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Newmann Problem in Spherical Coordinates

According to Section 2.10, the solution to the Newmann problem, in which the charge density is specified within some volume , and the normal derivative of the potential given on the bounding surface , takes the form

 (382)

where the Newmann Green's function is written

 (383)

Here, is solution of Laplace's equation (i.e., ) which is chosen so as to ensure that

 (384)

and

 (385)

The latter constraint holds when (or ) lies on . Note that we have chosen the arbitrary constant to which the potential is undetermined such that . It again follows from Sections 3.4 and 3.5 that

 (386)

where the and the are chosen in such a manner that the constraints (385) and (386) are satisfied.

As a specific example, suppose that the volume lies inside the spherical surface . The physical constraint that the Green's function remain finite at implies that the are all zero. Applying the constraint (385) at , we get

 (387)

Similarly, the constraint (386) leads to

 (388)

for . Hence, the unique Green's function for the problem becomes

 (389)

Finally, expanding and in the forms (378) and (379), respectively, Equations (383) and (390) yield

 (390)

and

 (391)

for .

Next: Laplace's Equation in Cylindrical Up: Potential Theory Previous: Dirichlet Problem in Spherical
Richard Fitzpatrick 2014-06-27