Question

A magnet of dipole moment M placed at origin as shown in figure. Then select correct alternative about magnetic field produced by it :

• A. $\frac{{\mu }_0}{8\pi }\frac{\sqrt{7}M}{a^3}$, at angle $\varphi ={30}^o$ with x - axis
• B. $\frac{{\mu }_0}{8\pi }\frac{\sqrt{7}M}{a^3}$, at angle $\varphi ={60}^o$ with x - axis
• C. $\frac{{\mu }_0}{4\pi }\frac{\sqrt{5}M}{a^3}$, at angle $\varphi =ta{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$ with x - axis
• D. $\frac{{\mu }_0}{8\pi }\frac{\sqrt{7}M}{a^3}$, at angle $\varphi =ta{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$ with x - axis

Answer 1

Answer: (D)

$\frac{{\mu }_0}{8\pi }\frac{\sqrt{7}M}{a^3}$, at angle $\varphi =ta{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$ with x - axis

Magnetic field produced by bar magnet

$B=\frac{{\mu }_{\circ }}{4M}\frac{M}{r^3}{\left(1+3{\cos }^2\theta \right)}^{1/2}$

$=\frac{{\mu }_{\circ }}{4M}\frac{M}{a^3}{\left(1+3{\left(\frac{1}{2}\right)}^2\theta \right)}^{1/2}$

$=\frac{{\mu }_{\circ }M}{4M{a}^3}\times \frac{\sqrt{7}}{2}$

$=\frac{{\mu }_{\circ }M\sqrt{7}}{8M{a}^3}$

$\varphi ={\tan }^{-1}\left(\frac{\tan \theta }{2}\right)$

$={\tan }^{-1}\left(\frac{\tan 60}{2}\right)$

hence, we get

$\varphi ={\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ with $x-axis$