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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cylindrical multipole moments ： ウィキペディア英語版 Cylindrical multipole moments Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as $\backslash ln\; \backslash \; R$. Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential. For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such as $(\backslash rho^,\; \backslash theta^)$ refer to the position of the line charge(s), whereas the unprimed coordinates such as $(\backslash rho,\; \backslash theta)$ refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector $\backslash mathbf$ has coordinates $(\; \backslash rho,\; \backslash theta,\; z)$ where $\backslash rho$ is the radius from the $z$ axis, $\backslash theta$ is the azimuthal angle and $z$ is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the $z$ axis. ==Cylindrical multipole moments of a line charge== The electric potential of a line charge $\backslash lambda$ located at $(\backslash rho^,\; \backslash theta^)$ is given by :$$ \Phi(\rho, \theta) = \frac \ln R = \frac \ln \left| \rho^ + \left( \rho^ \right)^ - 2\rho\rho^\cos (\theta-\theta^ ) \right| where $R$ is the shortest distance between the line charge and the observation point. By symmetry, the electric potential of an infinite linecharge has no $z$-dependence. The line charge $\backslash lambda$ is the charge per unit length in the $z$-direction, and has units of (charge/length). If the radius $\backslash rho$ of the observation point is greater than the radius $\backslash rho^$ of the line charge, we may factor out $\backslash rho^$ :$$ \Phi(\rho, \theta) = \frac \left\ e^ \right) \left( 1 - \frac e^ \right) \right\} and expand the logarithms in powers of :$$ \Phi(\rho, \theta) = \frac \left\ \left( \frac \right) \left( \frac \right)^ \left(\cos k\theta \cos k\theta^ + \sin k\theta \sin k\theta^ \right ) \right\} which may be written as :$$ \Phi(\rho, \theta) = \frac \ln \rho + \left( \frac \right) \sum_^ \frac \sin k\theta} = \frac \left( \rho^ \right)^ \cos k\theta^ , and $S\_\; =\; \backslash frac\; \backslash left(\; \backslash rho^\; \backslash right)^\; \backslash sin\; k\backslash theta^\; .$ Conversely, if the radius $\backslash rho$ of the observation point is less than the radius $\backslash rho^$ of the line charge, we may factor out $\backslash left(\; \backslash rho^\; \backslash right)^$ and expand the logarithms in powers of :$$ \Phi(\rho, \theta) = \frac \left\^ \left( \frac \right) \left( \frac \left(\cos k\theta \cos k\theta^ + \sin k\theta \sin k\theta^ \right ) \right\} which may be written as :$$ \Phi(\rho, \theta) = \frac \ln \rho^ + \left( \frac \right) \sum_^ \rho^ \left(I_ \cos k\theta + J_ \sin k\theta \right ) where the interior multipole moments are defined as $Q\; =\; \backslash lambda\; ,$ $I\_\; =\; \backslash frac$ \frac \right)^}, and $J\_\; =\; \backslash frac\; \backslash frac\; \backslash right)^\}.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cylindrical multipole moments」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.