Electrostatics

JEE Main level, single correct, medium

264 of 311

Question 264 of 311

An electric dipole has fixed dipole moment p⃗\vec{p}, which makes angle θ\theta with respect to x−x-axis. When subjected to an electric field E⃗1=Ei^,{\vec{E}}_{1}=E\hat{i}, it experiences a torque T⃗1=τk^{\vec{T}}_{1}=\tau \hat{k}. When subjected to another electric field E⃗2=3E1j^{\vec{E}}_{2}=\sqrt{3}{E}_{1}\hat{j} it experiences a torque T⃗2=−T⃗1{\vec{T}}_{2}=-{\vec{T}}_{1} . The angle θ\theta is:

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Answer: (D) 60o{60}^{o}

P⃗=Pcos⁡⁡θi^+Psin⁡⁡θj^\vec{P}=P\cos ⁡\theta \hat{i}+P\sin ⁡\theta \hat{j}

T⃗=(Pcos⁡⁡θi^+Psin⁡⁡θj^)×(Ei^)\vec{T}=\left(P\cos ⁡\theta \hat{i}+P\sin ⁡\theta \hat{j}\right)\times \left(E\hat{i}\right)

=−PEsin⁡⁡θk^=-PE\sin ⁡\theta \hat{k}

T⃗2=(Pcos⁡⁡θi^+Psin⁡⁡θj^)×(3Ej^){\vec{T}}_{2}=\left(P\cos ⁡\theta \hat{i}+P\sin ⁡\theta \hat{j}\right)\times \left(\sqrt{3}E\hat{j}\right)

=3PEcos⁡⁡θk^=\sqrt{3}PE\cos ⁡\theta \hat{k}

T⃗2=−T⃗1{\vec{T}}_{2}=-{\vec{T}}_{1}

∴sin⁡⁡θ=3cos⁡⁡θ∴\sin ⁡\theta =\sqrt{3}\cos ⁡\theta

∴tan⁡⁡θ=3∴\tan ⁡\theta =\sqrt{3}

∴θ=60o∴\theta ={60}^{o}

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