Question

A short magnet of moment $MA-m^2$ is placed on the $x$ axis at $x=-d$ m and another identical magnet is placed on $y=$$-d$ m. Their magnetic moments are directed in the positive and negative $x-$ directions respectively. The magnetic field at the origin (in $T$) is :

• A. $\frac{{\mu }_0}{4\pi }\times \frac{3M}{d^3}$
• B. $\frac{{\mu }_0}{2\pi }\times \frac{3M}{d^3}$
• C. $\frac{{\mu }_0}{8\pi }\times \frac{3M}{d^3}$
• D. $\frac{{\mu }_0}{2\pi }\times \frac{M}{d^3}$

Answer 1

Answer: (A)

$\frac{{\mu }_0}{4\pi }\times \frac{3M}{d^3}$

The position and orientation of the magnets is as shown in the figure.
From the formulae mentioned above:-
For the magnet placed on $x=-d$, the magnetic field on the origin is $\frac{{\mu }_0}{4\pi }(\frac{2M}{d^3})$ along $\hat{i}$.
For the magnet placed on $y=-d$, the magnetic field on the origin is $\frac{{\mu }_0}{4\pi }(\frac{M}{d^3})$ along $\hat{i}$.
So, the resultant magnetic field due to the magnets on the origin is $\frac{{\mu }_0}{4\pi }(\frac{2M}{d^3})\hat{i}+\frac{{\mu }_0}{4\pi }(\frac{M}{d^3})\hat{i}=\frac{{\mu }_0}{4\pi }(\frac{3M}{d^3})\hat{i}$