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,
$B_{\text{net}}=\sqrt{B_1^2+B_2^2}=\sqrt{{\left(\frac{{\mu }_0}{4\pi }\frac{2M}{R^3}\right)}^2+{\left(\frac{{\mu }_0}{4\pi }\frac{M}{R^3}\right)}^2}$
$B_{\text{net}}=\frac{{\mu }_0}{4\pi R^3}\cdot \sqrt{5}M$
Given, $B_{\text{net}}=\frac{{\mu }_0}{4\pi R^3}\cdot M_0$
By comparision Eq. (i) and Eq. (ii), we get
$M=\frac{M_0}{\sqrt{5}}$