SATHEE: Electrostatics 3 Question 20

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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Answer:

Correct Answer: 20. $\frac{Q (R + r)}{4 π ε_{0} (r^{2} + R^{2})}$

Solution:

Let $q_{1}$ and $q_{2}$ be the charges on them.

$\begin{matrix} σ_{1} = σ_{2} \\ ∴ \frac{q_{1}}{4 π r^{2}} = \frac{q_{2}}{4 π R^{2}} \Rightarrow \frac{q_{1}}{q_{2}} = \frac{r^{2}}{R^{2}} \end{matrix}$

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 $V = \frac{1}{4 π ε_{0}} \cdot \frac{q_{1}}{r} + \frac{1}{4 π ε_{0}} \cdot \frac{q_{2}}{R}$

potential due to $q_{2}$

$= \frac{Q (R + r)}{4 π ε_{0} (r^{2} + R^{2})}$

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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.

Answer:

Solution:

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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.