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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 Maxwell stress tensor ： ウィキペディア英語版 Maxwell stress tensor The Maxwell stress tensor (named after James Clerk Maxwell) is a second rank tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand. In the relativistic formulation of electromagnetism, the Maxwell's tensor appears as a part of the electromagnetic stress–energy tensor which is the electromagnetic component of the total stress–energy tensor. The latter describes the density and flux of energy and momentum in spacetime. ==Motivation== As outlined below, the electromagnetic force is written in terms of E and B, using vector calculus and Maxwell's equations symmetry in the terms containing E and B are sought for, and introducing the Maxwell stress tensor simplifies the result. : |- | Gauss's law for magnetism | $\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0$ |- | Maxwell–Faraday equation (Faraday's law of induction) | $\backslash nabla\; \backslash times\; \backslash mathbf\; =\; -\backslash frac$ |- | Ampère's circuital law (in vacuum) (with Maxwell's correction) | $\backslash nabla\; \backslash times\; \backslash mathbf\; =\; \backslash mu\_0\backslash mathbf\; +\; \backslash mu\_0\; \backslash epsilon\_0\; \backslash frac\backslash$ |} \times \mathbf\, |3= The time derivative can be rewritten to something that can be interpreted physically, namely the Poynting vector. Using the product rule and Faraday's law of induction gives :$\backslash frac\; (\backslash mathbf\backslash times\backslash mathbf)\; =\; \backslash frac\backslash times\; \backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash frac\; =\; \backslash frac\backslash times\; \backslash mathbf\; -\; \backslash mathbf\; \backslash times\; (\backslash boldsymbol\backslash times\; \backslash mathbf)\backslash ,$ and we can now rewrite f as :$\backslash mathbf\; =\; \backslash epsilon\_0\; \backslash left(\backslash boldsymbol\backslash cdot\; \backslash mathbf\; \backslash right)\backslash mathbf\; +\; \backslash frac\; \backslash left(\backslash boldsymbol\backslash times\; \backslash mathbf\; \backslash right)\; \backslash times\; \backslash mathbf\; -\; \backslash epsilon\_0\; \backslash frac\backslash left(\; \backslash mathbf\backslash times\; \backslash mathbf\backslash right)\; -\; \backslash epsilon\_0\; \backslash mathbf\; \backslash times\; (\backslash boldsymbol\backslash times\; \backslash mathbf),$ then collecting terms with E and B gives :$\backslash mathbf\; =\; \backslash epsilon\_0\backslash left((\backslash boldsymbol\backslash cdot\; \backslash mathbf\; )\backslash mathbf\; -\; \backslash mathbf\; \backslash times\; (\backslash boldsymbol\backslash times\; \backslash mathbf)\; \backslash right)\; +\; \backslash frac\; \backslash left(-\; \backslash mathbf\backslash times\backslash left(\backslash boldsymbol\backslash times\; \backslash mathbf\; \backslash right)\; \backslash right)$ - \epsilon_0\frac\left( \mathbf\times \mathbf\right). |4= A term seems to be "missing" from the symmetry in E and B, which can be achieved by inserting (∇ • B)B because of Gauss' law for magnetism: :$\backslash mathbf\; =\; \backslash epsilon\_0\backslash left((\backslash boldsymbol\backslash cdot\; \backslash mathbf\; )\backslash mathbf\; -\; \backslash mathbf\; \backslash times\; (\backslash boldsymbol\backslash times\; \backslash mathbf)\; \backslash right)\; +\; \backslash frac\; \backslash left(\backslash mathbf\; )\backslash mathbf\; -\; \backslash mathbf\backslash times\backslash left(\backslash boldsymbol\backslash times\; \backslash mathbf\; \backslash right)\; \backslash right)$ - \epsilon_0\frac\left( \mathbf\times \mathbf\right). Eliminating the curls (which are fairly complicated to calculate), using the vector calculus identity :$\backslash tfrac\; \backslash boldsymbol\; (\backslash mathbf\backslash cdot\backslash mathbf)\; =\; \backslash mathbf\; \backslash times\; (\backslash boldsymbol\; \backslash times\; \backslash mathbf)\; +\; (\backslash mathbf\; \backslash cdot\; \backslash boldsymbol)\; \backslash mathbf,$ leads to: :$\backslash mathbf\; =\; \backslash epsilon\_0\backslash left((\backslash boldsymbol\backslash cdot\; \backslash mathbf\; )\backslash mathbf\; +\; (\backslash mathbf\backslash cdot\backslash boldsymbol)\; \backslash mathbf\; \backslash right)\; +\; \backslash frac\; \backslash left(\backslash mathbf\; )\backslash mathbf\; +\; (\backslash mathbf\backslash cdot\backslash boldsymbol)\; \backslash mathbf\; \backslash right)\; -\; \backslash frac\; \backslash boldsymbol\backslash left(\backslash epsilon\_0\; E^2\; +\; \backslash frac\; B^2\; \backslash right)$ - \epsilon_0\frac\left( \mathbf\times \mathbf\right). |5= This expression contains every aspect of electromagnetism and momentum and is relatively easy to compute. It can be written more compactly by introducing the Maxwell stress tensor, :$\backslash sigma\_\; \backslash equiv\; \backslash epsilon\_0\; \backslash left(E\_i\; E\_j\; -\; \backslash frac\; \backslash delta\_\; E^2\backslash right)\; +\; \backslash frac\; \backslash left(B\_i\; B\_j\; -\; \backslash frac\; \backslash delta\_\; B^2\backslash right),$ and notice that all but the last term of the above can be written as the divergence of this: :$\backslash mathbf\; +\; \backslash epsilon\_0\; \backslash mu\_0\; \backslash frac\backslash ,\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf$, As in the Poynting's theorem, the second term in the left side of above equation can be interpreted as time derivative of EM field's momentum density and this way, the above equation will be the law of conservation of momentum in classical electrodynamics. where we have finally introduced the Poynting vector, :$\backslash mathbf\; =\; \backslash frac\backslash mathbf\backslash times\backslash mathbf.$ }} in the above relation for conservation of momentum, $\backslash nabla\; \backslash cdot\; \backslash mathbf$ is the momentum flux density and plays a role similar to $\backslash mathbf$ in Poynting's theorem. 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「Maxwell stress tensor」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. 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