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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 Matrix representation of Maxwell's equations ： ウィキペディア英語版 Matrix representation of Maxwell's equations In electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus. These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations.〔(Bialynicki-Birula, 1994, 1996a, 1996b)〕 A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium. For an inhomogeneous medium it necessarily requires 8 × 8 matrices.〔(Khan, 2002, 2005)〕 ==Introduction== Maxwell's equations in the standard vector calculus formalism, in an inhomogeneous medium with sources, are:〔(Jackson, 1998; Panofsky and Phillips, 1962)〕 :$$ \begin & \cdot \left( , t \right) = \rho\, \\ & \times \left( , t \right) - \frac \left( , t \right) = \, \\ & \times \left( , t \right) + \frac \left( , t \right) = 0\, \\ & \cdot \left( , t \right) = 0\,. \end The media is assumed to be linear, that is :$=\; \backslash epsilon\; \backslash ,,\backslash quad\; =\; \backslash mu$, where ε = ε(r, ''t'') is the permittivity of the medium and μ = μ(r, ''t'') the permeability of the medium (see constitutive equation). For a homogeneous medium ε and μ are constants. The speed of light in the medium is given by :$v\; (\; ,\; t)\; =\; \backslash frac\; ,\; t)\}\}$. In vacuum, ε0 = 8.85 × 10−12 C2·N−1·m−2 and μ0 = 4π × 10−7 H·m−1 One possible way to obtain the required matrix representation is to use the Riemann-Silberstein vector 〔(Silberstein, 1907a, 1907b, Bialynicki-Birula, 1996b)〕 given by :$$ \begin ^ \left( , t \right) & = \frac , t)} \left( , t \right) + \frac} \left( , t \right) \right) \\ ^ \left( , t \right) & = \frac , t)} \left( , t \right) - \frac} \left( , t \right) \right)\,. \end If for a certain medium ε = ε(r, ''t'') and μ = μ(r, ''t'') are constants (or can be treated as ''local'' constants under certain approximations), then the vectors F± (r, ''t'') satisfy :$$ \begin \frac ^ \left( , t \right) & = \pm v \times ^ \left( , t \right) - \frac ) \\ \cdot ^ \left( , t \right) & = \frac Thus by using the Riemann-Silberstein vector, it is possible to reexpress the Maxwell's equations for a medium with constant ε = ε(r, ''t'') and μ = μ(r, ''t'') as a pair of equations. 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「Matrix representation of Maxwell's equations」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.