A parallel-plate capacitor is filled with a dielectric material of resistivity ρ and dielectric constant K. The capacitor is charged 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.

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Solution

The capacitance of a parallel plate capacitor,
$C = \frac{K ε_{0} A}{d}$
The resistance of dielectric material,
$R = \frac{ρ d}{A}$

Time constant,
$τ = R C = ρ K ε_{0}$ ,
which is independent of the plate area or separation between the plates.

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