### 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 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).

Original language | English |
---|---|

Pages (from-to) | 485-494 |

Number of pages | 10 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 457 |

Issue number | 2006 |

DOIs | |

Publication status | Published - Feb 8 2001 |

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### Keywords

- Dirichlet's boundary-value problem
- Global extremum
- Laplace's equation
- Optimization
- Shaping

### ASJC Scopus subject areas

- General

### Cite this

**Optimal shape of a variable condenser.** / Kacimov, Anvar.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 457, no. 2006, pp. 485-494. https://doi.org/10.1098/rspa.2000.0677

}

TY - JOUR

T1 - Optimal shape of a variable condenser

AU - Kacimov, Anvar

PY - 2001/2/8

Y1 - 2001/2/8

N2 - 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).

AB - 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).

KW - Dirichlet's boundary-value problem

KW - Global extremum

KW - Laplace's equation

KW - Optimization

KW - Shaping

UR - http://www.scopus.com/inward/record.url?scp=57249114434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57249114434&partnerID=8YFLogxK

U2 - 10.1098/rspa.2000.0677

DO - 10.1098/rspa.2000.0677

M3 - Article

VL - 457

SP - 485

EP - 494

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2006

ER -