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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 Spherical multipole moments ： ウィキペディア英語版 Spherical multipole moments Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, ''i.e.'', as 1/''R''. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density $\backslash rho(\backslash mathbf\}$ refer to the position of charge(s), whereas the unprimed coordinates such as $\backslash mathbf$ refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector $\backslash mathbf,\; \backslash theta^,\; \backslash phi^)$ where $r^$ is the radius, $\backslash theta^$ is the colatitude and $\backslash phi^$ is the azimuthal angle. ==Spherical multipole moments of a point charge== The electric potential due to a point charge located at $\backslash mathbf)\; =$ \frac \frac = \frac \frac - 2 r^ r \cos \gamma}}. where $R\; \backslash \; \backslash stackrel\backslash \; \backslash left|\backslash mathbf\; -\; \backslash mathbf$ and $\backslash mathbf$ of the charge, we may factor out 1/''r'' and expand the square root in powers of using Legendre polynomials :$$ \Phi(\mathbf) = \frac \sum_^ \left( \frac \right)^ P_(\cos \gamma ) This is exactly analogous to the axial multipole expansion. We may express $\backslash cos\; \backslash gamma$ in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2) :$$ \cos \gamma = \cos \theta \cos \theta^ + \sin \theta \sin \theta^ \cos(\phi - \phi^) Substituting this equation for $\backslash cos\; \backslash gamma$ into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem :$$ P_(\cos \gamma) = \frac \sum_^ Y_(\theta, \phi) Y_^(\theta^, \phi^) where the $Y\_$ functions are the spherical harmonics. Substitution of this formula into the potential yields :$$ \Phi(\mathbf) = \frac \sum_^ \left( \frac \right)^ \left( \frac \right) \sum_^ Y_(\theta, \phi) Y_^(\theta^, \phi^) which can be written as :$$ \Phi(\mathbf) = \frac \sum_^ \sum_^ \left( \frac} \right) \sqrt} Y_(\theta, \phi) where the multipole moments are defined :$$ Q_ \ \stackrel\ q \left( r^ \right)^ \sqrt} Y_^(\theta^, \phi^). As with axial multipole moments, we may also consider the case when the radius $r$ of the observation point is less than the radius $r^$ of the charge. In that case, we may write :$$ \Phi(\mathbf) = \frac^ \left( \frac \left( \frac \right) \sum_^ Y_(\theta, \phi) Y_^(\theta^, \phi^) which can be written as :$$ \Phi(\mathbf) = \frac \sum_^ \sum_^ I_ r^ \sqrt} Y_(\theta, \phi) where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics :$$ I_ \ \stackrel\ \frac} \sqrt} Y_^(\theta^, \phi^) The two cases can be subsumed in a single expression if and $r\_>$ are defined to be the lesser and greater, respectively, of the two radii $r$ and $r^$; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion :$$ \Phi(\mathbf) = \frac \sum_^ \frac} \left( \frac \right) \sum_^ Y_(\theta, \phi) Y_^(\theta^, \phi^) 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「Spherical 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.