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(A) When two conducting spheres are brought into contact,charge flows until their potentials become equal.Let the radii be $R_1 = 10 \ cm$ and $R_2 =

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$2$

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(A) When two conducting spheres are brought into contact,charge flows until their potentials become equal.Let the radii be $R_1 = 10 \ cm$ and $R_2 = 20 \ cm$.Since the potentials are equal,$V_1 = V_2 \implies \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2}$.This implies the ratio of charges is $\frac{Q_1}{Q_2} = \frac{R_1}{R_2}$.The surface charge density is defined as $\sigma = \frac{Q}{4\pi R^2}$.Therefore,the ratio of surface charge densities is $\frac{\sigma_1}{\sigma_2} = \frac{Q_1}{4\pi R_1^2} 	imes \frac{4\pi R_2^2}{Q_2} = \frac{Q_1}{Q_2} 	imes \frac{R_2^2}{R_1^2}$.Substituting $\frac{Q_1}{Q_2} = \frac{R_1}{R_2}$,we get $\frac{\sigma_1}{\sigma_2} = \frac{R_1}{R_2} 	imes \frac{R_2^2}{R_1^2} = \frac{R_2}{R_1}$.Given $R_1 = 10 \ cm$ and $R_2 = 20 \ cm$,the ratio is $\frac{\sigma_1}{\sigma_2} = \frac{20}{10} = 2 : 1$.

$A$ conducting sphere of radius $10 \ cm$ is charged with $10 \ \mu C$. Another uncharged sphere of radius $20 \ cm$ is brought into contact with it and then separated. The ratio of the surface charge densities on the spheres will be ....... (in $: 1$)

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