Question

A charge $Q$ is distributed over two concentric conducting thin spherical shells radii $r$ and $R$ $( R > r )$. If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

• A. $\frac{1}{4 \pi \varepsilon_{0}} \frac{( R +2 r ) Q }{2\left( R ^{2}+ r ^{2}\right)}$
• B. $\frac{1}{4 \pi \varepsilon_{0}} \frac{( R + r )}{2\left( R ^{2}+ r ^{2}\right)} Q$
• C. $\frac{1}{4 \pi \varepsilon_{0}} \frac{( R + r )}{\left( R ^{2}+ r ^{2}\right)} Q$
• D. $\frac{1}{4 \pi \varepsilon_{0}} \frac{(2 R+r)}{\left(R^{2}+r^{2}\right)} Q$

Answer 1

Answer: (C)

Let the charges on inner and outer spheres are $Q_{1}$ and $Q_{2}$

Since charge density ' $\sigma^{\prime}$ is same for both spheres, so
$\sigma=\frac{Q_{1}}{4 \pi r^{2}}=\frac{Q_{2}}{4 \pi R^{2}} \Rightarrow \frac{Q_{1}}{Q_{2}}=\frac{r^{2}}{R^{2}}$
$Q _{1}+ Q _{2}= Q \Rightarrow \frac{ Q _{2} r ^{2}}{ R ^{2}}+ Q _{2}= Q$
$\Rightarrow Q _{2}=\frac{ QR ^{2}}{\left( r ^{2}+ R ^{2}\right)}$
$Q_{1}=\frac{r^{2}}{R^{2}} \cdot \frac{Q R^{2}}{\left(R^{2}+r^{2}\right)}=\frac{Q r^{2}}{\left(R^{2}+r^{2}\right)}$
Potential at centre $ ' O ^{\prime}=\frac{ kQ _{1}}{ r }+\frac{ kQ _{2}}{ R }$
$=k\left[\frac{ Qr ^{2}}{ r \left( R ^{2}+ r ^{2}\right)}+\frac{ QR ^{2}}{ R \left( R ^{2}+ r ^{2}\right)}\right]$
$=\frac{k Q(r+R)}{\left(R^{2}+r^{2}\right)}=\frac{1}{4 \pi \epsilon_{0}} \frac{(R+r)}{\left(R^{2}+r^{2}\right)} Q$