Two conducting spheres of radii $r_{1}$ and $r_{2}$ are charged such that they have the same electric field on their surfaces. The ratio of the electric potential at their centres is :

\sqrt{\frac{r_{1}}{r_{2}}}

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\frac{r_{1}}{r_{2}}

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\frac{r_{1}^{2}}{r_{2}^{2}}

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none of the above.

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Solution

The correct option is D $\frac{r_{1}}{r_{2}}$

Electric field on the surface of conducting sphere $= \frac{k Q}{R^{2}}$

according to given condition

$\frac{k q_{1}}{r_{1}^{2}} = \frac{k q_{2}}{r_{2}^{2}}$

$⟹ \frac{q_{1}}{q_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}}$ $- (1)$

ratio of potentials at centre

$= \frac{\frac{k q_{1}}{r_{1}}}{\frac{k q_{2}}{r_{2}}} = \frac{q_{1}}{q_{2}} \cdot \frac{r_{2}}{r_{1}}$

substituting $\frac{q_{1}}{q_{2}}$ from $(1)$

we get

ratio $= \frac{r_{1}}{r_{2}}$ Ans.

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