Problem 3.211

Two large parallel plates are located in vacuum. One of them serves as a cathode, a source of electrons whose initial velocity is negligible. An electron flow directed toward the op posite plate produces a space charge causing the potential in the gap between the plates to vary as $\varphi=a x^{4 / 3},$ where $a$ is a positive constant, and $x$ is ihe distance from the cathode. Find: (a) the volume density of the space charge as a function of $x$; (b) the current density.