Discussion of 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.
Once the question was published, I immediately received several responses.
Everybody stated, correctly, that the total charge inside the sphere
should be A, since this is the only way to produce outside
the sphere the potential A/r (this can be proven using Gauss law).
However, regarding the charge distribution inside the sphere we did
not receive satisfactory answers. Some people claimed (and even attempted
a mathematical proof) that the charge distribution inside the sphere should
be spherically symmetric. This is obviously wrong - consider the following example: Take a uniformly charged sphere (figure below - left), cut out a spherical
cavity inside the sphere (as shown on the figure below - right), and concentrate
all the charge which was in the piece which was cut into a single point which
will be placed in the center of the cut. Obviously, the point charge produces
a field outside the cut identical to the field which was produced by the
spherical area which was cut-out. Therefore, the field, and potential outside
the large sphere was unchanged by this cut-and-concentrate procedure.
However, the resulting charge distribution is NOT spherically
symmetric. Therefore, we conclude that the charge distribution does not
have to be spherically symmetric.
Nevertheless: Can we say something intelligent
(and constructive) about possible charge distributions?
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