Question

The radii of a spherical capacitor are equal to a and b(b > a). The space between them is filled with a dielectric of dielectric constant K and resistivity $\rho$. At t = 0, the inner electrode is given a charge $q_0$. The Charge q on the inner electrode as a function of time is given by $q=q_0{e}^{-\frac{t}{N\rho K{\epsilon }_0}};$ then N is___

Answer 1

$C=\frac{4\pi {\epsilon }_{0}abK}{b-a}$

$R={\int }_a^b\rho \frac{dr}{4\pi r^2}=\frac{\rho }{4\pi }\frac{(b-a)}{ab}$

$\tau =RC=\rho K{\epsilon }_0$

$q=q_0{e}^{\frac{-t}{\tau }}\Rightarrow q=q_0{e}^{-\frac{t}{\rho K{\epsilon }_0}}$