Page:AbrahamMinkowski1.djvu/20

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From that it follows

(49c) \begin{cases}
2\mathfrak{E'\dot{P}} & =2(\epsilon-1)\left\{ \mathfrak{E}'_{x}\mathfrak{\dot{E}}'_{x}+k^{-2}\mathfrak{E}'_{y}\mathfrak{\dot{E}}'_{y}+k^{-2}\mathfrak{E}'_{z}\mathfrak{\dot{E}}'_{z}\right\} \\
\\ & +4\mathfrak{\dot{q}}_{x}|\mathfrak{q}|k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}\right\} \\
\\ & -2\frac{\mathfrak{\dot{q}}_{y}}{|\mathfrak{q}|}\left\{ \mathfrak{E}'_{x}\mathfrak{P}_{y}-\mathfrak{E}'_{y}\mathfrak{P}_{x}\right\} -2\frac{\mathfrak{\dot{q}}_{z}}{|\mathfrak{q}|}\left\{ \mathfrak{E}'_{x}\mathfrak{P}_{z}-\mathfrak{E}'_{z}\mathfrak{P}_{x}\right\} ,\end{cases}

while (49a) gives

(49d) \frac{d}{dt}(\mathfrak{E'P})=2(\epsilon-1)\left\{ \mathfrak{E}'_{x}\mathfrak{\dot{E}}'_{x}+k^{-2}\mathfrak{E}'_{y}\mathfrak{\dot{E}}'_{y}+k^{-2}\mathfrak{E}'_{z}\mathfrak{\dot{E}}'_{z}\right\} +2\mathfrak{\dot{q}}_{x}|\mathfrak{q}|k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}\right\} .

Since one now has according to (48c)

\begin{array}{c}
\mathfrak{E}'_{x}\mathfrak{P}_{y}-\mathfrak{E}'_{y}\mathfrak{P}_{x}=|\mathfrak{q}|^{2}\mathfrak{E}'_{x}\mathfrak{P}_{y},\\
\mathfrak{E}'_{x}\mathfrak{P}_{z}-\mathfrak{E}'_{z}\mathfrak{P}_{x}=|\mathfrak{q}|^{2}\mathfrak{E}'_{x}\mathfrak{P}_{z},\end{array}

thus it is given

(49c) \begin{cases}
\mathfrak{E'\dot{P}-\dot{P}E'} & =2\mathfrak{\dot{q}}_{x}|\mathfrak{q}|k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}\right\} \\
 & -2\mathfrak{\dot{q}}_{y}|\mathfrak{q}|\mathfrak{E}'_{x}\mathfrak{P}_{y}-2\mathfrak{\dot{q}}_{z}|\mathfrak{q}|\mathfrak{E}'_{x}\mathfrak{P}_{z}.\end{cases}

The insertion of this expression and the corresponding magnetic term into (31), gives (instead of value (32) of momentum density) the corrected value

(50) \begin{cases}
c\mathfrak{g} & =\mathfrak{[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}\\
 & +\mathfrak{q}|\mathfrak{q}|^{2}k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}+\mathfrak{E}'_{y}\mathfrak{M}_{y}+\mathfrak{H}'_{z}\mathfrak{M}_{z}\right\} \end{cases}

That relation (18) is satisfied, can easily be verified.

If the value (50) of c\mathfrak{g} is inserted in the general formula (19) for the energy density, then it follows instead of (33)

(51) \begin{cases}
\psi & =\frac{1}{2}\mathfrak{E}^{2}+\frac{1}{2}\mathfrak{H}^{2}\\
 & +\frac{1}{2}\mathfrak{E'P}+\frac{1}{2}\mathfrak{H'M}+|\mathfrak{q}|^{2}k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}+\mathfrak{E}'_{y}\mathfrak{M}_{y}+\mathfrak{H}'_{z}\mathfrak{M}_{z}\right\} \end{cases}

One also obtains, because of (20), the corrected formula for the energy current

(52) \begin{cases}
\frac{\mathfrak{S}}{c} & =\mathfrak{[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}\\
 & +\mathfrak{q}|\mathfrak{q}|^{2}k^{-2}\left\{ \mathfrak{E}'_{y}\mathfrak{P}_{y}+\mathfrak{E}'_{z}\mathfrak{P}_{z}+\mathfrak{E}'_{y}\mathfrak{M}_{y}+\mathfrak{H}'_{z}\mathfrak{M}_{z}\right\} \end{cases}

From (50) and (52) one can see, that also in Lorentz's theory (when modified in the given way) the relation between the energy current and momentum density exists:

(53) \frac{\mathfrak{S}}{c}=c\mathfrak{g},

which we already encountered in Minkowski's theory.

This result was to be expected; after the equations connecting \mathfrak{D} and \mathfrak{B} with \mathfrak{E'} and \mathfrak{H'}, are brought into agreement, no essential difference exists any more between both theories from the standpoint of our system.