Some physical applications of the transformation by reciprocal radii
Date
1965-08
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Abstract
The present study deals with solving some potential
problems chiefly by means of the transformation by reciprocal
radii or by the method of inversion. It aims to illustrate the
power of this method by some examples which, though difficult to
solve by analytical means, easily yield to simple mathematical
treatment after the transformation is established.
A given potential problem is solved by transforming it,
whenever possible, into another problem which is either simple or
whose solution is known. Conversely, if a problem and its
solution are given, an entirely new problem can be formulated by
this transformation and its solution is readily obtained from the
solution of the original problem.
This makes this method a powerful tool in solving many
potential problems. Direct calculation of gravitational or
electric potentials oftentimes involve the evaluation of integrals
which do not lend easily to rapid progress. However, such an
analytical difficulty can be avoided by the method of inversion.
The following examples are included in this paper:
(1) The potential of a homogeneous circular wire, (2) the potential
of a spherical surface, (3) the potential of a spherical shell with
a surface density that varies inversely as the cube of its distance
from a point not on the sphere, (4) the balayag of a sphere,
(5) the solution of the Dirichlet problem for the exterior of a
sphere, and (6) the potential of two non-intersecting spherical
conductors influenced by a point-charge exterior to both.
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Keywords
Reciprocal radii