A parallel-plate capacitor is filled with a dielectric material having resistivity ρ and dielectric constant K. The capacitor is charge and disconnected from the charging source. The capacitor is slowly discharged through the dielectric. Show that the time constant of the discharge is independent of all geometrical parameters like the plate area or separation between the plates. Find this time constant.

Question

Philoid · Accepted Answer

Time constant for capacitor:

Where R is the resistance through which the capacitor is being charged/discharged and C is the capacitance.

Capacitance of a Capacitor in presence of a dielectric:
The capacitance of the capacitor is initially C₀ and then a dielectric medium of dielectric constant K is inserted between the plates. The new capacitance is

Also for parallel plate capacitors,

Where ϵ₀ is the permittivity of free space, A is the area of plate and l is the distance between the plates.

Resistance and Resistivity:

For a material of length l and uniform cross-section A and reistivity ρ, the resistance is given by:

Note the area of cross section of the material is the same as the area of the capacitor plates and the length of the material is the same as the distance of separation between the plates.

Now,