Two large parallel plates are located in vacumm. One of them serves as a cathoide, a source of electrodes whose initial velocity is neglible . An electrones flows direcee toward the opposite plate produces a space charge causing the potential int eh gap betwnen the plates to between the plates to very as `varphi = ax^(1//3)`, where `a` is positive constant, and `x` is the distnaace form the cathode. Find:
(a) the volume density of the space charge as a funciton of `x`,
(b) the current density.

Explanation

Between the plates `varphi = a x^(4//3)`
or, `(del varphi)/(del x) = a xx (4)/(3) x^(l//3)`
`(d^(2) varphi)/(del x^(2)) = (4)/(9) ax^(-2//3) = rho//epsilion_(0)`
or, `rho = - (4epsilon_(0) a)/(9) x^(-2//3)`
Let the charge on the electron be `-epsilon`,
then `(1)/(2) mv^(2) - epsilon varphi` = Const. = 0.
as the electron is initally emittied with neligble energy.
`v^(2) = (2 epsilon varphi)/(m), v = sqrt((2 e varphi)/(m))`
So, `j = -rho v = (4 epsilon_(0) a)/(9) sqrt((2 varphi)/(m)) x^(-2//3)`.
(`j` is measured from the anode to cathode, so the `-ve` sign)