A parallel-plate capacitor of plate area A and separation x is given a charge Q and is then removed from the charging source.

(a) Find the stored electrostatic energy as a function of x.

(b) Find the increase in energy dU due to an increase in plate separation dx from dU = (dU/dx) dx.

(c) If F is the force exerted by one plate on the other, the work needed to move one plate a distance dx is F dx = dU. Show that F = Q2/2є0A.

(d) Show that the force in part (c) equals ½EQ, where Q is the charge on one plate and E is the electric field between the plates. Discuss the reason for the factor ½ in this result.