Page:Grundgleichungen (Minkowski).djvu/25

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Now by putting \mathfrak{H}=\mathfrak{B}, the differential equation (29) is transformed into the same form as eq. (1) here when \mathfrak{m}-[\mathfrak{we}]=\mathfrak{M}-[\mathfrak{wE}]. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relativity postulate.

If we make use of (30) for non-magnetic bodies, and put accordingly \mathfrak{H}=\mathfrak{B}+[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}], then in consequence of (C) in §8,

(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])=\mathfrak{D}-\mathfrak{E}+(\mathfrak{w}[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}])

,

i.e. for the direction of \mathfrak{w}

(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{w}}=(\mathfrak{D}-\mathfrak{E})_{\mathfrak{w}},

and for a perpendicular direction \mathfrak{\bar{w}},

(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{\bar{w}}}=(1-\mathfrak{w}^{2})(\mathfrak{D}-\mathfrak{E})_{\mathfrak{\bar{w}}},

i.e. it coincides with Lorentz's assumption, if we neglect \mathfrak{w}^2 in comparison to 1.

Also to the same order of approximation, Lorentz's form for \mathfrak{F} corresponds to the conditions imposed by the relativity principle [comp. (E) § 8] — that the components of \mathfrak{F_{w}}, \mathfrak{F_{\bar{w}}} are equal to the components of \sigma(\mathfrak{E}+(\mathfrak{wB}]) multiplied by \sqrt{1-\mathfrak{w}^{2}} or \frac{1}{\sqrt{1-\mathfrak{w}^{2}}} respectively.

§ 10. Fundamental Equations of E. Cohn.

E. Cohn[1] assumes the following fundamental equations

(31) \begin{array}{c}
curl\ (M+[\mathfrak{wE}])=\frac{\partial\mathfrak{E}}{\partial t}+\mathfrak{w}\ div\ \mathfrak{E}+\mathfrak{F},\\
\\-curl\ (\mathfrak{E}-[\mathfrak{wM}])=\frac{\partial\mathfrak{M}}{\partial t}+\mathfrak{w}\ div\ \mathfrak{M},\end{array}
(32) \mathfrak{F}=\sigma E,\ \mathfrak{E}=\epsilon E-[\mathfrak{w}M],\ \mathfrak{M}=\mu M+[\mathfrak{w}E],

where E, M are the electric and magnetic field intensities (forces), \mathfrak{E,M} are the electric and magnetic polarisation (induction).

  1. Gött. Nachr. 1901, w. 74 (also in Ann. d. Phys. 7 (4), 1902, p. 29).