Question

A capacitor filled by dielectric slab (k=4.5) is connected to a 12.0V battery and charged up. The capacitor is then disconnected from the battery and the slab is removed.

a) Find the amount by which the potential difference across the plates changes.

b) Find the difference between the new stored energy and the original amount and specify whether this energy difference is an increase or decrease.

Answer 1

A capacitor filled by dielectric slab (k=4.5) is connected to a 12.0V battery and charged up. The capacitor is then disconnected from the battery and the slab is removed.

k = 4.5

V = 12 V

(a)

Capacitance, C, of the capacitor with a dielectric slab is given by,

C = Q/V = k A / d

Q = k AV / d

where Q is the charge on the plates, V the potential difference, A the plate area, d is the separation between the plates and is the permittivity of the free space.

Since the capacitor is charged up before the removal of the dielectric slab, the charge Q on the capacitor remains the same. But, due to the removal of the dielectric slab, the capacitance of the capacitor decreases by a factor of k. That is the new capacitance of the capacitor after the removal of the dielectric slab is,

C_new = C / k = A / d

To accommodate this change in the capacitance, the potential difference across the plates changes and this change is given by,

V_new = Q / C_new

V_new = (k AV / d) / ( A / d)

V_new = kV = 4.5 * 12 = 54 V

That is, potential difference across the plates increases by an amount k.

(b)

Energy stored in the capacitor is given by,

U = (1/2) C V²

This is the energy stored in the capacitor before the removal of the dielectric slab.

After the removal of the slab, the energy stored in the capacitor is given by,

U_new = (1/2) C_new V_new²

U_new = (1/2) (C/k) (kV)²

U_new = k [ (1/2) C V² ]

U_new = k U

Thus, the difference between the new stored energy and the original amount is,

U = U_new - U

U = (k - 1) U

U = (4.5 - 1) U

U = 3.5 U

This difference is certainly an increase in the new store energy than the original amount.