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 Saffman-Taylor 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).
|الصفحات (من إلى)||485-494|
|دورية||Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|
|المعرِّفات الرقمية للأشياء|
|حالة النشر||Published - فبراير 8 2001|
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