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  4. 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 (σ1 / σ2) is

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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 $\left(\sigma_{1} / \sigma_{2}\right)$ is

NEETNEET 2021Electrostatic Potential and Capacitance

Solution:

$\frac{ Q _{1}}{4 \pi \varepsilon_{0} R _{1}}=\frac{ Q _{2}}{4 \pi \varepsilon_{0} R _{2}}= V$
$\therefore \frac{ Q _{1}}{ Q _{2}}=\frac{ R _{1}}{ R _{2}}$
$\therefore \frac{\sigma_{1}}{\sigma_{2}}=\frac{ Q _{1} / 4 \pi R _{1}^{2}}{ Q _{2} / 4 \pi R _{2}^{2}}$
$=\frac{ Q _{1}}{ Q _{2}}\left(\frac{ R _{2}}{ R _{1}}\right)^{2}=\left(\frac{ R _{1}}{ R _{2}}\right)\left(\frac{ R _{2}}{ R _{1}}\right)^{2}$
$\Rightarrow \frac{\sigma_{1}}{\sigma_{2}}=\frac{ R _{2}}{ R _{1}}$