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):

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

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.

$\mathfrak{E}=\mathfrak{\bar{d}}, \mathfrak{B}=\mathfrak{\bar{h}}{,}$|undefined

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

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

$$\bar{\varrho}$$ 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: