Page:Die elektromagnetische Masse des Elektrons.djvu/1

The electromagnetic mass of the electron.

W. Kaufmann (Göttingen)

At the meeting of natural scientists in Hamburg^[1], I was able to report about an experiment with the result that the ratio $\epsilon /\mu$ of Becquerel rays would decrease with increased velocity, and $\mu$ would increase if one assumes $\epsilon$ as constant, namely it increases the quicker, the more the velocity $(q)$ would approach the speed of light $(c)$. Such a behavior is theoretically given from the equation of energy of a quickly moving electric charge. At that time, it was achieved to bring the results into accordance with the theoretical formula derived by Searle^[2]; but only under the assumption, that the greatest part of the mass of the moving electron would be of mechanical origin, and the rest of electromagnetic origin. However, soon after publication of the experiments of that time, it was shown by Abraham,^[3], that Searle's formula for the field energy of the moving electron, only allows to calculate (without further ado) the electromagnetic mass in the case of acceleration in the direction of motion, while on the other hand, for the transverse acceleration (as it was the case in my experiments) an expression for mass applies which deviates from Searle's formula. If $\beta =q/c$, $\epsilon$ is the charge of the electron in E.M.E, $\mu _{0}$^[4] is the value of electromagnetic mass for small velocities, then it is according to Abraham:

1)	${\frac {\epsilon }{\mu }}={\frac {\epsilon }{\mu _{0}}}{\frac {4}{3}}{\frac {1}{\psi (\beta )}}$

where

2)	$\psi (\beta )={\frac {1}{\beta ^{2}}}\left[{\frac {1+\beta ^{2}}{2\beta }}\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right],$

($\psi (\beta )={\tfrac {4}{3}}$ for $\beta =0$; $\psi (\beta )=\infty$ for $\beta =1$).

A comparison of my experimental results with his formula that was already undertaken by Abraham, gave no good agreement, since the mass varies faster as it was required by the theory, so that a possibly added mechanical mass had to be set as negative.

In what follow, a more rational value for the evaluation of the results shall be shown, and at the same time by the aid of new experimental material, the full agreement between observation and theory shall be proven.

In the observation of that time, the absolute values of $q$ and $\epsilon \mu$ have been determined by using the absolute values of the electric and magnetic field, though already at that time, the possible errors were estimated to ca. 5%; these are errors which are much higher than the relative errors when measuring the plates.

Due to the large variability of $\psi (\beta )$ for $\beta$ nearly equal to 1, a small error of $\beta$ means a very large error for $\mu$ (for $\beta =0,96$ or 0,98, it is for instance $\psi (\beta )=3,141$, i.e., to an error of 2% in determining of $\beta$, an error of 19% for $\mu$ is corresponding.).

A rational utilization of the curve measured on the plate is only achieved by comparing the relative values with each other; one may not determine the curve constants directly by measuring the apparatus dimensions and the field, but one has to determine the most probable values by the method of least squares.

Let $y$ and $z$ be the magnetic deflections measured on the plate. From them, two other quantities $\eta$ and $\zeta$ can be derived, which have a simple relation to $\epsilon /\mu$ and $q$. $\eta$ and $\zeta$ are approximately proportional to $y$ and $z$; the deviations from proportionality

1. Verhdl. D. Naturf. u. Ärzte Hamburg 1901. II. 1. 45. Gött. Nachr. 1901. H. 2.
2. Phil. Mag (5) 44. 340, 1897.
3. Gött. Nachr. 1902. H. 1.
4. If $a$ is the radius of the electron, then $\mu _{0}={\tfrac {2}{3}}{\tfrac {\epsilon ^{2}}{a}}b$ under the assumption of surface charge.