A conducting sphere of radius $10\, cm$ is charged $10\,\mu \,C$. Another uncharged sphere of radius $20\, cm$ is allowed to touch it for some time. After that if the sphere are separated, then surface density of charges, on the spheres will be in the ratio of
AIIMS 2002, Medium
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(c) After redistribution new charges on spheres are $Q{'_1} = \left( {\frac{{10}}{{10 + 20}}} \right) \times 10 = \frac{{10}}{3}\,\mu C$
and $Q{'_2} = \left( {\frac{{20}}{{10 + 20}}} \right) \times 10 = \frac{{20}}{3}\,\mu C$
Ratio of charge densities $\frac{{{\sigma _1}}}{{{\sigma _2}}} = \frac{{Q'_1}}{{Q'_2}} \times \frac{{{r_2}^2}}{{{r_1}^2}}$
$ = \frac{{10/3}}{{20/3}} \times {\left( {\frac{{20}}{{10}}} \right)^2} = \frac{2}{1}$$\left\{ {\sigma = \frac{Q}{{4\pi {r^2}}}} \right\}$
and $Q{'_2} = \left( {\frac{{20}}{{10 + 20}}} \right) \times 10 = \frac{{20}}{3}\,\mu C$
Ratio of charge densities $\frac{{{\sigma _1}}}{{{\sigma _2}}} = \frac{{Q'_1}}{{Q'_2}} \times \frac{{{r_2}^2}}{{{r_1}^2}}$
$ = \frac{{10/3}}{{20/3}} \times {\left( {\frac{{20}}{{10}}} \right)^2} = \frac{2}{1}$$\left\{ {\sigma = \frac{Q}{{4\pi {r^2}}}} \right\}$
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