Page:AbrahamMinkowski1.djvu/27

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That all three theories give the same values for the ponderomotive force in resting bodies, is caused (in the sense of our system) by the fact, that the equations connecting \mathfrak{D} and \mathfrak{B} with \mathfrak{E'} and \mathfrak{H'}, are in agreement (including the terms linear in \mathfrak{q}). The notation of Lorentz's theory may be employed in the discussion of the force in resting bodies.

If one sets value (61b) for \mathfrak{W}, then the ponderomotive force (61) can be decomposed into two parts

(62) \begin{cases}
\mathfrak{K}_{e}=\mathfrak{E}\rho-\frac{1}{2}\mathfrak{E}^{2}\nabla\epsilon+(\epsilon\mu-1)\left[\mathfrak{E}\frac{\partial\mathfrak{H}}{\partial l}\right],\\
\\\mathfrak{K}_{m}=[i\mathfrak{B}]-\frac{1}{2}\mathfrak{H}^{2}\nabla\mu+(\epsilon\mu-1)\left[\frac{\partial\mathfrak{E}}{\partial l}\mathfrak{H}\right],\end{cases}

which are to be interpreted as the contributions of the electric and magnetic field.

From the main equations for resting bodies

\begin{array}{l}
\mathrm{curl}\mathit{\mathfrak{H}}=\frac{\partial\mathfrak{D}}{\partial l}+i,\\
\\\mathrm{curl}\mathit{\mathfrak{E}}=-\frac{\partial\mathfrak{B}}{\partial l}\end{array}

one derives, by introduction of the electric and magnetic polarization

\begin{array}{l}
\mathfrak{P=D-E}=(\epsilon-1)\mathfrak{E},\\
\mathfrak{M=B-H}=(\mu-1)\mathfrak{H}\end{array}

the two following relations

\begin{array}{rl}
0= & -[\mathfrak{P}\mathrm{curl}\mathfrak{E}]-\mu(\epsilon-1)\left[\mathfrak{E}\frac{\partial\mathfrak{H}}{\partial l}\right]\\
\\{}[i\mathfrak{B}]= & [i\mathfrak{H}]-[\mathfrak{M}curl\mathfrak{H}]-\epsilon(\mu-1)\left[\frac{\partial\mathfrak{E}}{\partial l}\mathfrak{H}\right]\end{array}

With respect to them, expressions (62) go over into

(62a) \begin{cases}
\mathfrak{K}_{e}=\mathfrak{E}\rho-[\mathfrak{P}\mathrm{curl}\mathfrak{E}]-\frac{1}{2}\mathfrak{E}^{2}\nabla(\epsilon-1)+\left[\mathfrak{E}\frac{\partial\mathfrak{M}}{\partial l}\right],\\
\\\mathfrak{K}_{m}=[i\mathfrak{H}]-[\mathfrak{M}\mathrm{curl}\mathfrak{H}]-\frac{1}{2}\mathfrak{H}^{2}\nabla(\mu-1)+\left[\frac{\partial\mathfrak{P}}{\partial l}\mathfrak{H}\right].\end{cases}

Since it furthermore holds

(63) \begin{array}{l}
\frac{1}{2}(\epsilon-1)\nabla\mathfrak{E}^{2}+\frac{1}{2}\mathfrak{E}^{2}\nabla(\epsilon-1)=\frac{1}{2}\nabla(\epsilon-1)\mathfrak{E}^{2}=\frac{1}{2}\nabla(\mathfrak{PE}),\\
\\\frac{1}{2}(\epsilon-1)\nabla\mathfrak{E}^{2}=(\mathfrak{P}\nabla)\mathfrak{E}+[\mathfrak{P}\mathrm{curl}\mathfrak{E}];\\
\\\frac{1}{2}(\mu-1)\nabla\mathfrak{H}^{2}+\frac{1}{2}\mathfrak{H}^{2}\nabla(\mu-1)=\frac{1}{2}\nabla(\mu-1)\mathfrak{H}^{2}=\frac{1}{2}\nabla(\mathfrak{MH}),\\
\\\frac{1}{2}(\mu-1)\nabla\mathfrak{H}^{2}=(\mathfrak{M}\nabla)\mathfrak{H}+[\mathfrak{M}\mathrm{curl}\mathfrak{H}];\end{array}

it eventually becomes

(63) \begin{cases}
\mathfrak{K}_{e}=(\mathfrak{P}\nabla)\mathfrak{E}+\mathfrak{E}\rho+\left[\mathfrak{E}\frac{\partial\mathfrak{M}}{\partial l}\right]-\frac{1}{2}\nabla(\mathfrak{PE}),\\
\\\mathfrak{K}_{m}=(\mathfrak{M}\nabla)\mathfrak{H}+[i\mathfrak{H}]+\left[\frac{\partial\mathfrak{P}}{\partial l}\mathfrak{H}\right]-\frac{1}{2}\nabla(\mathfrak{MH}).\end{cases}