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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 Axial multipole moments ： ウィキペディア英語版 Axial multipole moments Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the ''z''-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as $\backslash frac$. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density $\backslash lambda(z)$ localized to the ''z''-axis. ==Axial multipole moments of a point charge== The electric potential of a point charge ''q'' located on the ''z''-axis at $z=a$ (Fig. 1) equals :$$ \Phi(\mathbf) = \frac \frac = \frac \frac - 2 a r \cos \theta}}. If the radius ''r'' of the observation point is greater than ''a'', we may factor out $\backslash frac$ and expand the square root in powers of using Legendre polynomials :$$ \Phi(\mathbf) = \frac \sum_^ \left( \frac \right)^ P_(\cos \theta ) \equiv \frac \sum_^ M_ \left( \frac(\cos \theta ) where the axial multipole moments $M\_\; \backslash equiv\; q\; a^$ contain everything specific to a given charge distribution; the other parts of the electric potential depend only on the coordinates of the observation point P. Special cases include the axial monopole moment $M\_=q$, the axial dipole moment $M\_=q\; a$ and the axial quadrupole moment $M\_\; \backslash equiv\; q\; a^$. This illustrates the general theorem that the lowest non-zero multipole moment is independent of the origin of the coordinate system, but higher multipole multipole moments are not (in general). Conversely, if the radius ''r'' is less than ''a'', we may factor out $\backslash frac$ and expand in powers of using Legendre polynomials :$$ \Phi(\mathbf) = \frac \sum_^ \left( \frac \right)^ P_(\cos \theta ) \equiv \frac \sum_^ I_ r^ P_(\cos \theta ) where the interior axial multipole moments $I\_\; \backslash equiv\; \backslash frac\{a^\{k+1\}\}$ contain everything specific to a given charge distribution; the other parts depend only on the coordinates of the observation point P. 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「Axial 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.