Two isolated metallic solid spheres of radii and are charged such that both have same charge density . The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is , the ratio is: [2023]

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Solution

Q.
Two isolated metallic solid spheres of radii $R$ and $2 R$ are charged such that both have same charge density $σ$ . The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $σ'$ , the ratio $\frac{σ'}{σ}$ is: [2023]

1 $\frac{9}{4}$

2 $\frac{4}{3}$

3 $\frac{5}{3}$

4 $\frac{5}{6}$

Ans.
(4)

$\frac{Q_{1}'}{4 π ε_{0} R} = \frac{Q_{2}'}{4 π ε_{0} (2 R)}$

$∴$ $Q_{2}' = 2 Q_{1}'$

$Q_{1}' + Q_{2}' = Q_{1} + Q_{2}$

$∴$ $\frac{Q_{2}'}{2} + Q_{2}' = 20 π R^{2} σ$

$∴$ $\frac{3}{2} Q_{2}' = 20 π R^{2} σ$

$∴$ $\frac{Q_{2}'}{4 π {(2 R)}^{2}} = \frac{2}{3} \frac{20 π R^{2} σ}{16 π R^{2}}$

$∴$ $\frac{σ'}{σ} = \frac{5}{6}$

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