Question

Two short magnets of equal dipole moments $M$ are fastened perpendicularly at their centres as shown below (point $P$ is in the plane of the magnets)

The magnetic field (magnitude) at $P$, 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. $\frac{\mu_{0}}{4 \pi} \frac{2 M}{d^{3}}$
• D. $\frac{\mu_{0}}{4 \pi} \frac{2 \sqrt{2} M}{d^{3}}$

Answer 1

Answer: (D)

$M$ can be broken of both the magnets in terms of their components, one in direction of bisector and other perpendicular to it
Parallel components will be added whereas perpendicular components will be cancel out
So $\vec{B}_{\text {net }}=\frac{2 \mu_{0}}{4 \pi}\left(\frac{2 M}{d^{3}}\right) \cos 45^{\circ}=\frac{\mu_{0}}{4 \pi} \frac{2 \sqrt{2} M}{d^{3}}$