Page:Electromagnetic phenomena.djvu/12

$\int \left({\mathfrak {d}}_{y}^{'2}+{\mathfrak {d}}_{z}^{'2}\right)dS'={\frac {2}{3}}\int b^{'2}dS'={\frac {e^{2}}{6\pi }}\int _{R}^{\infty }{\frac {dr}{r^{2}}}={\frac {e^{2}}{6\pi R}}$,

and

${\mathfrak {G}}_{x}={\frac {e^{2}}{6\pi c^{2}R}}klw$.

It must be observed that the product kl is a function of w and that, for reasons of symmetry, the vector ${\mathfrak {G}}$ has the direction of the translation. In general, representing by ${\mathfrak {w}}$ the velocity of this motion, we have the vector equation

${\mathfrak {G}}={\frac {e^{2}}{6\pi c^{2}R}}kl{\mathfrak {w}}$.

(28)

Now, every change in the motion of a system will entail a corresponding change in the electromagnetic momentum and will therefore require a certain force, which is given in direction and magnitude by

${\mathfrak {F}}={\frac {d{\mathfrak {G}}}{dt}}$.

(29)

Strictly speaking, the formula (28) may only be applied in the case of a uniform rectilinear translation. On account of this circumstance — though (29) is always true — the theory of rapidly varying motions of an electron becomes very complicated, the more so, because the hypothesis of § 8 would imply that the direction and amount of the deformation are continually changing. It is even hardly probable that the form of the electron will be determined solely by the velocity existing at the moment considered.

Nevertheless, provided the changes in the state of motion be sufficiently slow, we shall get a satisfactory approximation by using (28) at every instant. The application of (29) to such a quasi-stationary translation, as it has been called by Abraham,^[1] is a very simple matter. Let, at a certain instant, ${\mathfrak {j}}_{1}$, be the acceleration in the direction of the path, and ${\mathfrak {j}}_{2}$, the acceleration perpendicular to it. Then the force ${\mathfrak {F}}$ will consist of two components, having the directions of these accelerations and which are given by

${\mathfrak {F}}_{1}=m_{1}{\mathfrak {j}}_{1}$ and ${\mathfrak {F}}_{2}=m_{2}{\mathfrak {j}}_{2}$

if

$m_{1}={\frac {e^{2}}{6\pi c^{2}R}}{\frac {d(klw)}{dw}}$ and $m_{2}={\frac {e^{2}}{6\pi c^{2}R}}kl$.

(30)

Hence, in phenomena in which there is an acceleration in the

1. Abraham, Wied. Ann, 10 (1903), p. 105.