Page:AbrahamMinkowski1.djvu/9

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§ 5. Determination of momentum density and energy density.


The various theories of electrodynamics of moving bodies, are differing by the relations assumed between the four vectors \mathfrak{E'H'DB} arising in the main equations. However, before we pass to the discussion of special theories, we want to pursue the general developments; there, only a quite general presupposition shall be made about the form of these relations: The vectors \mathfrak{E'H'DB} shall be connected by equations, which namely contain the velocity vector \mathfrak{w} itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

\mathrm{div}\mathfrak{S}'=c\{\mathfrak{H}'\mathrm{curl}\mathfrak{E}'-\mathfrak{E}'\mathrm{curl}\mathfrak{H}'\}

this becomes with respect to the two first main-equations:

\mathfrak{JE}'+\mathrm{div}\mathfrak{S}'=-\mathfrak{E}'\frac{\partial'\mathfrak{D}}{\partial t}-\mathfrak{H}'\frac{\partial'\mathfrak{B}}{\partial t}

From main equation (III) and relation (14) it follows:

(14a) \frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}-P'=\mathfrak{E}'\frac{\partial'\mathfrak{D}}{\partial t}+\mathfrak{H}'\frac{\partial'\mathfrak{B}}{\partial t}

a condition, which one can also be written in accordance with (4):

(14b) \frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}-P'=\mathfrak{E}'\frac{\delta\mathfrak{D}}{\delta t}+\mathfrak{H}'\frac{\delta\mathfrak{B}}{\delta t}-\mathfrak{E}'(\mathfrak{D}\nabla)\mathfrak{w}-\mathfrak{H}'(\mathfrak{B}\nabla)\mathfrak{w}

and which finally, by using main equation (V), passes into:

(15) \frac{\delta\psi}{\delta t}-\mathfrak{w}\frac{\delta\mathfrak{g}}{\delta t}=\mathfrak{E}'\frac{\delta\mathfrak{D}}{\delta t}+\mathfrak{H}'\frac{\delta\mathfrak{B}}{\delta t}-\frac{1}{2}[\mathfrak{E'D+H'B}\}\mathrm{div}\mathfrak{w}

This relation serves to determine the densities of energy and momentum in their dependence from the electromagnetic vectors.

They read with respect to (2a):

(15a) \dot{\psi}=\mathfrak{w\dot{g}}+(\psi-\mathfrak{wg})\mathrm{div}\mathfrak{w}=\mathfrak{E'\dot{D}+H'\dot{B}}+\frac{1}{2}[\mathfrak{E'D+H'B}\}\mathrm{div}\mathfrak{w}

Since the way of time differentiation now employed, is satisfying the calculation rules, it follows when it is set for brevity's sake:

(16) \psi-\mathfrak{wg}=\varphi
(17) \dot{\varphi}+\mathfrak{g\dot{w}}-\mathfrak{E'\dot{D}-H'\dot{B}}+\left\{ \varphi-\frac{1}{2}\mathfrak{E'D}-\frac{1}{2}\mathfrak{H'B}\right\} \mathrm{div}\mathfrak{w}=0

As mentioned in the beginning of the paragraph, the relations which connect \mathfrak{D,B} with \mathfrak{E'H'} shall contain the velocity vector \mathfrak{w}, though not its derivative with respect to time and space. The same is to be demanded from the expressions,