question_answer

Two short magnets of equal dipole moments M are fastened perpendicularly at their centre (figure). The magnitude of the magnetic field at a distance d from the centre on the bisector of the right angle is

A)            \[\frac{{{\mu }_{0}}}{4\pi }\frac{M}{{{d}^{3}}}\]

B)            \[\frac{{{\mu }_{0}}}{4\pi }\frac{M\sqrt{2}}{{{d}^{3}}}\]

C)            \[\frac{{{\mu }_{0}}}{4\pi }\frac{2\sqrt{2}M}{{{d}^{3}}}\]

D)             \[\frac{{{\mu }_{0}}}{4\pi }\frac{2M}{{{d}^{3}}}\]

Correct Answer: C

Solution :

                   Resultant magnetic moment of the two magnets is                    \[{{M}_{net}}=\sqrt{{{M}^{2}}+{{M}^{2}}}=\sqrt{2}M\] Imagine a short magnet lying along OP with magnetic moment equal to \[M\sqrt{2}\]. Thus point P lies on the axial line of the magnet.            \[\therefore \]Magnitude of magnetic field at P is given by \[B=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\sqrt{2}M}{{{d}^{3}}}\]