Application: Capacitor with space charge between plates. The parallel plate capacitor in Figure 6.41 is given. In addition to the data in Problem 6.1, there is also a charge density everywhere inside the material such that p/ε = 1, where p is the charge density and ε is the permittivity of the dielectric between the plates.

(a) Find the potential distribution everywhere within the capacitor using four equal divisions.

(b) Repeat the solution in (a) with eight equal divisions. Show that the division chosen is important. Which division gives a better result? Compare with the analytical solution.

Problem 6.1

One dimensional geometry. A parallel plate capacitor is shown in Figure 6.41. The capacitor may be viewed as infinite in extent. With d = 1 m and free space between the plates, calculate:

(a) The potential distribution everywhere within the capacitor using the finite difference method with four equal divisions.

(b) Repeat the solution in (a) with eight equal divisions.

(c) Calculate the potential distribution using direct integration. Show by comparison with (a) and (b) that the division does not matter in this case. Why?