Abstract
We study the classic problem of the capacitance of a circular parallel plate capacitor. At small separations between the plates, it is initially considered in 19th century by Kirchhoff who found the leading and the subleading term in the capacitance. Despite a large interest in the problem, one and a half century later, analytically was found only the second subleading term. Using the recent advances in the asymptotic analysis of Fredholm integral equations of the second kind with finite support, here we study the one governing the circular capacitor, known as the Love equation. We found analytically many new subleading terms in the capacitance at small separations. We also calculated the asymptotic expansion at large separations, thus providing the two simple expressions which practically describe the capacitance at all distances. The approach described here could be used to find exact analytical expansions for the capacitance to an arbitrary number of terms in both regimes of small and large separations.
Highlights
Capacitance is one of the basic concepts in electrodynamics
The coefficients in front of logarithms in Eq (12) can be fixed. Such a procedure parallels the one of Popov [24], who was working with the ansatz for f
In this paper we calculated the capacitance of the circular plate capacitor
Summary
Capacitance is one of the basic concepts in electrodynamics. For a capacitor, it denotes the ratio between the charge on one of the plates and the potential difference between them. The capacitance purely depends on the geometry. The standard simplification in the textbooks is a parallel plate capacitor in a vacuum with the characteristic plate size much larger than their separation. In this case, the capacitance has the familiar form S C = 0κ. S denotes the surface of the plates, κ is their separation, while the constant 0 is the vacuum permittivity. The expression (1) should be understood only as a result valid in the limit κ → 0+ where the edge effects are neglected
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