Page:AbrahamMinkowski1.djvu/6

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then one obtains another form of the momentum- and energy theorems

(8) \begin{cases}
\mathfrak{K}_{x}=\frac{\partial X'_{x}}{\partial x}+\frac{\partial X'_{y}}{\partial y}+\frac{\partial X'_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{x}}{\partial t},\\
\\\mathfrak{K}_{y}=\frac{\partial Y'_{x}}{\partial x}+\frac{\partial Y'_{y}}{\partial y}+\frac{\partial Y'_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{y}}{\partial t},\\
\\\mathfrak{K}_{z}=\frac{\partial Z'_{x}}{\partial x}+\frac{\partial Z'_{y}}{\partial y}+\frac{\partial Z'_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{z}}{\partial t}.\end{cases}
(9) \mathfrak{wK}+Q=-\mathrm{div}\{\mathfrak{S}-\mathfrak{w}\psi\}-\frac{\delta\psi}{\delta t}

Here, the vector

\mathfrak{S}-\mathfrak{w}\psi

represents the "relative energy current". The system of "relative stresses"

(10) \left\{ \begin{array}{ccccc}
X'_{x}=X_{x}+\mathfrak{w}_{x}\mathfrak{g}_{x}, &  & X'_{y}=X_{y}+\mathfrak{w}_{y}\mathfrak{g}_{x}, &  & X'_{z}=X_{z}+\mathfrak{w}_{z}\mathfrak{g}_{x},\\
Y'_{x}=Y_{x}+\mathfrak{w}_{x}\mathfrak{g}_{y}, &  & Y'_{y}=Y_{y}+\mathfrak{w}_{y}\mathfrak{g}_{y}, &  & Y'_{z}=Y_{z}+\mathfrak{w}_{z}\mathfrak{g}_{y},\\
Z'_{x}=Z_{x}+\mathfrak{w}_{x}\mathfrak{g}_{z}, &  & Z'_{y}=Z_{y}+\mathfrak{w}_{y}\mathfrak{g}_{z}, &  & Z'_{z}=Z_{z}+\mathfrak{w}_{z}\mathfrak{g}_{z},\end{array}\right.

is so defined, that (6) and (8) lead to the equal values of the ponderomotive force.

From (6a) and (10), the relations follow

\begin{array}{l}
Y'_{x}-X'_{y}=\mathfrak{w}_{x}\mathfrak{g}_{y}-\mathfrak{w}_{y}\mathfrak{g}_{x},\\
Z'_{y}-Y'_{z}=\mathfrak{w}_{y}\mathfrak{g}_{z}-\mathfrak{w}_{z}\mathfrak{g}_{y},\\
X'_{z}-Z'_{x}=\mathfrak{w}_{z}\mathfrak{g}_{x}-\mathfrak{w}_{x}\mathfrak{g}_{z},\end{array}

which can be written vectorially

(11) \mathfrak{R}'=[\mathfrak{wg}]

\mathfrak{R}' is the unit volume related to the torque of relative stresses; it vanishes in ordinary mechanics, since the momentum vector is in agreement with the velocity vector in terms of direction. In electromagnetic mechanics it is not to be neglected in general, but it will be compensated (when related to a fixed point of moment) by that torque which stems from the entrained momentum.

We can imagine the relative energy current as being decomposed into two parts, one of them represents the energy transfer caused by the relative stresses, and the other one the "relative radiation"[1], which in optics, for example, can be measured by heat production in a black surface:

(12) \begin{cases}
\mathfrak{S}_{x}-\mathfrak{m}_{x}\psi=\mathfrak{S}'_{x}-\left\{ \mathfrak{w}_{x}X'_{x}+\mathfrak{w}_{y}Y'_{x}+\mathfrak{w}_{z}Z'_{x}\right\} ,\\
\mathfrak{S}_{y}-\mathfrak{m}_{y}\psi=\mathfrak{S}'_{y}-\left\{ \mathfrak{w}_{x}X'_{y}+\mathfrak{w}_{y}Y'_{y}+\mathfrak{w}_{z}Z'_{y}\right\} ,\\
\mathfrak{S}_{z}-\mathfrak{m}_{z}\psi=\mathfrak{S}'_{z}-\left\{ \mathfrak{w}_{x}X'_{z}+\mathfrak{w}_{y}Y'_{z}+\mathfrak{w}_{z}Z'_{z}\right\} .\end{cases}

Vector \mathfrak{S}' we call the "relative ray".

  1. M. Abraham, l. c. 8), p. 324.