The magnetic field at a point X on the axis of a short bar magnet is equal to the magnetic field at another point Y on its equatorial line. The ratio of the distances of X and Y from the centre of the magnet is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 2^{1/3} \)
A short bar magnet behaves as a magnetic dipole of moment \( m \). At a distance \( r \) much larger than its size, the field on its axis is \( \mu_0\,2m/4\pi r^{3} \). On its equatorial line it is only \( \mu_0 m/4\pi r^{3} \), half as large at the same distance.
Let X be at distance \( x \) and Y at distance \( y \) from the centre. Equating the two fields:
\( \displaystyle \frac{\mu_0}{4\pi}\,\frac{2m}{x^{3}}=\frac{\mu_0}{4\pi}\,\frac{m}{y^{3}} \)
\( \displaystyle \frac{x^{3}}{y^{3}}=2 \)
\( \displaystyle \Rightarrow\; \frac{x}{y}=2^{1/3} \)