Currents Limited by Space Charge between Concentric Spheres

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.