Q. A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is $60^{\circ}$ and one of the fields has a magnitude of $1.2 \times 10^{-2}\, T$. If the dipole comes to stable equilibrium at an angle of $30^{\circ}$ with this field, then the magnitude of the field is
Magnetism and Matter
Solution:
Here,
$\theta = 60^{\circ}$,
$B_{1} = 1.2 \times 10^{-2}\,T$
$\theta_{1} = 30$ and
$\theta_{2} = 60^{\circ } - 30^{\circ} = 30^{\circ}$
in stable equilibrium, torques due to two fields must be balanced i.e. $\tau_{1} = \tau_{2}$
$\Rightarrow \quad MB_{1}\, sin\, \theta_{1} = MB_{2}\, sin \,\theta_{2}$
or $\quad B_{2} = B_{1} \frac{ sin \,\theta _{1}}{ sin \,\theta _{2}}$
$= B_{2} = \frac{ sin \,30^{\circ}}{ sin \,30^{\circ}} = B_{1}$
$= 1.2 \times 10^{-2}\,T$
