Two charged spherical conductors of radius and are connected by a wire. Then the ratio of surface charge densities of the spheres () is [2021]

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Solution

Q.
Two charged spherical conductors of radius $R_{1}$ and $R_{2}$ are connected by a wire. Then the ratio of surface charge densities of the spheres ( $σ_{1} / σ_{2}$ ) is [2021]

1 $\frac{R_{1}^{2}}{R_{2}^{2}}$

2 $\frac{R_{1}}{R_{2}}$

3 $\frac{R_{2}}{R_{1}}$

4 $\sqrt{(\frac{R_{1}}{R_{2}})}$

Ans.
(3)

When the two spheres are connected by conducting wire, the potential of both the spheres becomes same.

$V_{1} = V_{2} \Rightarrow \frac{k q_{1}}{R_{1}} = \frac{k q_{2}}{R_{2}}$

$\Rightarrow \frac{q_{1}}{q_{2}} = \frac{R_{1}}{R_{2}}$ ...(i)

Ratio of surface charge densities

$\frac{σ_{1}}{σ_{2}} = \frac{\frac{Q}{4 π R_{1}^{2}}}{\frac{Q}{4 π R_{2}^{2}}} = \frac{R_{2}^{2}}{R_{1}^{2}} \times \frac{q_{1}}{q_{2}} = {(\frac{R_{2}}{R_{1}})}^{2} \times \frac{R_{1}}{R_{2}} (From (i))$

$\Rightarrow \frac{σ_{1}}{σ_{2}} = \frac{R_{2}}{R_{1}}$

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