Page:AbrahamMinkowski1.djvu/22

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quantities \zeta,\ \eta have a somewhat different meaning here

(54b) \begin{cases}
\zeta=\frac{1}{2}\left\{ \mathfrak{E}_{x}^{'2}+k^{-2}\left(\mathfrak{E}_{y}^{'2}+\mathfrak{E}_{z}^{'2}\right)\right\} ,\\
\eta=\frac{1}{2}\left\{ \mathfrak{H}_{x}^{'2}+k^{-2}\left(\mathfrak{H}_{y}^{'2}+\mathfrak{H}_{z}^{'2}\right)\right\} ,\end{cases}

This result also holds for Lorentz's theory in the form which we gave to it in § 10; because all expressions, which only contain the vectors \mathfrak{E'H'DB}, are in this theory identical with the corresponding expressions of Minkowski's theory.

Now, since equation (54) contradicts relation (18), and since we won't allow a change in the values of momentum density and energy density, we consider it necessary to correct the value of quantity P' given in (V), namely by

-\zeta\dot{\epsilon}-\eta\dot{\mu}

then the considerations of § 5 indeed exactly lead to relation (54) instead of relation (18).

This view finds support in the theory of electrostriction[1]. In the simplest case present at fluids and gases, where \epsilon and \mu only depend on the density \sigma, one has

-\zeta\dot{\epsilon}-\eta\dot{\mu}=-\dot{\sigma}\left\{ \zeta\frac{d\epsilon}{d\sigma}+\eta\frac{d\mu}{d\sigma}\right\}

this becomes in consequence of the continuity condition of matter

-\zeta\dot{\epsilon}-\eta\dot{\mu}=\mathrm{div}\mathfrak{w}\left\{ \zeta\sigma\frac{d\epsilon}{d\sigma}+\eta\sigma\frac{d\mu}{d\sigma}\right\}

If one considers definition (13) of quantity P', then one sees that this increase corresponds to a growth of relative normal stresses X'_{x},\ Y'_{y},\ Z'_{z} by

(55) -p'=\zeta\sigma\frac{d\epsilon}{d\sigma}+\eta\sigma\frac{d\mu}{d\sigma}.

In case \epsilon and \mu are increasing with growing density, the additional pressure p' becomes negative, i.e. the fluid tends to contract in the electric and magnetic field. In the case of rest, (55) together with (54a) and (54b) gives the ordinary approach for the theory of electrostriction.

At solid bodies, more general considerations are necessary in order to represent the dependence of electric and magnetic constants from the state of deformation. H. Hertz[2] has generally calculated the corresponding supplementary stresses from the standpoint of his theory; on the other hand, E. Cohn as well as H. Minkowski didn't introduce such supplementary stresses. This simplification, which is allowed in the light of the insignificance of these supplementary stresses, we will still allow to ourselves too.

  1. F. Pockels [Enzyklopädie der mathematischen Wissenschaften, Vol. V, 2, Article 16, Nr. 4].
  2. H. Hertz, Über die Grundgleichungen der Elektrodynamik für bewegte Körper [Gesammelte Werke, Bd. II, pp. 256-285], p. 280.