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 $q_0$. Choose the correct options:

• A. Charge q on the inner electrode as a function of time is given by $q=q_0{e}^{-\frac{1}{{pK}_{{\epsilon }_0}}}$
• B. Ina short time, the charge on the inner electrode will become zero
• C. After a long time , the charge on the outer sphere will become $q_0$
• D. The total amount of heat generated during the spreading of charge will be given by $H=(\frac{1}{a}-\frac{1}{b})\frac{q_0^2}{8\pi {\epsilon }_{0}K}$

Answer 1

Answer: (D)

$C=\frac{4\pi {\epsilon }_{0}abK}{b-a}$
$R={\int }_a^b\rho \frac{dr}{4\pi r^2}=\frac{\rho }{4\pi }\frac{(b-a)}{ab}$
$\tau =RC=\rho K{\epsilon }_0$
$q=q_0{e}^{-t/\tau }$
$q=q_0{e}^{\frac{\tau }{p{K}_{{\epsilon }_0}}}$
$t=\infty ,q=0$
Amount of heat generated
$H=(\frac{1}{a}-\frac{1}{b})\frac{q_0^2}{8\pi {\epsilon }_{0}K}$