Limiting current between concentric spheres; calculation of the function $\alpha =f\left(\frac{r}{r_0}\right)$ in the space charge equation $i=\left(\frac{4\sqrt{2}}{9}\right)\frac{\sqrt{\left(\frac{e}{m}\right)}{V}^{\frac{3}{2}}}{{\alpha }^2}$.—The coefficients of the first six terms of a series for $\alpha$ were determined, and ${\alpha }^2$ calculated from this series. The results were checked by an integration method which was also used to calculate values in the region where the series failed. For an emitter of radius $r_0$ inside a collector of radius $r$, values of ${\alpha }^2$ when $log\left(\frac{r}{r_0}\right)>\mathrm{}6.4\mathrm{}$ are given by the equation $\frac{1}{2}{\alpha }^2=0.112\mathrm{} log \left(\frac{logr}{r_0}\right)+\frac{1}{3}log \left(\frac{r}{r_0}\right)+\mathrm{0.152.}\mathrm{}$ Where the collector is the inside sphere, values of ${\alpha }^2$ for $\frac{r_0}{r}>9\mathrm{}$ are given by the equation ${\left(\frac{1}{2}{\alpha }^2\right)}^{\frac{2}{3}}=1.11\mathrm{} \left(\frac{r_0}{r}\right)-1.64\mathrm{}$. It is shown that when the collector is the inside sphere the potential distribution near the collector is unaltered if the emitter is replaced by a non-emitting sphere with a diameter.677 times the original diameter.

Limiting current between coaxial cylinders and between concentric spheres.—Equations are derived for the current in terms of the radius of curvature of the emitter. It is shown that at a surface in space four-fifths of the distance from the emitter to the collector the current density is independent of the radius of curvature when $\frac{r}{r_0} or \frac{r_0}{r}<2\mathrm{}$; and in the case of coaxial cylinders with the emitter inside this holds true even when $\frac{r}{r_0}=20\mathrm{}$.

• Received 9 February 1924

DOI:https://doi.org/10.1103/PhysRev.24.49