An electron in the ground state of a hydrogen atom is revolving in an anticlockwise direction in a circular orbit of radius $r$. The atom is placed in a uniform magnetic field $B$ in such a way that the magnetic moment of the orbital electron makes an angle of $30^{\circ}$ with the magnetic field. The torque experienced by the orbital electron is

$A$ thin wire of length $L$ made of an insulating material is bent to form a circular loop and a positive charge $q$ is given so that it is distributed uniformly around the circumference of the loop. The loop is then rotated with an angular speed $\omega$ about an axis passing through its centre. If a uniform magnetic field $B$ directed parallel to the plane of the loop is applied,then the magnitude of the magnetic torque on the loop is:

$A$ coil in the shape of an equilateral triangle of side $2 \,cm$ is suspended from a vertex such that it hangs in a vertical plane between the poles of a permanent magnet producing a horizontal magnetic field of $100 \times 10^{-3} \,T$. The magnetic field is parallel to the plane of the coil. For the moment of couple acting on the coil to be $2 \sqrt{3} \times 10^{-5} \,Nm$, the current to be passed through the coil is (in $A$)

$A$ uniform current-carrying ring of mass $m$ and radius $R$ is suspended by a massless string as shown. $A$ uniform magnetic field $B_0$ exists in the region to keep the ring in a horizontal position. Determine the current $I$ in the ring.

$A$ conducting ring of mass $2 \ kg$ and radius $0.5 \ m$ is placed on a smooth horizontal plane. The ring carries a current $i = 4 \ A$. $A$ horizontal magnetic field $B = 10 \ T$ is switched on at time $t = 0$ as shown in the figure. The initial angular acceleration of the ring will be:

Derive an expression for the torque acting on a current-carrying loop which subtends an angle $\theta$ with a uniform magnetic field.