Electrostatics 3 Question 20
20. A charge $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R(>r)$ such that the surface densities are equal. Find the potential at the common centre.
(1981, 3M)
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Solution:
- Let $q _1$ and $q _2$ be the charges on them.
$$ \begin{gathered} \sigma _1=\sigma _2 \\ \therefore \quad \frac{q _1}{4 \pi r^{2}}=\frac{q _2}{4 \pi R^{2}} \Rightarrow \frac{q _1}{q _2}=\frac{r^{2}}{R^{2}} \end{gathered} $$
i.e., charge on them is distributed in above ratio.
or $q _1=\frac{r^{2}}{r^{2}+R^{2}} Q$ and $q _2=\frac{R^{2}}{r^{2}+R^{2}} Q$
Potential at centre $V=$ potential due to $q _1+$
or $\quad V=\frac{1}{4 \pi \varepsilon _0} \cdot \frac{q _1}{r}+\frac{1}{4 \pi \varepsilon _0} \cdot \frac{q _2}{R}$
potential due to $q _2$
$$ =\frac{Q(R+r)}{4 \pi \varepsilon _0\left(r^{2}+R^{2}\right)} $$
