$A$ circular coil of radius $4\, cm$ has $50$ turns. In this coil,a current of $2\, A$ is flowing. It is placed in a magnetic field of $0.1\, Wb/m^2$. The amount of work done in rotating it through $180^\circ$ from its equilibrium position will be ........ $J$.

$(a)$ $A$ circular coil of $30$ turns and radius $8.0 \; cm$ carrying a current of $6.0 \; A$ is suspended vertically in a uniform horizontal magnetic field of magnitude $1.0 \; T$. The field lines make an angle of $60^{\circ}$ with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning.
$(b)$ Would your answer change,if the circular coil in $(a)$ were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

$A$ small coil $C$ with $N = 200$ turns is mounted on one end of a balance beam and introduced between the poles of an electromagnet as shown in the figure. The cross-sectional area of the coil is $A = 1.0 \, cm^2$,and the length of the arm $OA$ of the balance beam is $l = 30 \, cm$. When there is no current in the coil,the balance is in equilibrium. On passing a current $I = 22 \, mA$ through the coil,the equilibrium is restored by putting an additional counterweight of mass $\Delta m = 60 \, mg$ on the balance pan. Find the magnetic induction $B$ at the spot where the coil is located. (in $, T$)

$A$ current loop in a magnetic field:

$(a)$ $A$ 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?

The figure represents four positions of a current-carrying coil in a magnetic field directed towards the right. $\hat{n}$ represents the direction of the area vector of the coil. The correct order of potential energy is: