Question 40 of 311

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

A$\dfrac{1}{4{\pi ϵ}_{0}}\dfrac{\left(R+r\right)}{2\left({R}^{2}+{r}^{2}\right)}Q$B$\dfrac{1}{4{\pi ϵ}_{0}}\dfrac{\left(2R+r\right)}{\left({R}^{2}+{r}^{2}\right)}Q$C$\dfrac{1}{4{\pi ϵ}_{0}}\dfrac{\left(R+2r\right)Q}{2\left({R}^{2}+{r}^{2}\right)}$D$\dfrac{1}{4{\pi ϵ}_{0}}\dfrac{(R+r)}{\left({R}^{2}+{r}^{2}\right)}Q$

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Answer: (D) $\dfrac{1}{4{\pi ϵ}_{0}}\dfrac{(R+r)}{\left({R}^{2}+{r}^{2}\right)}Q$

${Q}_{1}=\sigma 4{\pi r}^{2}$
${Q}_{2}=\sigma 4{\pi r}^{2}$
$\begin{matrix}∴ & Q=\sigma 4\pi \left({r}^{2}+{R}^{2}\right) \\ ∴ & \sigma =\dfrac{Q}{4\pi \left({r}^{2}+{R}^{2}\right)}\end{matrix}$
${V}_{C}=\dfrac{{KQ}_{1}}{r}+\dfrac{{KQ}_{2}}{R}$
$=\dfrac{{K}_{0}4{\pi r}^{2}}{r}+\dfrac{{K}_{0}4{\pi R}^{2}}{R}$
$=\dfrac{KQ4\pi (r+R)}{4\pi \left({r}^{2}+{R}^{2}\right)}$
$=\dfrac{KQ(r+R)}{\left({r}^{2}+{R}^{2}\right)}$