The expression for the torque acting on a coil having area of cross-section $A$,number of turns $n$,placed in a magnetic field of strength $B$,making an angle $\theta$ with the normal to the plane of the coil,when a current $i$ is flowing in it,will be

$A$ galvanometer coil has $500$ turns and each turn has an average area of $3 \times 10^{-4} \ m^{2}$. If a torque of $1.5 \ Nm$ is required to keep this coil parallel to the magnetic field when a current of $0.5 \ A$ is flowing through it,the strength of the magnetic field (in $T$) is:

$(a)$ $A$ 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.)

$A$ uniform,constant magnetic field $\vec B$ is directed at an angle of $45^o$ to the $x-$ axis in the $xy-$ plane. $PQRS$ is a rigid square wire frame carrying a steady current $I_0,$ with its centre at the origin $O.$ At time $t = 0,$ the frame is at rest in the position shown in the figure,with its sides parallel to the $x$ and $y$ axis. Each side of the frame is of mass $M$ and length $L.$ Find the torque $\vec \tau$ acting on the frame.

$A$ current-carrying loop is free to turn in a uniform magnetic field. The loop will come into equilibrium when its plane is inclined at

Four wires,each of length $2.0\,m$,are bent into four loops $P, Q, R$ and $S$ and then suspended in a uniform magnetic field. If the same current is passed in each,then the torque will be maximum on the loop: