# 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:

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