Solution of boundary value problems with Laplace’s equation for ellipsoids and elliptical cylinders
| Autor: | Leonard Eyges |
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| Rok vydání: | 1980 |
| Předmět: | |
| Zdroj: | Journal of Mathematical Physics. 21:571-581 |
| ISSN: | 1089-7658 0022-2488 |
| DOI: | 10.1063/1.524456 |
| Popis: | If a dielectric ellipsoid or elliptical cylinder is placed in a uniform applied field it is well known that the field inside remains uniform, and is changed only by a depolarization factor that multiplies each applied field component. This paper generalizes this result. Namely, if for the three‐dimensional case the potential Φapp of the applied field can be expanded in the neighborhood of the ellipsoid as Φapp=Jl=0LJm=−llDlm rl Ylm(ϑ,φ) where l goes from zero to a maximum value L, then it is shown that the resultant potential inside the ellipsoid, Φint, is Φint=Jl=0LJm=−ll ClmrlYlm(ϑ,φ) where the coefficients Clm are found explicitly and there is no Clm with l≳L. For a dielectric constant e, the limits of the above solution as e→∞ and e→o are considered and are shown to yield respectively the solutions to the Dirichlet problem with potential zero on the boundary (grounded perfect conductor) and the Neumann problem with normal derivative of the potential zero on the boundary (ideal fluid flow). The homogen... |
| Databáze: | OpenAIRE |
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