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Two charged spherical conductors of radius $R_1$ and $R_2$ ....

Two charged spherical conductors of radius $R_1$ and $R_2$ are connected by wire. Then the ratio of surface charge densities of the spheres $\frac {\sigma_1}{\sigma_2}$ is:

(1) $\frac {R_1}{R_2}$
(2) $\frac {R_2}{R_1}$
(3) $\sqrt \frac {R_1}{R_2}$
(4) $\sqrt \frac {R_2}{R_1}$

Solution

$V_1=V_2$

$\therefore k\frac {Q_1}{R_1}=k\frac {Q_2}{R_2}$

$\Rightarrow \frac {\sigma_1 . A_1}{R_1}=\frac {\sigma_2 . A_2}{R_2}$

$\Rightarrow \frac {\sigma_1 . 4\pi {R_1}^2}{R_1}=\frac {\sigma_2 . 4\pi {R_2}^2}{R_2}$

$\Rightarrow \sigma_1 . R_1 = \sigma_2 . R_2 $

$\Rightarrow \frac {\sigma_1}{\sigma_2} = \frac {R_2}{R_1}$

Answer: (2)