In a vacuum diode, a hot flat cathode at $x = 0$ gives off electrons. They cross to a flat anode at $x = d = 4.0\ \text{mm}$, which is held at $200\ \text{V}$ above the cathode. The electrons leave the cathode almost at rest. A steady stream of them fills the gap, and their charge changes the potential there. Between the plates it is $\displaystyle V(x) = V_0\left(\frac{x}{d}\right)^{4/3}, \qquad V_0 = 200\ \text{V}$ The plates are large, so $V$ depends on $x$ only. Take $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}$. What is the total charge in the space between the plates, per square metre of plate?