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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 Ritz's Equation ： ウィキペディア英語版 Ritz's Equation In 1908, Walter Ritz published Recherches critiques sur l'Électrodynamique générale,〔Ritz, Walter (1908), ''Annales de Chimie et de Physique'', Vol. 13, pp. 145-275〕 (See English translation〔(''Critical Researches on General Electrodynamics,'' Introduction and First Part (1980) Robert Fritzius, editor; Second Part (2005) Yefim Bakman, Editor. )〕) a lengthy criticism of Maxwell-Lorentz electromagnetic theory, in which he contended that the theory's connection with the luminiferous aether (see Lorentz ether theory) made it "essentially inappropriate to express the comprehensive laws for the propagation of electrodynamic actions." Ritz proposed a new equation, derived from the principles of the ballistic theory of electromagnetic waves. This equation relates the force between two charged particles with a radial separation r relative velocity v and relative acceleration a, where ''k'' is an undetermined parameter from the general form of Ampere's force law as proposed by Maxwell. This equation obeys Newton's third law and forms the basis of Ritz's electrodynamics. :$\backslash mathbf\; =\; \backslash frac\; \backslash left$ ==Derivation of Ritz's Equation== On the assumption of an emission theory, the force acting between two moving charges should depend on the density of the messenger particles emitted by the charges ($D$), the radial distance between the charges (ρ), the velocity of the emission relative to the receiver, ($U\_x$ and $U\_r$ for the ''x'' and ''r'' components, respectively), and the acceleration of the particles relative to each other ($a\_x$). This gives us an equation of the form: ::$F\_x\; =\; eD\; \backslash left(A\_1\; cos(\backslash rho\; x)+\; B\_1\; \backslash frac\; +\; C\_1\; \backslash frac\; \backslash right)$. where the coefficients $A\_1$, $B\_1$ and $C\_1$ are independent of the coordinate system and are functions of $u^2/c^2$ and $u^2\_\backslash rho\; /c^2$. The stationary coordinates of the observer relate to the moving frame of the charge as follows ::$X+x(t\text{'})\; =\; X\text{'}+x\text{'}(t\text{'})-(t-t\text{'})v\text{'}\_x$ Developing the terms in the force equation, we find that the density of particles is given by ::$D\; \backslash alpha\; \backslash frac\; =\; -\; \backslash frac\; dS\; dn$ The tangent plane of the shell of emitted particles in the stationary coordinate is given by the Jacobian of the transformation from $X\text{'}$ to $X$: ::$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \backslash left(1\; +\backslash frac\; \backslash right)$ We can also develop expressions for the retarded radius $\backslash rho$ and velocity using Taylor series expansions ::$\backslash rho\; =\; r\; \backslash left(1+\backslash frac\backslash right)^$ ::$\backslash rho\_x\; =\; r\_x\; +\; \backslash frac$ ::$U\_\backslash rho\; =\; v\_r\; -\; v\text{'}\_r\; +\backslash frac$ With this substitutions, we find that the force equation is now ::$F\_x\; =\; \backslash frac\; \backslash left(1+\backslash frac\backslash right)\; \backslash left(cos(rx)\; \backslash left(1-\backslash frac\backslash right)\; +\; A\backslash left(\backslash frac\backslash right)-B\backslash left(\backslash frac\backslash right)-C\backslash left(\backslash frac\backslash right)\backslash right)$ Next we develop the series representations of the coefficients ::$A\; =\; \backslash alpha\_0\; +\; \backslash alpha\_1\; \backslash frac\; +\; \backslash alpha\_2\; \backslash frac+...