On the Currents Carried by Electrons of Uniform Initial Velocity

The following problem is treated: Electrons enter the space between two infinite parallel planes with uniform velocity at right angles to the planes. The current is studied for all possible values (positive and negative) of the potential difference between the planes. The complete solution can be obtained if this boundary condition for the current is accepted: The number of electrons entering the discharge space must be equal to or smaller than a given number $N_0$ per ${\mathrm{cm}}^2$ per sec., and is for each potential as high as the potential permits. The problem depends on two dimensionless parameters, a reduced plate distance ${\xi }_0$ (involving the current) and ${\eta }_0=\frac{V_0}{E_0}$, where $V_0$ is the impressed potential difference and $e{E}_0$ the energy of the electrons. For each value of ${\xi }_0$ the current is space-charge limited below a critical value of ${\eta }_0$. The limitation, however, is not due to the appearance of a potential minimum low enough to stop the electrons, but is due to the fact that the solution does not exist unless the current is limited. The space-charge limited characteristic is a simple generalization of Child and Langmuir's well-known formula [see Eqs. (28) or (35)]. The deviations from this formula are very considerable if $E_0\sim V_0$ (see Fig. 2). The behavior of the potential distribution is graphed for some typical cases of positive, zero, and negative potentials $V_0$. Currents can pass with zero or negative potentials only below critical values of the reduced plate distance.