Page:AbrahamMinkowski1.djvu/27

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)

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