# Page:AbrahamMinkowski1.djvu/26

this expression of the electromagnetic force of inertia deviates from our value (58a), according to (4), by

${\displaystyle ({\mathfrak {g}}\nabla ){\mathfrak {w}}}$

Thus the difference between the force expression of E. Cohn and the one obtained here, altogether amounts to

${\displaystyle 2({\mathfrak {g}}\nabla ){\mathfrak {w}}+{\mathfrak {w}}\mathrm {div} {\mathfrak {g}}}$

where ${\displaystyle {\mathfrak {g}}}$ is determined by (27). It is probably too small, to be accessible to experimental tests.

We now pass to the theory of Minkowski. It was already mentioned in § 9, that the close relationship between momentum density and energy current, which takes place in this theory according to the results of the present investigation, is not assumed in Minkowski's approach. Consequently, also value (60) of the ponderomotive force deviates from Minkowski's approach; especially the term (60) is missing there, which already comes into play in the case of rest. It was already alluded to by A. Einstein and J. Laub[1], that the force which according to Lorentz shall act in the magnetic field upon the polarization stream, is missing in Minkowski's approach. Now, no experimental confirmation for the existence of this force was provided, however, the conviction of its existence is based on the analogy which exists between conduction current and polarization current according to the concepts of the theory of electrons; this analogy is so useful, that one won't deny that force without weighty reasons. Our force expressions, as it can be see from eq. 63, contains that force; that it doesn't contradict the principle of relativity, was noticed by us at the end of § 9.

In the case of rest, where ${\displaystyle {\mathfrak {E,H}}}$ has to be written instead of ${\displaystyle {\mathfrak {E',H'}}}$, the ponderomotive force becomes

 (61) ${\displaystyle {\mathfrak {K}}={\mathfrak {E}}\rho +[{\mathfrak {iB}}]-{\frac {1}{2}}{\mathfrak {E}}^{2}\nabla \epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla \mu +{\frac {\partial {\mathfrak {W}}}{\partial l}}}$

Vector ${\displaystyle {\mathfrak {W}}}$ has the following values in the different theories:

A) Theory of H. Hertz.

Here, it follows from (22) and (24)

 (61a) ${\displaystyle {\mathfrak {W}}=[{\mathfrak {DB}}]=\epsilon \mu [{\mathfrak {EH}}]}$

B) Theories of E. Cohn, H. A. Lorentz and H. Minkowski.

In all three theories, as it follows from (27), (32), (40a), it becomes

 (61b) ${\displaystyle {\begin{array}{c}cg=[EH]\\{\mathfrak {W=[DB]-[EH]}}=(\epsilon \mu -1)[{\mathfrak {EH}}]\end{array}}}$
1. A. Einstein and J. Laub, Über die im elektromagnetischen Felde auf ruhende Körper ausgeübten ponderomotorischen Kräfte [Annalen der Physik, Bd. XXVI (1908), pp. 541-550].