Question

Two short bar magnets of equal dipole moments $M$ each are fastened mutually perpendicular to each other at their centers, as shown in the figure. The magnitude of the magnetic field at $P$ at a distance $d$ from their common center as shown in the figure is:

• A. $\frac{{\mu }_0}{4\pi }\frac{M}{d^3}$
• B. $\frac{{\mu }_0}{4\pi }\frac{2\sqrt{2}M}{d^3}$
• C. $\frac{{\mu }_0}{4\pi }\frac{2M}{d^3}$
• D. $\frac{{\mu }_0}{2\pi }\frac{M}{d^3}$

Answer 1

The correct option is B $\frac{{\mu }_0}{4\pi }\frac{2\sqrt{2}M}{d^3}$

Magnetic field due to bar magnet of magnetic moment $\overrightarrow{M}$ for equatorial position:
$B=\frac{2{\mu }_{0}M}{4\pi d^3}$,
where $\overrightarrow{M}$ is magnetic moment of magnet
The resultant magnetic moment of combination is:
$\overrightarrow{M}=M_1\hat{i}+M_2\hat{j}=M\hat{i}+M\hat{j}=2M\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
Magnetic field due to bar magnet of magnetic moment $\sqrt{2}\overrightarrow{M}$ for equatorial position:
$B_{net}=\frac{2\sqrt{2}{\mu }_{0}M}{4\pi d^3}$ along resultant of magnetic moment of two magnets.