Chapter 2.5, Problem 2.53P

(a))

To determine

The Poisson^’s equation for the region between the plates.

(b))

To determine

The speed of electron at point $x$ .

(c))

To determine

The relation between $\rho$ and $v$ .

(d))

To determine

The expression of differential equation for $V$ .

(e))

To determine

The equation for $V$ as a function of $x$ , $v_0$ and $d$ .Plot the graph of $V\left(x\right)$ and compare it to the potential without space charge. Express $\rho$ and $v$ as a function of $x$ .

(f))

To determine

The proof that shows $I=K{V}_0{}^{3/2}$ .