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A ring of radius R carries a non-uniform charge of linear density $λ = λcosθ$ (see in the figure) Magnitude of the net dipole moment of the ring is:

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$\begin{aligned} {πR}^{2} λ_{0} \end{aligned}$

$\begin{array}{r} 2 {πR}^{2} λ_{0} \end{array}$

$\frac{{πR}^{2}}{2} λ_{0}$

$4 {πR}^{2} λ_{0}$

answer is A.

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Detailed Solution

Charge = $λ_{0} Rcosθ . dθ$

dipole moment = $λ_{0} cosθRdθ \times 2 Rcosθ$

net dipolemoment

$= 2 \int_{0}^{π / 2} 2 λ_{0} R^{2} c o s^{2} θ d θ = (4 λ_{0} R^{2}) π / 4 = π λ_{0} R^{2}$

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