Page:Electromagnetic phenomena.djvu/2

of electrons, unless it be modified by some new hypothesis, would undoubtedly require the existence of such a couple. In order to see this, it will suffice to consider a condenser with aether as dielectricum. It may be shown that in every electrostatic system, moving with a velocity ${\mathfrak {w}}$,^[1] there is a certain amount of electromagnetic momentum. If we represent this, in direction and magnitude, by a vector ${\mathfrak {G}}$, the couple in question will be determined by the vector product^[2]

$\left[{\mathfrak {G}}.{\mathfrak {w}}\right]$.

(1)

Now, if the axis of z is chosen perpendicular to the condenser plates, the velocity w having any direction we like, and if U is the energy of the condenser, calculated on the ordinary way, the components of ${\mathfrak {G}}$ are given^[3] by the following formulae, which are exact up to the first order:

${\mathfrak {G}}_{x}={\frac {2U}{c^{2}}}{\mathfrak {w}}_{x},\quad {\mathfrak {G}}_{y}={\frac {2U}{c^{2}}}{\mathfrak {w}}_{y},\quad {\mathfrak {G}}_{z}=0$.

Substituting these values in (1), we get for the components of the couple, up to terms of the second order,

${\frac {2U}{c^{2}}}{\mathfrak {w}}_{y}{\mathfrak {w}}_{z},\quad -{\frac {2U}{c^{2}}}{\mathfrak {w}}_{x}{\mathfrak {w}}_{z},\quad 0$.

These expressions show that the axis of the couple lies in the plane of the plates, perpendicular to the translation. If α is the angle between the velocity and the normal to the plates, the moment of the couple will be ${\frac {U}{c^{2}}}w^{2}\sin \ 2\alpha$; it tends to turn the condenser into such a position that the plates are parallel to the Earth's motion.

In the apparatus of Trouton and Noble the condenser was fixed to the beam of a torsion-balance, sufficiently delicate to be deflected by a couple of the above order of magnitude. No effect could however be observed.

§ 2. The experiment of which I have spoken are not the only reason for which a new examination of the problems connected with the motion of the Earth is desirable. Poincaré^[4] has objected

1. A vector will be denoted by a German letter, its magnitude by the corresponding Latin letter.
2. See my article: Weiterbildung der Maxwell'schen Theorie. Elektronentheorie. Encyclopädie V 14, § 21, a. (This article will be quoted as M.E.)
3. M. E. § 56, c.
4. Poincaré, Rapports du Congrés de physique de 1900, Paris, 1, p. 22, 23