Question

A Unifrom volume charge density $\rho _{v0}$ is inside of a parallel-plate capacitor. ( The charge density is floating in free space between the plates.) The bottom plate of the capacitor (at x=0) is at zero volts. The top plate of the capacitor ( at x = h) is also at zero volts. The plates are large compared to the separation h, so fringing may be neglected.

a) find the potential $\phi (x)$ at any point inside the capacitor.

b) find the surface charge densities $\rho _{s}^{top}$ and $\rho _{s}^{bottom}$ on the top and bottom plates.

Answer 1

By Poisson's equation d^2 (x)/dx^2 = -p v_0/ E_0 taking integration d (x)/dx = -p v_0/ E_0 x + k_1 k_1 is constant taking integration again (x) = -p v_o/E_o x^2/z + k_1 x + k_2 k_2 is n constant (0) = 0 rightarrow k_2 = 0 (h) = 0 rightarrow p v_0/E_o h^2/ 2 = k_1 rightarrow k_1 p v_o h/2 E_o rightarrow k_1 = p v_o h/2 E_o rightarrow (x) = p v_o/E_o x^2