$ ::$B\; =\; \backslash beta\_0\; +\; \backslash beta\_1\; \backslash frac\; +\; \backslash beta\_2\; \backslash frac+...$ ::$C\; =\; \backslash gamma\_0\; +\; \backslash gamma\_1\; \backslash frac\; +\; \backslash gamma\_2\; \backslash frac+...$ With these substitutions, the force equation is now ::$F\_x\; =\; \backslash frac\; \backslash left(\backslash frac+\backslash alpha\_2\; \backslash frac\backslash right)\; cos(rx)\; -\; \backslash beta\_0\; \backslash frac-\backslash alpha\_0\; \backslash frac\; +\; \backslash left(\backslash frac\backslash right)(\backslash alpha\_0-2\backslash gamma\_0)\; \backslash right)$ Since the equation must reduce to the Coulomb force law when the relative velocities are zero, we immediately know that $\backslash alpha\_0\; =\; 1$. Furthermore, to obtain the correct expression for electromagnetic mass, we may deduce that $2\backslash gamma\_0\; -1\; =\; 1$ or $\backslash gamma\_0\; =1$. To determine the other coefficients, we consider the force on a linear circuit using Ritz's expression, and compare the terms with the general form of Ampere's law. The second derivative of Ritz's equation is ::$d^2\; F\_x\; =\; \backslash sum\_\backslash frac\; \backslash left(\backslash frac+\backslash alpha\_2\; \backslash frac\backslash right)\; cos(rx)\; -\; \backslash beta\_0\; \backslash frac-\backslash alpha\_0\; \backslash frac\; +\; \backslash frac\backslash right)$ Consider the diagram on the right, and note that $dq\; v\; =\; I\; dl$, :$\backslash sum\_\; de\_i\; de\_j\text{'}\; =\; 0$ :$\backslash sum\_\; de\_i\; de\_j\text{'}\; u^2\_x\; =\; -2dq\; dq\text{'}\; w\_x\; w\text{'}\_x$ ::::::$=\; -2\; I\; I\text{'}\; ds\; ds\text{'}\; cos\backslash epsilon$ :$\backslash sum\_\; de\_i\; de\_j\text{'}\; u^2\_r\; =\; -2dq\; dq\text{'}\; w\_r\; w\text{'}\_r$ ::::::$=\; -2\; I\; I\text{'}\; ds\; ds\text{'}\; cos(rds)cos(rds)$ :$\backslash sum\_\; de\_i\; de\_j\text{'}\; u\_x\; u\_r\; =\; -dq\; dq\text{'}\; (w\_x\; w\text{'}\_r+w\text{'}\_x\; w\_r)$ ::::::$=\; -I\; I\text{'}\; ds\; ds\text{'}\; \backslash left()$ :$\backslash sum\_\; de\_i\; de\_j\text{'}\; a\text{'}\_r\; =\; 0$ :$\backslash sum\_\; de\_i\; de\_j\text{'}\; a\text{'}\_x\; =\; 0$ Plugging these expressions into Ritz's equation, we obtain the following ::$d^2\; F\_x\; =\; \backslash frac\; \backslash left$ Comparing to the original expression for Ampere's force law ::$d^2\; F\_x\; =\; -\; \backslash frac\; \backslash left$ we obtain the coefficients in Ritz's equation ::$\backslash alpha\_1\; =\; \backslash frac$ ::$\backslash alpha\_2\; =\; -\backslash frac$ ::$\backslash beta\_0\; =\; \backslash frac$ From this we obtain the full expression of Ritz's electrodynamic equation with one unknown :$\backslash mathbf\; =\; \backslash frac\; \backslash left$ In a footnote at the end of Ritz's section on (''Gravitation'' ) (English translation) the editor says, "Ritz used ''k'' = 6.4 to reconcile his formula (to calculate the angle of advancement of perihelion of planets per century) with the observed anomaly for Mercury (41") however recent data give 43.1", which leads to ''k'' = 7. Substituting this result into Ritz’s formula yields exactly the general relativity formula." Using this same integer value for ''k'' in Ritz's electrodynamic equation we get: :$\backslash mathbf\; =\; \backslash frac\; \backslash left$ 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「Ritz's Equation」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.