Page:AbrahamMinkowski1.djvu/19

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and for the transverse components

(47b) \begin{cases}
k^{2}\mathfrak{D}_{y}= & \left(k^{2}\epsilon_{y}+[\mathfrak{q}]^{2}\right)\mathfrak{E}'_{y}-[\mathfrak{qH}']_{y},\\
k^{2}\mathfrak{B}_{y}= & \left(k^{2}\mu_{y}+[\mathfrak{q}]^{2}\right)\mathfrak{H}'_{y}+[\mathfrak{qE}']_{y}.\end{cases}

If we compare (45a) and (47a) on one hand, and on the other hand (45b) and (47b), then we see that the equations that connect \mathfrak{D,B} and \mathfrak{E',H'}, are in agreement in both theories, when one sets in Lorentz's theory

(48a) \epsilon_{x}=\epsilon,\ \mu_{x}=\mu;
(48b) \epsilon_{y}-1=k^{-2}(\epsilon-1),\ \mu_{y}-1=k^{-2}(\mu-1)

Then, the longitudinal and transverse components of the electric and magnetic polarization, become according to (31)

(48c) \mathfrak{P}_{x}=(\epsilon-1)\mathfrak{E}'_{x},\ \mathfrak{P}_{y}=k^{-2}(\epsilon-1)\mathfrak{E}'_{y},\ \mathfrak{P}_{z}=k^{-2}(\epsilon-1)\mathfrak{E}'_{z};
(48d) \mathfrak{M}_{x}=(\mu-1)\mathfrak{H}'_{x},\ \mathfrak{M}_{y}=k^{-2}(\mu-1)\mathfrak{H}'_{y},\ \mathfrak{M}_{z}=k^{-2}(\mu-1)\mathfrak{H}'_{z};

That the electric polarization of a body being isotropic when at rest, must be influenced by its motion in the way indicated by (48c) – in case the relativity postulate should be compatible with Lorentz's theory – has already been spoken out by Lorentz. If one assumes the symmetry of electric and magnetic vectors, then the corresponding behavior is given for the magnetic polarization.

The presupposition made in § 8, that \epsilon and \mu should be independent from velocity, has become irrelevant now. Thus the values for momentum density, energy density and energy current, which were found there, must be corrected. The quantities included in (31)

(49) \begin{cases}
\mathfrak{E'\dot{P}-P\dot{E}'}=2\mathfrak{E'\dot{P}}-\frac{d}{dt}(\mathfrak{E'P}),\\
\\\mathfrak{H'\dot{M}-M\dot{H}'}=2\mathfrak{H'\dot{M}}-\frac{d}{dt}(\mathfrak{H'M}).\end{cases}

are not to be neglected any more. It follows from (48c)

(49a) \mathfrak{E'P}=(\epsilon-1)\left\{ \mathfrak{E}_{x}^{'2}+k^{-2}\left(\mathfrak{E}_{y}^{'2}+\mathfrak{E}_{z}^{'2}\right)\right\} .

Furthermore one obtains, under consideration of the transverse acceleration and rotation of the polarization ellipsoid caused by it, the expressions for the components of \mathfrak{\dot{P}}

(49b) \begin{cases}
\mathfrak{\dot{P}}_{x}=(\epsilon-1)\mathfrak{\dot{E}}'_{x}-\frac{\mathfrak{\dot{q}}_{y}}{|\mathfrak{q}|}\mathfrak{P}_{y}-\frac{\mathfrak{\dot{q}}_{z}}{|\mathfrak{q}|}\mathfrak{P}_{z},\\
\\\mathfrak{\dot{P}}_{y}=k^{-2}(\epsilon-1)\mathfrak{\dot{E}}'_{y}+2\mathfrak{\dot{q}}_{x}|\mathfrak{q}|k^{-2}\mathfrak{P}_{y}-\frac{\mathfrak{\dot{q}}_{y}}{|\mathfrak{q}|}\mathfrak{P}_{x},\\
\\\mathfrak{\dot{P}}_{z}=k^{-2}(\epsilon-1)\mathfrak{\dot{E}}'_{z}+2\mathfrak{\dot{q}}_{x}|\mathfrak{q}|k^{-2}\mathfrak{P}_{z}+\frac{\mathfrak{\dot{q}}_{z}}{|\mathfrak{q}|}\mathfrak{P}_{x},\end{cases}.