Question

Two short magnets of equal dipole moments $M$ are fastened perpendicularly at their centres which lies at origin. Let two magnets lie along $x$-axis and $y$-axis respectively. The magnitude of the magnetic field at a distance $R$ from the centre on the $y$-axis is $\frac{\mu_{0}}{4 \pi} \cdot \frac{M}{R^{3}}$ Assuming $R \gg l$ (magnet length), the magnitude of $M$ is

• A. $\frac{M_{0}}{2 \sqrt{2}}$
• B. $\frac{M_{0}}{2}$
• C. $\frac{M_{0}}{\sqrt{5}}$
• D. $\frac{M_{0}}{\sqrt{2}}$

Answer 1

Answer: (C)

Two magnet are joined as shown in figure. The magnetic field at point $P$ due to magnet $M_{1}$ is on axial line, while that due to $M_{2}$ is on equatorial line.

So, net magnetic field at a point $P$ at a distance $R$ on $Y$-axis,
$B_\text{net}=\sqrt{B_{1}^{2}+B_{2}^{2}}=\sqrt{\left(\frac{\mu_{0}}{4 \pi} \frac{2 M}{R^{3}}\right)^{2}+\left(\frac{\mu_{0}}{4 \pi} \frac{M}{R^{3}}\right)^{2}} $
$B_\text{net}=\frac{\mu_{0}}{4 \pi R^{3}} \cdot \sqrt{5} M$
Given, $B_\text{net}=\frac{\mu_{0}}{4 \pi R^{3}} \cdot M_{0}$
By comparision Eq. (i) and Eq. (ii), we get
$M=\frac{M_{0}}{\sqrt{5}}$