Question

The plates of an air-filled parallel-plate capacitor with a plate area of 15.0 cm² and a separation of 8.95 mm are charged to a 170-V potential difference. After the plates are disconnected from the source, a porcelain dielectric with

κ = 6.5

is inserted between the plates of the capacitor.

(a) What is the charge on the capacitor before and after the dielectric is inserted?

Qi	=	C
Qf	=	C

(b) What is the capacitance of the capacitor after the dielectric is inserted?
F

(c) What is the potential difference between the plates of the capacitor after the dielectric is inserted?
V

(d) What is the magnitude of the change in the energy stored in the capacitor after the dielectric is inserted?
J

Answer 1

Given

capacitor with  

A = 15.0 cm^2 , separation of 8.95 mm are charged to a

dV = 170-V potential difference.

k = 6.5

we know that the capacitance of a parallel plate capacitor is C = epsilon not *A/d

C = 8.854*10^-12*(15*10^-4)/(8.95*10^-3)F

C = 1.4839106145251*10^-12 F

C = 1.483911*10^-12 F

charge on the capacitor before the dielectric inserted is  

Q = C*V

Q = 1.483911*10^-12*170 C = 2.5226487*10^-10 C

after the dielectric inserted  

the capacitance changes ( increases) by k times  

C' = k*C = 6.5*1.483911*10^-12 F = 9.6454215*10^-12 F

when the capacitor is disconnected from the battery there is no change in the charee  

We know Q = C*V

Capacitance increases so as the potential difference decreases by k times to have constant

a) so

Qi = 2.5226487*10^-10 C

Qf = 2.5226487*10^-10 C

b)the capacitance of the capacitor after the dielectric is inserted is c' = k*C =

c' = 6.5*1.483911*10^-12 F

C' = 9.6454215*10^-12 F

c) the potential difference between the plates of the capacitor after the dielectric is inserted is decreased to 1/k times

dV' = (1/k*V) = 170/6.5 V = 26.153846 V

d)

d) before dielectric inserted

the energy stored is U = 0.5*C*V^2

U = 0.5* 9.6454215*10^-12*170^2 J

U = 1.39376341*10^-7 J

after the dielectric inserted

the energy stored is U = 0.5*C*V^2 = 0.5*C'*dv^2 = 0.5* 9.6454215*10^-12*26.153846^2 J

U' = 3.29884826*10^-9 J

now the energy difference is

U-U' = 1.39376341*10^-7 - 3.29884826*10^-9 J

dU = 1.3607749274*10^-7 J