$A$ circular coil of radius $9 \, cm$ carrying a current of $2 \, A$ is free to rotate about an axis in its plane perpendicular to an external magnetic field of $\pi \times 10^{-2} \, T$. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a time period of $\frac{1}{3} \, s$. If the moment of inertia of the coil about its axis of rotation is $9 \times 10^{-5} \, kg \cdot m^2$, the number of turns of the coil is . . . . . .

$(a)$ $A$ current-carrying circular loop lies on a smooth horizontal plane. Can a uniform magnetic field be set up in such a manner that the loop turns around itself (i.e.,turns about the vertical axis)?
$(b)$ $A$ current-carrying circular loop is located in a uniform external magnetic field. If the loop is free to turn,what is its orientation of stable equilibrium? Show that in this orientation,the flux of the total field (external field $+$ field produced by the loop) is maximum.
$(c)$ $A$ loop of irregular shape carrying current is located in an external magnetic field. If the wire is flexible,why does it change to a circular shape?

Write an expression for the torque acting on a current-carrying loop suspended in a uniform magnetic field.

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

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

$A$ circular coil having $N$ turns and radius $R$ carrying current $I$ is held in the $z-x$ plane in a magnetic field $B\hat{k}$. The torque on the coil due to the magnetic field in $N-m$ is: