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A neutral small metal sphere is placed at a large distance from a point 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 5.

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Detailed Solution

Due to induction, the sphere behaves as dipole whose dipole moment is directly proportional to inducing field generated by point charge. So $p = \frac{a}{r^{2}},$ a is a constant and r is the distance between them.

$F_{0} = \frac{b q p}{r^{3}} = \frac{b a q}{r^{5}} \Rightarrow F = \frac{b a q}{{(2 r)}^{5}} = \frac{b a q}{2^{5} r^{5}} = 2^{- 5} F_{0}$

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