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:

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