Abstract
As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.
Highlights
The calculation of the capacitance of a parallel-plate condenser appears in any electrostatics textbook
The second approach, which constitutes the natural follow-up of the intuitive calculations of Kirchhoff and Thomson, attempts to solve Laplace’s equation directly, with the high-aspect-ratio singularity being addressed from the outset using singular perturbations
Making use of matched asymptotic expansions, we have analysed the electrostatics of a parallelplate capacitor in the limit of small separation
Summary
The calculation of the capacitance of a parallel-plate condenser appears in any electrostatics textbook. The second approach, which constitutes the natural follow-up of the intuitive calculations of Kirchhoff and Thomson, attempts to solve Laplace’s equation directly, with the high-aspect-ratio singularity being addressed from the outset using singular perturbations This direct approach was carried out by Shaw [28] for both strip and disk capacitors. The restricted derivation which follows may serve as a convenient introduction for non-experts, supplementing both fundamental analyses of capacitance calculations [15, 16, 32] as well as less mathematical discussions of the fringing effect [6, 10, 20, 24, 26] This educational goal will be accomplished in the present paper using matched asymptotic expansions [12], where the electric potential is solved in different subdomains (defined via appropriate limit processes). The electrostatic attraction is determined using two different approaches, one direct, using the Maxwell stress concept, and one indirect, using the principle of virtual work
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