Page:Das Relativitätsprinzip und seine Anwendung.djvu/15

Now we compare this procedure with the inverse case, that the magnet $ab$ is at rests and plates $c,d$ are moving with opposite velocity. Then according to the relativity principle, everything must be quite the same as in the first case. Indeed, one immediately finds from the ordinary law of induction, exactly the mount of charge upon plate $d$ previously given. But now this charge upon $d$ must produce the opposite equal charge upon plane $a$ of the resting magnet by electrostatic induction, and the corresponding must hold for $b$ and $c$. Since no current can flow $({\mathfrak {C}}=0)$, then in both cases (whether the magnet is moving and the plate are at rest, or vice versa) the same charges must be present upon the magnet. Thus we have to consider, as to how it comes that in the first treated case, the opposite charge arises upon plane $a$ of the moving magnet, than upon plate $d$; this is only possible by the polarization $\textstyle {{\mathfrak {P}}_{x}=-{\frac {v}{c}}{\mathfrak {M}}_{y}}$ emerging during the motion. Because one has

${\mathfrak {D}}_{x}={\mathfrak {E}}_{x}+{\mathfrak {P}}_{x}={\frac {v}{c}}{\mathfrak {B}}_{y}-{\frac {v}{c}}{\mathfrak {M}}_{y};$

since ${\mathfrak {P}}$ (thus the term $[{\mathfrak {P}}\cdot {\mathfrak {w}}]$) is to be neglected in the velocity of first order, it becomes

${\mathfrak {B}}-{\mathfrak {M}}={\mathfrak {H}}{,}$

though ${\mathfrak {H}}$ is zero, because the plate is assumed to be infinitely extended. From that if follows

${\mathfrak {D}}_{x}=0{,}$

no dielectric displacement arises in the moving plate, thus the charge upon $a$ agrees with that upon $d$, as required by the relativity principle.

The last remark concerns the circumstance, that the motion of Earth cannot have an influence upon electromagnetic processes according to the relativity principle. However, Liénard alluded to a phenomenon, where such an influence (namely amounting to first order) shall be expected; also Poincaré has discussed this case in his book Electricité et Optique. It is about the ponderomotive force upon an conductor. In order to determine it, one will make the obvious assumption for the force acting upon the conduction electrons per unit force:

${\mathfrak {E}}_{1}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {v}}\cdot {\mathfrak {B}}];$

then the force caused by Earth's motion upon the conductor in the direction of motion, amounts to

${\frac {1}{c^{2}}}({\mathfrak {C}}_{l}\cdot {\mathfrak {E}}){\mathfrak {w}}_{z};$