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Visualising Potential

Two metal sph...

Question

Two metal spheres (radii $r_{1}, r_{2}$ with $r_{1} < r_{2}$ ) are very far apart but are connected by a thin wire. If their combined charge is Q, then what is their common potential?

k Q / (r_{1} + r_{2})

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k Q / (r_{1} - r_{2})

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- k Q / (r_{1} + r_{2})

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- k Q / r_{1} r_{2}

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Solution

The correct option is C $k Q / (r_{1} + r_{2})$
Potential due to a sphere of radius $r$ having charge $q$ is given by $V = \frac{k q}{r}$
Let the charge of the spheres be $q_{1}$ and $q_{2}$ .
After connecting both spheres, potential of both of them must be equal.
$V_{c} = V_{1} = V_{2}$
$∴$ $V_{c} = \frac{k q_{1}}{r_{1}} = \frac{k q_{2}}{r_{2}}$
Thus we get $\frac{q_{1}}{q_{2}} = \frac{r_{1}}{r_{2}}$ .......... (i)
Total charge remains constant so,
$q_{1} + q_{2} = Q$ ....... (ii)
From Eqs. (i) and (ii), we get $q_{1} + \frac{r_{2}}{r_{1}} q_{1} = Q$
$⟹ q_{1} = \frac{Q r_{1}}{r_{1} + r_{2}}$
Thus common potential $V_{c} = \frac{k q_{1}}{r_{1}}$
$⟹ V_{c} = \frac{k Q}{r_{1} + r_{2}}$

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