Page:Das Relativitätsprinzip und seine Anwendung.djvu/11

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:

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