The Effect of A Uniform Magnetic Field on the Motion of Electrons Between Coaxial Cylinders.

Equations of motion for electrons are developed starting from a cylindrical cathode and moving toward a co-axial cylindrical anode, in a uniform magnetic field parallel to the common axis. The electrons will reach the anode if the ratio of potential difference to magnetic field is greater than a critical value, and will fail to reach it if the ratio is less than this value.In the case of a small cathode in the axis of an anode of radius $R$ at potential $V$, the critical magnetic field is $H=\sqrt{\frac{8m}{e}}\frac{{V}^{\frac{1}{2}}}{R}$For a small anode of radius ${r}_{0}$ at potential $V$, in the axis of a cathode of radius ${R}_{0}$, the critical field is $H=\frac{{r}_{0}}{{R}_{0}}\sqrt{\frac{8m}{e}}\frac{{V}^{\frac{1}{2}}}{{R}_{0}}+\frac{{2mv}_{0}}{e{R}_{0}}$In this case the initial velocity ${v}_{0}$ of the electrons cannot be neglected. This equation also applies, with appropriate ${R}_{0}$ and $\frac{e}{m}$, to positive ions produced by electrons from an internal cathode.If the radii of both cylinders are large the solution reduces to the familiar one of plane parallel plates.The equation of the path of the electrons is deduced, on the assumption that the space charge distribution is the same as without magnetic field. The path is given by $r={R(sin\frac{2}{3} \ensuremath{\theta})}^{\frac{3}{2}}$. This is a close approximation to the true path, as calculated from the space charge distribution recently worked out by Langmuir. Experimental curves showing current at constant potential as a function of magnetic field, for different anode diameters, voltages, filament temperatures, degrees of symmetry, etc., are in agreement with theory within the limit of experimental error.The internal cathode tube offers a means of measuring $\frac{e}{m}$, and the internal anode a means of measuring the distribution of initial velocities.