Question

A dielectric-filled parallel-plate capacitor has plate area A = 30.0 cm2 , plate separation d = 8.00 mm and dielectric constant k = 4.00. The capacitor is connected to a battery that creates a constant voltage V = 5.00 V . Throughout the problem, use ϵ0 = 8.85×10⁻¹² C2/N⋅m2 .

Part A- Find the energy U1 of the dielectric-filled capacitor.

Part B- The dielectric plate is now slowly pulled out of the capacitor, which remains connected to the battery. Find the energy U2 of the capacitor at the moment when the capacitor is half-filled with the dielectric.

Part C- The capacitor is now disconnected from the battery, and the dielectric plate is slowly removed the rest of the way out of the capacitor. Find the new energy of the capacitor, U3.

Part D- In the process of removing the remaining portion of the dielectric from the disconnected capacitor, how much work W is done by the external agent acting on the dielectric?

Answer 1

Area A = 30.0 cm^2 = 0.003 m^2, d = 8.0 mm = 0.008 m

Part - A -  
C = k * εo A /d = (4x 8.85x10^-12x 0.003) / 0.008 = 1.33 x 10^-11 F

U1 = Energy stored = 0.5CV^2 = 0.5x1.33x10^-11x5^2 = 1.662 x 10^-10 J

Part -B

U2 = Energy stored = [1+k] /2k* U1 = [(1+4) / (2x4)]x1.662 x 10^-10 = 1.04 x 10^-10 J

And, charge residing in this new capacitor =2* U2/ V = 2*1.04x10^-10 /5  
Q= 4.16 x 10^-11 couloumb.

Part -C

After fully removing the dielectric,
Now the capacitance C /k = (1.33 x 10^-11)/4= 3.32 x 10^-12 F
U3 = Energy stored = Q^2 / (2 x3.32x10^-12) = (4.16x10^-11)^2 / (2 x3.32x10^-12) =  2.61 x 10^-10 J

Part -D

U3-U2 = 2.61x10^-10 - 1.04 x 10^-10 = 1.57 x 10^-10 J

So, work done by the external agent = 1.57 x 10^-10 J.