A point charge q is placed at a distance $l$ from centre of an uncharged conducting sphere of radius $r (l > > r)$

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Induced surface charge density $σ$ as a function of $θ$ is given by $\frac{q \cos θ}{4 π l^{2}}$

Induced surface charge density $σ$ as a function of $θ$ is given by $\frac{3 q \cos θ}{4 π l^{2}}$

Induced dipole moment of sphere is given by $\frac{q r^{3}}{l^{2}}$

Induced dipole moment of sphere is given by $\frac{2 q r^{3}}{l^{2}}$

answer is A, C.

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Detailed Solution

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$\frac{k q}{l^{2}} \sin θ = \frac{k p \sin θ}{r^{3}}$
$p = \frac{q r^{3}}{r^{2}}$
$E_{n e t} = \frac{2 k p \cos θ}{r^{3}} + \frac{k q \cos θ}{l^{2}}$
$\frac{σ}{\in_{0}} = \frac{2 k q \cos θ}{l^{2}} + \frac{k q \cos θ}{l^{2}} = \frac{3 k q \cos θ}{l^{2}}$
$σ = \frac{3 q \cos θ}{4 π l^{2}}$