charge Distributions
For the case of an axisymmetric charge distribution (i.e., a charge distribution that is independent of the azimuthal angle
), we
can neglect the spherical harmonics of nonzero order (i.e., the nonaxisymmetric harmonics) in Equation (335), which reduces to the
following expression for the general axisymmetric Green's function:

(364) 
Here, use have been made of the fact that [see Equation (309)]

(365) 
In this case, the general solution to Poisson's equation, (337), reduces to

(366) 
where
Consider the potential generated by a charge
distributed uniformly in a thin ring of radius
that lies in the

plane, and
is centered at the origin. It follows that

(369) 
Hence, for
we obtain
and
. On the other hand, for
we get
and
. Thus,

(370) 
where
represents the lesser of
and
, whereas
represents the greater.
Next: Dirichlet Problem in Spherical
Up: Potential Theory
Previous: Multipole Expansion
Richard Fitzpatrick
20140627