Q: A parallel – plate capacitor, filled with a dielectric of dielectric constant k, is charged to a potential V_{0}. It is now disconnected from the cell and the slab is removed. If it now discharges, with time constant τ , through a resistance, then find time after which the potential difference across it will be V_{0} ?

Sol: When slab is removed, the potential difference across capacitor increases to k V_{0}

$\large C V_0 = k C V_0 e^{\frac{-t}{\tau}}$ ; as q_{0} = k C V_{0}

$\large \frac{1}{k} = e^{\frac{-t}{\tau}}$

$\large k = e^{\frac{t}{\tau}}$

$\large lnk = \frac{t}{\tau}$

$\large t = \tau lnk $