Page:AbrahamMinkowski1.djvu/25

If one now considers, that the calculation of the ponderomotive force is based on the presuppositions

${\displaystyle {\begin{array}{l}{\frac {1}{c}}{\dot {\epsilon }}={\frac {\partial \epsilon }{\partial l}}+({\mathfrak {q}}\nabla )\epsilon =0,\\\\{\frac {1}{c}}{\dot {\mu }}={\frac {\partial \mu }{\partial l}}+({\mathfrak {q}}\nabla )\mu =0,\end{array}}}$

and that from (3) and the known calculation rule

${\displaystyle \nabla ({\mathfrak {qW}})=({\mathfrak {q}}\nabla ){\mathfrak {W}}+[\mathrm {{\mathfrak {q}}curl} {\mathfrak {W}}]+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}],}$

it is given

${\displaystyle {\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]={\frac {\partial {\mathfrak {W}}}{\partial l}}+{\mathfrak {W}}\mathrm {div} {\mathfrak {q}}+\nabla ({\mathfrak {qW}})-[\mathrm {{\mathfrak {q}}curl} {\mathfrak {W}}]}$

and that it furthermore follows with respect to (3a)

${\displaystyle {\begin{array}{c}{\mathfrak {q}}\left\{{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]\right\}\\\\=-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\partial ({\mathfrak {qW}})}{\partial l}}+\mathrm {div} {\mathfrak {q}}({\mathfrak {qW}})=-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\delta ({\mathfrak {qW}})}{\delta l}}\end{array}}}$

then one finally obtains, for the energy which is emanated in unit time by unit volume, the formula

 (60a) ${\displaystyle q+{\mathfrak {qK}}=\{i+\rho {\mathfrak {q}}\}\left\{{\mathfrak {E}}'-[{\mathfrak {qB}}]\right\}+\zeta {\frac {\partial \epsilon }{\partial l}}+\eta {\frac {\partial \mu }{\partial l}}-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\delta ({\mathfrak {qW}})}{\delta l}}}$

Also here, neglecting the meaning (which somewhat deviates for magnitudes of second order) of ${\displaystyle \zeta }$ and ${\displaystyle \eta }$, the different theories only differ by the value of the vector ${\displaystyle {\mathfrak {W}}}$, when considered from the standpoint of our system.

Now one imagines, that the value is set every time for ${\displaystyle {\mathfrak {W}}}$, which it has in the relevant theory, and then compare our expression (60) of ponderomotive force with the one obtained by other authors.

The value of the ponderomotive force given by E. Cohn is slightly deviating from our value. This partially stems from the fact, that E. Cohn's approach for the relative stresses is not entirely identical with (Va); he namely sets at this place ${\displaystyle \epsilon {\mathfrak {E}}'}$ instead of ${\displaystyle {\mathfrak {D}}}$, probably with the intention to remove the torque ${\displaystyle {\mathfrak {R'}}}$ of the relative stresses. The difference in the value of the force contribution stemming from the relative stresses, which is caused by that, can immediately be found

${\displaystyle ({\mathfrak {g}}\nabla ){\mathfrak {w}}+{\mathfrak {w}}\mathrm {div} {\mathfrak {g}}}$

Only then we have seen the vanishing of ${\displaystyle {\mathfrak {R'}}}$ as necessary, when – as in the theory of Hertz – no electromagnetic momentum comes into play. E. Cohn, however, also accounts for the second part of the force which is connected with vector ${\displaystyle {\mathfrak {g}}}$, namely

${\displaystyle {\mathfrak {K}}_{2}=-{\frac {\partial '{\mathfrak {g}}}{\partial t}}}$