A dielectric-filled parallel-plate capacitor has plate area A = 25.0 cm2 , plate

Question

A dielectric-filled parallel-plate capacitor has plate area A = 25.0 cm2 , plate separation d = 10.0 mm and dielectric constant k = 2.00. The capacitor is connected to a battery that creates a constant voltage V = 15.0 V . Throughout the problem, use 0 = 8.85×1012 C2/Nm2

1. Find the energy U1 of the dielectric-filled capacitor.

2. 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.

3. 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.

4. 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? Express your answers in joules

Explanation / Answer

A = (25 cm²) (100 cm m)² = 0.0025 m²

d = 0.0010 m

C = (ø)•()•A d
C = [8.85 ×10^(-12)]•(2)•(0.0025) (0.001)
C = 4.425 ×10^(-12) F

U = (½)•C•V²
U = (½)•[8.85 ×10^(-12)]•(15²)
U = 4.98 ×10^(-10) Joules

______________________________________...

At the halfway point there are essentially two capacitors in parallel

C = (ø)•A d...(air region)
C = [8.85 ×10^(-12)]•(0.00125) (0.001)
C = 1.106 ×10^(-12) F
C = 1.106 pF

C = (ø)•()•A d...(dielectric region)
C = [8.85 ×10^(-12)]•(2)•(0.00125) (0.001)
C = 2.212 ×10^(-12) F
C = 2.212 pF

C = C + C
C = 1.106 + 2.212 = 3.318 pF

U_2 = (½)•C•V²
U_2 = (½)•[3.318 ×10^(-12)]•(15²)
U_2 = 3.735 ×10^(-10) Joules

______________________________________...

Q is conserved in this case:

Q = C•V
Q = [3.32 ×10^(-12)]•(15)
Q = 4.98 ×10^(-11) C



C = (ø)•A d...(completely air)
C = [8.85 ×10^(-12)]•(0.0025) (0.001)
C = 2.21 ×10^(-12) F
C = 2.21 pF

U_3 = (½)•C•V²
U_3 = (½)•C•(Q C)²
U_3 = (½)•Q² C
U_3 = (½)•[4.98 ×10^(-11)]² 2.21×10^(-12)
U_3 = 5.61 X 10-10J
_________________________________

Work = U = ( 5.61 X 10-10) (37.3 x 10-9J)

Work = 1.875 X 10-10 J

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