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):

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

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

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

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

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

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

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

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

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

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