Page:Das Relativitätsprinzip und seine Anwendung.djvu/11

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In this way, one is led to the following equations (which are in agreement with those of ordinary Maxwellian theory):

\begin{array}{l} \operatorname{div}\ \mathfrak{D}=\varrho_{l}{,}\\ \operatorname{div}\ \mathfrak{B}=0{,}\\ \operatorname{rot}\ \mathfrak{H}=\frac{1}{c}(\mathfrak{C}+\mathfrak{\dot{D}}){,}\\ \operatorname{rot}\ \mathfrak{E}=-\frac{1}{c}\mathfrak{\dot{B}}. \end{array}

Herein, \mathfrak{D} is the dielectric displacement, \mathfrak{B} the magnetic induction, \mathfrak{H} the magnetic force, \mathfrak{E} the electric force, \mathfrak{C} the electric current, \varrho_{l} the density of the observable electric charges. If one indicates the average formation by overlines, then it is e.g.

\mathfrak{E}=\mathfrak{\bar{d}}, \mathfrak{B}=\mathfrak{\bar{h}}{,}

where \mathfrak{d}, \mathfrak{h} have the earlier meaning; furthermore it is

\begin{array}{l} \mathfrak{D}=\mathfrak{E}+\mathfrak{P}{,}\\ \mathfrak{H}=\mathfrak{B}-\mathfrak{M}-\frac{1}{c}[\mathfrak{P}\cdot\mathfrak{w}]{,} \end{array}

where \mathfrak{P} is the electric moment, \mathfrak{M} the magnetization per unit volume, and \mathfrak{w} the velocity of matter. In the derivation of these formulas, one separates the electrons into three kinds. The first kind, the polarization electrons, produce the electric moment \mathfrak{P} by their displacement; the second kind, the magnetization electrons, produce the magnetic moment \mathfrak{M} by their orbits; the third kind, the conduction electrons, are freely moving in matter and produce the observable charge density \varrho_{l} and the current \mathfrak{C}. The latter is still to be separated into two parts; if \mathfrak{u} is the relative velocity of the electrons towards matter, then the total velocity of the electrons is \mathfrak{v}=\mathfrak{w}+\mathfrak{u}, thus the current transported by them

\mathfrak{C}=\overline{\varrho\mathfrak{v}}=\varrho\mathfrak{w}+\overline{\varrho\mathfrak{u}};

\bar{\varrho} is the observable charge \varrho_{l}, \bar{\varrho}\mathfrak{w} the convection current, \overline{\varrho\mathfrak{u}} the actual conduction current \mathfrak{C}_{l}.

Transformation formulas exist for all these magnitudes, of which some may be given:

\mathfrak{C}'_{x}=\mathfrak{C}_{x},\ \mathfrak{C}'_{y}=\mathfrak{C}_{y},\ \mathfrak{C}'_{z}=a\mathfrak{C}_{z}-bc\varrho_{l},\ \varrho'_{l}=a\varrho_{l}-\frac{b}{c}\mathfrak{C}_{z}{,}

\begin{array}{l} \mathfrak{P}'_{x}=a\mathfrak{P}_{x}-\frac{b}{c}(\mathfrak{w}_{z}\mathfrak{P}_{x}-\mathfrak{w}_{x}\mathfrak{P}_{z})+b\mathfrak{M}_{y}{,}\\ \mathfrak{P}'_{y}=a\mathfrak{P}_{y}-\frac{b}{c}(\mathfrak{w}_{z}\mathfrak{P}_{y}-\mathfrak{w}_{y}\mathfrak{P}_{z})-b\mathfrak{M}_{x}{,}\\ \mathfrak{P}'_{z}=\mathfrak{P}_{z}. \end{array}