Question

A spherical capacitor has an inner sphere of radius $12cm$ and an outer sphere of radius $13cm$. The outer sphere is earthed, and the inner sphere is given a charge of $2.5\mu C$. The space between the concentric spheres is filled with a liquid of dielectric constant $32$. Determine the potential of the inner sphere :

• A. $400V$
• B. $450V$
• C. $500V$
• D. $300V$

Answer 1

The correct option is B $450V$

As outer sphere is earthed so its potential is zero. Let the potential of inner sphere is $V$.

Using Gauss's law the field at a distance $r$ from the center in between the sphere is:

$E.4\pi r^2=\frac{Q}{ϵ}\Rightarrow E=\frac{Q}{4\pi k{ϵ}_0{r}^2}$ as dielectric constant, $(k=\frac{ϵ}{{ϵ}_0})$

Potential difference between the spheres is:
$dV=-Edr=-\frac{Q}{4\pi k{ϵ}_0{r}^2}dr$

${\int }_V^0=-\frac{Q}{4\pi k{ϵ}_0}{\int }_b^a\frac{dr}{r^2}=\frac{Q}{4\pi {ϵ}_0}[\frac{1}{a}-\frac{1}{b}]$

$V=\frac{Q(b-a)}{4\pi k{ϵ}_{0}ab}=\frac{9\times {10}^9\times 2.5\times {10}^{-6}(0.13-0.12)}{32\times 0.12\times 0.13}=450V$