A dielectric-filled parallel-plate capacitor has plate area A = 15.0 cm2 , plate

Question

A dielectric-filled parallel-plate capacitor has plate area A = 15.0 cm2 , plate separation d = 6.00 mm and dielectric constant k = 5.00. The capacitor is connected to a battery that creates a constant voltage V = 10.0 V . Throughout the problem, use 0 = 8.85×1012 C2/Nm2 .

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

B) The dielectric plate is now slowly pulled out of the capacitor, which remains connected to the battery. Find the energy U2of the capacitor at the moment when the capacitor is half-filled with the dielectric.

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.

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?

Explanation / Answer

A = (15 cm^2) (100 cm m)^2 = 0.0015 m^2

d = 0.006 m
C = e0*(er)*A d
C = [8.85 *10^(-12)*(5)*(0.0015) (0.006) F
C = 8.85 *10^(-12) *1.25
C = 11.06 pF

U1 = (1/2)*C*(V)^2
U1 = (1/2)*[11.06*10^(-12)*(10)^2]
U1 = 553.125*10^(-12) Joules
U1 = 553.125pJ

At the halfway point there are essentially two capacitors in parallel

C1 = (e0)*A d(air region)
C1 = [8.85 *10^(-12)]*(0.00075)/ (0.006)
C1 =1.10 *10^(-12) F
C1 = 1.10pF

C2 = (e0)*(er)*A d(dielectric region)
C2 = [8.85 *10^(-12)]*(5)*(0.00075) (0.006)
C2 = 5.50*10^(-12) F
C2 = 5.50 pF

Ct = C1 + C2
Ct = 1.10 + 5.50 = 6.60 pF

U2 = (1/2)*Ct*V^2
U2 = (1/2)*[6.60 *10^(-12)]*(10)^2

U2 =330 *10^(-12) Joules
U2 = 330 pJ

Q is conserved in this case

Q = C*V
Q = [6.60 *10^(-12)]*(10)
Q = 66.0 *10^(-12) C
Q = 66.0 pC


C = (e0)*A d(completely air)
C = [8.85 *10^(-12)]*(0.0015) (0.006)
C = 2.21*10^(-12) F
C = 2.21 pF

U3 = (1/2)*C*(V)^2
U3 = (1/2)*C*(Q C)^2
U3 = (1/2)*(Q)^2 C
U3 = (1/2))[66*10^(-12)]^2 2.21*10^(-12)
U3 = 985.5 pJ

Work = delta U = (985.52 pJ) (330 pJ)

Work =655.52 pJ

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