ON THE PROBLEM OF CLOSELY SEPARATED CIRCULAR DISCS AT EQUAL POTENTIAL.
STANFORD UNIV CALIF DEPT OF MATHEMATICS
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A pair of identical and coaxially aligned circular discs is maintained at unit potential, inducing a common axisymmetric charge distribution on their respective surfaces. An approximate solution of the related boundary value problem is sought when the ratio epsilon of separation against diameter is small, with particular emphasis given to the asymptotic development of the total charge C on each disc. A crude estimate therefore, C approximately equal to 1pi, obtains readily on the basis of an intuitive argument which identifies the potential field for narrowly separated discs with that of a coincident pair and assigns to each one-half of the density or total charge found on the limiting surface at the assigned potential. More precise results are obtained, firstly, by using the particulars of the single disc field distribution in a complementary pair of variational principles, and secondly, by relying on a uniformly accurate characterization of the potential over the whole plane containing one of the discs, which evidences different behaviors for points close to and far from the edge, respectively.
- Electricity and Magnetism