Question

Two short electric dipoles \(A\) and \(B\) having dipole moments \(p_1\) and \(p_2\) respectively are placed with their axes mutually perpendicular as shown in the figure. The resultant electric field at a point \(x\) is making an angle of \(60^\circ\) with the line joining points \(O\) and \(x\). The ratio of dipole moments \( \dfrac{p_2}{p_1} \) is:

• A. \( \dfrac{\sqrt3}{2} \)
• B. \( 2\sqrt3 \)
• C. \( \dfrac{1}{\sqrt3} \)
• D. \( \sqrt3 \)

Answer 1

The Correct Option is D

Concept: Electric field of a short dipole:

• Along axial line: \[ E_{\text{axial}}=\frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3} \]
• Along equatorial line: \[ E_{\text{equatorial}}=\frac{1}{4\pi\varepsilon_0}\frac{p}{r^3} \]

From the figure:

• Dipole \(A\) produces axial field at point \(x\).
• Dipole \(B\) produces equatorial field at point \(x\).

Step 1:Write electric fields} \[ E_A=\frac{1}{4\pi\varepsilon_0}\frac{2p_1}{r^3} \] \[ E_B=\frac{1}{4\pi\varepsilon_0}\frac{p_2}{r^3} \] These fields are perpendicular.
Step 2:Use the resultant angle} Given resultant makes \(60^\circ\) with the horizontal line. \[ \tan 60^\circ=\frac{E_B}{E_A} \] \[ \sqrt3=\frac{\frac{p_2}{4\pi\varepsilon_0 r^3}}{\frac{2p_1}{4\pi\varepsilon_0 r^3}} \] \[ \sqrt3=\frac{p_2}{2p_1} \]
Step 3:Find the ratio} \[ \frac{p_2}{p_1}=\sqrt3 \] \[ \boxed{\sqrt3} \]