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Q.
Two charged spheres of radii $R_1 \, and \, R_2$ have equal surface charge density. The ratio of their potential is
Odisha JEEOdisha JEE 2009Electrostatic Potential and Capacitance
Solution:
Given two spheres of radii $R _{1}$ and $R _{2}$ have equal surface charge density.
To find the ratio of their potentials at an equidistant external point.
Let us suppose that the sphere with radius $R _{1}$ has charge $q _{1}$ on it, and that the sphere with radius $R _{2}$ has a charge $q _{2}$ on it. Now, they have same charge densities, so $\frac{ q _{1}}{4 \pi R _{1}{ }^{2}}=\frac{ q _{2}}{4 \pi R _{2}{ }^{2}}$
So, $\frac{ q _{1}}{ q _{2}}=\left(\frac{ R _{1}}{ R _{2}}\right)^{2} $...(i)
Potential in first sphere, $V _{1}=\frac{ k q _{1}}{ R _{1}}$
Potential in second sphere, $V _{2}=\frac{ kq }{ R _{2}}$
Thus, $\frac{ V _{1}}{ V _{2}}=\frac{ q _{1}}{ R _{1}} \cdot \frac{ R _{2}}{ q _{2}}$
$
=\frac{ q _{1}}{ q _{2}}\left(\frac{ R _{1}}{ R _{2}}\right)^{2}
$
From (i), $\frac{V_{1}}{V_{2}}=\left(\frac{R_{1}}{R_{2}}\right)^{2} \times \frac{R_{1}}{R_{2}}=\frac{R_{1}}{R_{2}}$