Question

The capacitance of a spherical condenser is $1mF$. If the spacing between the two spheres is $1mm$, then the radius of the outer sphere is

• A. $30cm$
• B. $6m$
• C. $5cm$
• D. $3m$

Answer 1

The correct option is A $30cm$

$\begin{array}{c}Fromthequestion\\ C=\frac{4\pi {\epsilon }_0}{\left[\frac{1}{r_{in}}-\frac{1}{r_{out}}\right]}=\frac{4\pi \epsilon }{\left[\frac{1}{r_1}-\frac{1}{r_1+0.001}\right]}\\ 1\times {10}^{-6}=\frac{4\times 3.14\times 8.854\times {10}^{-12}}{\left[\frac{1}{r_1}-\frac{1}{r_1+0.001}\right]}\\ \left[\frac{1}{r_1}-\frac{1}{r_1+0.001}\right]=\frac{4\times 3.14\times 8.854\times {10}^{-12}}{1\times {10}^{-6}}\\ \frac{\left[\left(r_1+0.001\right)-r_1\right]}{r_1\times \left(r_1+0.001\right)}=4\times 3.14\times 8.854\times {10}^{-6}\\ r_1\times \left(r_1+0.001\right)=\frac{4\times 3.14\times 8.854\times {10}^{-6}}{0.001}\\ r_1^2+0.001r_1-4\times 3.14\times 8.854\times {10}^{-3}=0\\ r_1^2+0.001r_1-0.1112=0\\ r_1=0.333mor{r}_1=-334m\\ Since,itcannotbenegative\\ Thereforemradiusofoutersphere=r_1+0.001\\ r_{outer}=0.334m\\ or,r_1=33.4cm\end{array}$

Hence, the option $A$ is the correct answer.