Answer to the Question 01/97

The question was:
An electrostatic potential has been measured everywhere outside a sphere of radius R. It was found that the potential is spherically symmetric, i.e. depends only on the distance r from the center of the sphere, and is given by the expression A/r, where A is some constant. No measurement of the potential inside the sphere has been performed. What can you say about the charge distribution inside the sphere.

(5/97) Many people noted correctly that the total charge of the sphere must be A (in Gaussian units). This can be proven directly using Gauss law. At the same time it was demonstrated (see Discussion) that the charge distribution does not have to be spherically symmetric.


(6/97) Itamar Borukhov from Tel Aviv U. wrote that presenting the potential at point r as a volume integral over rho(r')/|r-r'| where rho(r') is the density of the charge, and by expanding 1/|r-r'| in powers of r and integrating over r', we get a multipole expansion of the potential. Since we know that the resulting potential is spherically symmetric, all the multipoles must vanish. Thus the statement which can be made is: the charge distribution is such that all multipoles (except the monopole) vanish.


(6/97) Y. Kantor: So now we know that the multipoles vanish. It may look strange, since the charge density is not constant. However, we must keep in mind that multipoles are "kind-of" expansion of the charge density into functions rl Ylm(theta,phi), where r is the distance from the center, theta,phi are the polar angles and Y is the spherical-harmonic function. This is not a complete set of functions within a sphere. (It is only a complete set for harmonic functions, and the density is not a harmonic function.) So now we somewhat reduced the list of functions which can create the potential A/r. Can we rephrase the restrictions in a more constructive form, i.e. say something like: "the most general charge density producing such a potential is..."?

(9/97) Koenraad Audenaert from University of Gent (RUG), Belgium (email: ka@elis.rug.ac.be) gave a constructive solution: He has shown that the charge density must be an expansion of terms of the form Ylm(theta,phi)Pk(0,2)(2r-1) where Pk(0,2)(2r-1) is Jacobian polynomial. For detailed derivation see the following postscript file.
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