A parallel plate capacitor is filled by a dielectric

Question:

A parallel plate capacitor is filled by a dielectric whose relative permittivity varies with the applied voltage (U) as ε = αU where α = 2V-1. A similar capacitor with no dielectric is charged to U0 = 78 V. It is then connected to the uncharged capacitor with the dielectric. Find the final voltage on the capacitors.

Solution:

Since the capacitors are connected in parallel, the potential difference across the capacitors is the same. The final voltage is assumed to be U. C is the capacitance of the capacitor without dielectric, then the charge is given as Q1 = CU

The initial charge is given as

Q0 = CU0

The conversion of charges is

Q0 = Q1 + Q2

CU0 = CU + αCU2

αU2 + U – U0 = 0

Solving the equation, we get U = 6V