Currents Limited by Space Charge between Coaxial Cylinders

Limiting current between coaxial cylinders; calculation of the function $\beta =\mathbf{f}\left(\frac{\mathbf{r}}{{\mathbf{r}}_{\circ }}\right)$ in the space charge equation $i=\frac{\left(\frac{2\sqrt{2}}{9}\right)\sqrt{\left(\frac{e}{m}\right)}{V}^{\frac{3}{2}}}{\left(r{\beta }^2\right)}$. Two different infinite series were obtained for $\beta$ and the coefficients for fourteen terms of each were determined. These two sets were checked against each other. Thus values of ${\beta }^2$ were obtained accurate to 1 in 10,000. These were checked by an integration method which was also used to calculate values in the region where the series failed. For a cathode of radius $r_0$ inside an anode of radius $r$, ${\beta }^2$ increases from 1 at $\frac{r}{r_0}=11.2\mathrm{}$ to a maximum value 1.0946 at $\frac{r}{r_0}=42\mathrm{}$, decreases to a minimum value 0.9990 at $\frac{r}{r_0}=30,000\mathrm{}$, and becomes 1 at $\frac{r}{r_0}=\infty$. The customary assumption ${\beta }^2=1\mathrm{}$ has therefore led to errors up to 9.5 per cent in previous calculations, but this error is usually about equal and opposite to that introduced by neglecting the effect of initial velocities. For the cathode outside the anode, ${\beta }^2$ is given very closely by the equation ${\beta }^2=4.6712\mathrm{} \left(\frac{r_0}{r}\right) {\left[{log}_{10}\right(\frac{r_0}{r})-{log}_{10}\sqrt{2}]}^{\frac{3}{2}}$, for $\frac{r_0}{r}>10\mathrm{}$. The empirical constant √2 is interpreted to mean that the potential distribution near the anode is unaltered if the hot cathode is replaced by a cold cylinder having one half the cross-section of the original cathode. The correction for initial velocities is less for a cylindrical cathode inside an anode than for parallel planes. In the inverted case it is much greater than in the case for parallel planes, and the effect of the tangential component of the initial velocity may greatly decrease the current that flows.