Optimal shape of a variable condenser

Abstract

The shape of a condenser of maximal cross‐sectional area at a given capacity is derived in an analytical explicit form. Optimization is performed in the class of practically arbitrary curves by solution of the Dirichlet boundary‐value problem for a complex coordinate and expansions of the Cauchy integral kernels in Chebyshev polynomials. The criterion (area) becomes a quadratic form of the Fourier coefficients and both the necessary and sufficient extremum conditions are rigorously satisfied such that a global and unique extremum is achieved. The resulting curve coincides with the Polubarinova‐Kochina contour of a concrete dam of constant hydraulic gradient, which in its own term coincides with the Taylor‐Saffman bubble. In the limit of high capacitance, the Polubarinova‐Kochina contour tends to the SaffmanTaylor finger, which in its own turn coincides with the Morse‐Feshbach condenser contour of constant field intensity. Thus the contour found is of a minimal breakdown danger in the dielectric between charged surfaces (non‐isoperimetric optimum) and of maximal confined area (isoperimetric extremum).