Question

The capacitance of two concentric spherical shells of radii $R_1$ and $R_2\left(R_2>R_1\right)$ when inner sphere is earthed will be:

• A. $4\pi {\epsilon }_0{R}_2$
• B. $4\pi {\epsilon }_0\left(R_2-R_1\right)$
• C. $4\pi {\epsilon }_0\frac{R_1{R}_2}{\left(R_2-R_1\right)}$
• D. $4\pi {\epsilon }_0{R}_1$

Answer 1

Answer: (C)

We know that concentric spheres shell
$\left(R_2>R_1\right)$
$V_{R_1}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R_1}$
$V_{R_2}=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R_2}$
$V=V_{R_1}-V_{R_2}=\frac{1}{4\pi {\epsilon }_0}Q\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
If the capacitance of two concentric spherical shells be C then
$C=\frac{Q}{V}=\frac{Q}{\frac{Q}{4\pi {\epsilon }_0}\left(\frac{R_2-R_1}{R_1{R}_2}\right)}$
$=4\pi {\epsilon }_0\frac{\left(R_1{R}_2\right)}{\left(R_2-R_1\right)}$