A neutral metal sphere is placed at a large distance from a charge. The magnitude of coulomb electrostatic interaction force between the charge and sphere is $F_{0}$ . If the distance between them is doubled, the new magnitude of coulomb electrostatic interaction force between the charge and sphere is $2^{- n} F_{0}$ . Find the value of n. (Assuming that the radius of sphere is very small as compared to the distance between them)

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answer is 14.

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Detailed Solution

Due to induction, the sphere behaves as dipole whose dipole moment is directly proportional to inducting field generated by point charge. So
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$p = \frac{a}{r^{2}}$ , a is a constant and r is the distance between them.
$F_{0} = \frac{b q p}{r^{3}} = \frac{b a q}{r^{5}} \Rightarrow F = \frac{b a q}{{(2 r)}^{5}} = \frac{b a q}{2^{5} r^{5}} = 2^{- 5} F_{0}$