Page:AbrahamMinkowski1.djvu/28

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These formulas for the electric and magnetic force contribution which were given here from (61) and (61b), may be compared with the approaches made by A. Einstein and J. Laub[1] for the ponderomotive force in resting bodies. The first three terms in \mathfrak{K}_{e} and \mathfrak{K}_{m} can also be found there; the first terms are interpreted as the forces of the field upon electrically and magnetically polarized volume elements, and the vector product of i and \mathfrak{H} as the force of the magnetic field upon the electric conduction currents; the previously mentioned force of the magnetic field upon the electric polarization stream is added to it, and the force corresponding to it, which acts upon the magnetic polarization stream in the electric field. Though the two last terms of expressions (63) are missing in the approach of the mentioned authors, which is connected with the fact, that their values of the fictitious normal stresses are somewhat deviating from the ones that are assumed otherwise. If it is about a force upon a section, at whose boundary \mathfrak{P} and \mathfrak{M} are zero, then those two terms are dropping; because the surface integrals provided by them, vanishes.

In this case which happens not seldom, one may employ the approach of Einstein and Laub. Though the consequence drawn by those authors, namely that vector \mathfrak{B} is not decisive for the force upon the conduction currents, seems not correct to me. Because we have seen, that it is just vector \mathfrak{B} in (61), that determines the force upon the conductor. However, the force acting in magnetic fields upon a conducting and at the same time magnetized wire, is not at all to be calculated from neither the vector product of i and \mathfrak{B}, nor from that of i and \mathfrak{H}; one rather has to add the force -\frac{1}{2}\mathfrak{H}^{2}\nabla\mu to the vector product, which is located at the transition layer between wire and air, while the force (\mathfrak{M}\nabla)\mathfrak{H} is added, which acts in the magnetized wire, in case the field is not coincidentally homogeneous there. Neglecting this very special case, the force acting at the magnetized volume elements of a homogeneous conducting wire, is to be set equal to the vector product of i and \mathfrak{B}, though not to that of i and \mathfrak{H}.

Incidentally, formulas (63) in no way have such a principal importance, as the initial expressions (61) for the ponderomotive force. The latter was given from the system of electrodynamics which also includes moving bodies; yet the former could hardly be so generalized, in order to determine the ponderomotive force in moving bodies as well.

Ospedaletti Ligure, December 1908.

Max Abraham.

  1. A. Einstein and J. Laub, l. c. 15), p. 549.