Page:Zur Elektrodynamik bewegter Systeme II.djvu/2

$\int \limits _{\circ }{\mathsf {M}}_{s}ds={\frac {d}{dt}}\int {\mathfrak {E}}_{N}dS+\int \Lambda _{N}dS$

II

${\begin{cases}{\mathfrak {E}}=\epsilon {\mathsf {E}}-[u{\mathsf {M}}]\\\\{\mathfrak {M}}=\mu {\mathsf {M}}+[u{\mathsf {E}}]\\\\\Lambda =\lambda ({\mathsf {E-K}})\end{cases}}$

III

$\Sigma ={\mathsf {EM}}$

IV

$w={\frac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}})+{\frac {1}{2}}({\mathsf {M}}\cdot {\mathfrak {M}})+(u\cdot \Sigma )$

V

Here, E and M denote the two field intensities;

$\epsilon ,\mu ,\lambda$ scalar constants, K a constant vector;

$u$ the velocity of matter;

$\Sigma$ the radiation relative to matter;

$w$ the electromagnetic energy in unit volume;

$S$ a surface which continuously goes through the same material particle, $s$ its boundary curve, $N$ the perpendicular of $dS$.

In vacuum, the values apply:

$u=0,\ \lambda =0,\ \epsilon =1,\ \mu =1.$

By that it is said, that the speed of light in vacuum is chosen as unity. In the preceding, the entirety of our presuppositions is contained. The equations claim applicability for arbitrary velocities $u$ as far as Maxwell's equations apply for $u=0$.

From our equations it follows, for example, that in case $u=0$, the radiation is propagating with the same velocity in all directions. Thus they presuppose a reference system to which this actually applies. That such a references system is existing with respect to the fixed stars, is without question. To what extent it is defined by our equations, shall be examined later.

Equations I and II, related to the unit of area (supposed as infinitely small), shall be written:

$-{\mathsf {P(E)}}={\frac {\overline {d{\mathfrak {M}}}}{dt}}$

I'

${\mathsf {P(M)}}={\frac {\overline {d{\mathfrak {E}}}}{dt}}+\Lambda$.

II'

Consequently, the meaning of the newly introduced symbols is:

${\frac {\overline {dA}}{dt}}={\frac {dA}{dt}}+\Gamma (u)A-(A\cdot \nabla )u$

(1)

$={\frac {\partial A}{\partial t}}+\Gamma (A)u-{\mathsf {P}}[uA)$

(2)