A parallel-plate capacitor consists of two square conducting plates of side length…
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
A parallel-plate capacitor consists of two square conducting plates of side length L separated by a distance d≪L. The capacitor is initially charged to total charge Q0 and then disconnected from the charging source so that the charge on the plates remains constant. A rectangular dielectric slab of identical thickness d and length L (so it completely fits between the plates) is slowly pushed a distance x (measured from the left edge of the plates) into the region between the plates, as shown in the side view described below.
Side view description (textual):
• The left plate is at x=0 and the right plate at x=L.
• The slab enters from the left; at a general moment, a length x of the slab is between the plates while the remaining (L−x) is air.
• The dielectric constant of the slab is κ (uniform and isotropic).
Neglect edge effects, assume the electric field is uniform in each region, and treat d as much smaller than L so fringing may be ignored.
(a) Express the capacitance C(x) of the system as a function of the insertion distance x.
(b) Using the fact that the charge on the plates remains equal to Q0, derive an expression for the electrical energy U(x) stored in the capacitor as a function of x.
(c) Determine the magnitude and direction of the net force F(x) exerted on the dielectric slab when its insertion distance is x.
(d) Calculate the total work Wext that an external agent must do to insert the slab completely (x=0→L). State whether this work is positive, negative, or zero and justify your answer physically.