Q126 / 160 · TS EAMCET 2014 · Physics

Two concentric hollow spherical shells have radii $r$ and $R(R \gg r)$. A charge $Q$ is distributed on them such that the surface charge densities are equal. The electric potential at the centre is

  • Correct answer
    A
    $\frac{Q(R+r)}{4 \pi \varepsilon_0\left(R^2+r^2\right)}$
  • B
    $\frac{Q\left(R^2+r^2\right)}{4 \pi \varepsilon_0(R+r)}$
  • C
    $\frac{Q}{(R+r)}$
  • D
    zero

Solution

We know that, the surface charge densities is $$ \sigma=\frac{Q}{4 \pi\left(r^2+R^2\right)} $$ The electric potential at the centre $$ \begin{aligned} V & =\frac{1}{4 \pi \varepsilon_0}\left[\frac{\sigma \times 4 \pi r^2}{r}+\frac{\sigma \times 4 \pi R^2}{R}\right] \\ & =\frac{1}{4 \pi \varepsilon_0} \times \sigma \times 4 \pi\left[\frac{r^2}{r}+\frac{R^2}{R}\right] \\ & =\frac{\sigma}{\varepsilon_0}(r+R) \end{aligned} $$ and $\quad V=\frac{Q}{\varepsilon_0 \cdot 4 \pi\left(r^2+R^2\right)} \cdot(r+R)$ $$ \Rightarrow \quad V=\frac{Q}{4 \pi \varepsilon_0} \cdot \frac{(r+R)}{\left(r^2+R^2\right)} $$

Hence, the correct answer is option: 'A'.

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