Electrostatics

JEE Main level, single correct, medium

236 of 311

Question 236 of 311

Two short electric dipoles AA and BB having dipole moment p1p_1 and p2p_2 respectively are placed with their axis mutually perpendicular as shown in the figure. The resultant electric field at a point xx is making an angle of 60°60° with the line joining points OO and xx. The ratio of the dipole moments p2/p1p_2/p_1 is _______.

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Answer: (B) 232\sqrt{3}

Let the distance of point xx from the origin OO be rr.

Point xx lies on the axial line of dipole A. The electric field at xx due to dipole A is directed along the horizontal axis and its magnitude is given by:
EA=14πϵ02p1r3E_A = \dfrac{1}{4\pi\epsilon_0} \dfrac{2p_1}{r^3}

Point xx lies on the equatorial line of dipole B. The electric field at xx due to dipole B is directed perpendicular to the horizontal axis and its magnitude is given by:
EB=14πϵ0p2r3E_B = \dfrac{1}{4\pi\epsilon_0} \dfrac{p_2}{r^3}

The resultant electric field makes an angle of 60∘60^{\circ} with the line OxOx (the horizontal axis). Therefore, the tangent of this angle is the ratio of the perpendicular component of the electric field to the parallel component:
tan⁡60∘=EBEA\tan 60^{\circ} = \dfrac{E_B}{E_A}

Substituting the expressions for EAE_A and EBE_B:
3=14πϵ0p2r314πϵ02p1r3\sqrt{3} = \dfrac{\dfrac{1}{4\pi\epsilon_0} \dfrac{p_2}{r^3}}{\dfrac{1}{4\pi\epsilon_0} \dfrac{2p_1}{r^3}}

3=p22p1\sqrt{3} = \dfrac{p_2}{2p_1}

p2p1=23\dfrac{p_2}{p_1} = 2\sqrt{3}

Answer: 232\sqrt{3}

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