Two spheres $A$ and $B$ of radius $a$ and $b$ respectively are at same electric potential. The ratio of the surface charge densities of $A$ and $B$ is
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(b) Given electric potential of spheres are same i.e. ${V_A} = {V_B}$
$==>$ $\frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}}}{a} = $$\frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_2}}}{b} \Rightarrow $$\frac{{{Q_1}}}{{{Q_2}}} = \frac{a}{b}$......$(i)$
as surface charge density $\sigma = \frac{Q}{{4\pi {r^2}}}$
$==>$ $\frac{{{\sigma _1}}}{{{\sigma _2}}} = \frac{{{Q_1}}}{{{Q_2}}} \times \frac{{{b^2}}}{{{a^2}}} = \frac{a}{b} \times \frac{{{b^2}}}{{{a^2}}} = \frac{b}{a}$
$==>$ $\frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}}}{a} = $$\frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_2}}}{b} \Rightarrow $$\frac{{{Q_1}}}{{{Q_2}}} = \frac{a}{b}$......$(i)$
as surface charge density $\sigma = \frac{Q}{{4\pi {r^2}}}$
$==>$ $\frac{{{\sigma _1}}}{{{\sigma _2}}} = \frac{{{Q_1}}}{{{Q_2}}} \times \frac{{{b^2}}}{{{a^2}}} = \frac{a}{b} \times \frac{{{b^2}}}{{{a^2}}} = \frac{b}{a}$
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