Two short magnets P and Q of magnetic moments, m1 and m2, respectively, are placed one over another with their magnetic axes mutually perpendicular to each other. It is found that the resultant field at a point on the axial line of magnet P is inclined at 30° with this axis. The ratio of their magnetic moments, is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{m_1}{m_2} = \frac{\sqrt{3}}{2} \)
The magnets are at the same place with perpendicular axes, so a point on the axial line of P lies on the equatorial line of Q, at the same distance r from both. For a short magnet the axial field is twice the equatorial field at the same distance:
\( \displaystyle B_P = \frac{\mu_0}{4\pi}\frac{2m_1}{r^{3}}\ \text{(along the axis of P)} \)
\( \displaystyle B_Q = \frac{\mu_0}{4\pi}\frac{m_2}{r^{3}}\ \text{(parallel to the axis of Q)} \)
The two fields are perpendicular, and the resultant makes 30° with the axis of P:
\( \displaystyle \tan 30^\circ = \frac{B_Q}{B_P} = \frac{m_2}{2m_1} \)
\( \displaystyle \frac{1}{\sqrt{3}} = \frac{m_2}{2m_1} \)
\( \displaystyle \frac{m_1}{m_2} = \frac{\sqrt{3}}{2} \)