Page:Über die scheinbare Masse der Ionen.djvu/1

H. A. Lorentz (Leiden)

On the apparent mass of the ions.

It is known that by observations of cathode rays we were able to derive the ratio ${\tfrac {e}{m}}$, i.e. the ratio between the charge of an ion $e$ and its mass $m$. The question arises, what is meant by that mass. In any case we must attribute an apparent mass to the ion, as it generates a certain energy in the ether by virtue of its motion. This apparent mass will be denoted by $m_{0}$. It is possible that the ion also possesses a real mass in the ordinary sense of the word; in this case, $m_{0}<m$. If this is not the case, then $m_{0}=m$.

So we have the inequality

${\frac {e}{m_{0}}}>{\frac {e}{m}}{,}$

when there still is a real mass besides the apparent mass; otherwise

${\frac {e}{m_{0}}}={\frac {e}{m}}.$

So we want to write

${\frac {e}{m_{0}}}\geqq {\frac {e}{m}}{,}$

where ${\tfrac {e}{m}}=10^{7}$ is.

Now

$m_{0}={\frac {8}{3}}\pi R\sigma e{,}$

if we conceive the ion as a sphere, $R$ is the radius of this sphere, and $\sigma$ means the surface density of the charge.

This formula allows for an interesting conclusion on the radius of the ions. If, namely, we substitute for $m_{0}$ the now specified value into the inequality, we obtain an inequality for the radius. We have

$4\pi R^{2}\cdot \sigma =e{,}$

thus

$m_{0}={\frac {8}{3}}\pi R\sigma e={\frac {8}{3}}\pi Re\cdot {\frac {e}{4\pi R^{2}}}={\frac {2e^{2}}{3R}}$

and thus

${\frac {e}{m_{0}}}={\frac {3R}{2e}}{,}$

and

${\frac {3R}{2e}}\geq 10^{7}$

and

$R>10^{7}\cdot {\frac {2}{3}}e.$

The magnitude $e$ is unfortunately not known. If we take the charge of an ion in a cathode ray to be as great as in an electrolytic hydrogen, and presuppose the size of a hydrogen molecule, we obtain for $R$ a magnitude of order