Insulators — Question 3

Question 3

A leaky dielectric gives a geometry-independent discharge time

A parallel-plate capacitor is filled by a uniform dielectric of permittivity $\varepsilon$ and finite conductivity $\sigma$. Plate area is A, separation d, and edge effects are negligible. The source is disconnected after charging.

1. Find capacitance and leakage resistance.

2. Derive the voltage-decay time constant and explain why changing A or d does not change it.

Question 3 - Solution

A leaky dielectric gives a geometry-independent discharge time

(a) Method and calculation

$$\boxed{C=\varepsilon A/d,\quad R_\ell=d/(\sigma A)}.$$$$I_\ell=V/R_\ell,\quad C\dot V=-I_\ell\Rightarrow\dot V=-V/(R_\ell C).$$

(b) Interpretation and further calculation

$$\boxed{V(t)=V_0e^{-t/\tau},\quad\tau=R_\ell C=\varepsilon/\sigma}.$$$$\boxed{\partial\tau/\partial A=\partial\tau/\partial d=0}\quad(\varepsilon,\sigma\text{ fixed}).$$

Physical check

The cancellation applies to bulk leakage through the same homogeneous dielectric that sets capacitance. Surface leakage or an external resistor need not share it.