Question

A small conducting sphere of radius $r$ is lying concentrically inside a bigger hollow conducting sphere of radius $R$. The bigger and smaller spheres are charged with $Q$ and $q (Q > q)$ and are insulated from each other. The potential difference between the spheres will be

• A. $\frac {1}{4\pi\epsilon_0} \left(\frac {Q}{R}-\frac {q}{r}\right)$
• B. $\frac {1}{4\pi\epsilon_0} \left(\frac {q}{R}-\frac {Q}{r}\right)$
• C. $\frac {1}{4\pi\epsilon_0} \left(\frac {q}{r}-\frac {Q}{R}\right)$
• D. $\frac {1}{4\pi\epsilon_0} \left(\frac {q}{r}-\frac {q}{R}\right)$

Answer 1

Answer: (D)

$\frac {1}{4\pi\epsilon_0} \left(\frac {q}{r}-\frac {q}{R}\right)$

Answer 2

The potential $V_{1}$ of smaller sphere is given by
$V_{1}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{r}+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}\,\,\,\,\,\,\,\,\,....(i)$
The potential $V_{2}$ of bigger sphere is given by

So, the potential difference between the plates
$V=V_{1}-V_{2}$
or $\,\,\, V =\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{r}+\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q}{R}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{R}$
$=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{R}$
$=\frac{1}{4 \pi \varepsilon_{0}}\left(\frac{q}{r}-\frac{q}{R}\right)$