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Two identical spheres having charges $+ q_{1}$ and $+ q_{2}$ are kept at a distance "d". These two are brought in contact with an insulated handle and again separated to the same distance. Find the force $F_{2}$ between them in terms of initial force $F_{1}$ .

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Solution

Force between the two charges ( $+ q_{1} a n d + q_{2})$ , if they are at a distance 'd' is
$F_{1} = \frac{1}{4 π ϵ_{0}} \frac{q_{1} q_{2}}{d^{2}} . . . . . (1)$
When the metal spheres are in contact the charge flow from one sphere to another till both the spheres acquire the same charge.
Here $q_{1} a n d q_{2}$ are of same sign, hence after contact each sphere will have a charge $\frac{(q_{1} + q_{2})}{2}$
The new charge $Q = \frac{q_{1} + q_{2}}{2}$
Now the force between them,
$F_{2} = \frac{1}{4 π ϵ_{0}} \frac{[\frac{q_{1} + q_{2}}{2}]^{2}}{d^{2}} . . . . . (2)$
From equations (1) and (2)
$F_{2} = F_{1} \frac{(q_{1} + q_{1})^{2}}{4 q_{1} q_{2}}$

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