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.
Given
Dielectric of capacitor 2 = ε = αU
Value of constant a=2 V-1
Voltage applied to capacitor 1 = U0=78 V
We know that the charge stored in a capacitor and the voltage applied across are related by the equation
Where C is the capacitance of the capacitor and V is the applied voltage. Let the final voltage of the capacitors be U, and capacitance of the capacitor with no dielectric be C, then the capacitance of the capacitor with dielectric is then εC, then the final charges stored in these capacitors after being connected will be,
and
respectively.
Now from conservation of electric charge we have:
Where Q is the total charge distributed among the capacitors, which comes from the original capacitor charged by a voltage of 78V. Thus,
Solving for U we get
Thus, the final voltage is 6 V on each capacitor.