(a) Derive the expression for the torque acting on a current carrying loop placed in a magnetic field.
Torque on current carrying loop in magnetic field :

(b) Explain the significance of a radial magnetic field when a current carrying coil is  kept in it.

Consider a coil PQRS placed in a magnetic field. Let 0 be the angle between the plane of the coil and direction of B.
When current flow through coil each side experiences a force. The forces on vertical side will constitute a couple.
Moment of torque = One of forces × Perpendicular distance between lines of action of force
$\begin{array}{l}\mathrm{\tau }=\mathrm{F}\times \mathrm{QT}=\mathrm{F}\times \mathrm{bsin}\mathrm{\theta }\\ =\mathrm{IlBbsin}\mathrm{\theta }\\ \mathrm{\tau }=\mathrm{I}\left(\mathrm{lb}\right)\mathrm{Bsin}\mathrm{\theta }\\ \mathrm{\tau }=\mathrm{I}(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}})(\because \text{ Area }=\mathrm{lb})\end{array}$
If there are N turns, then $\mathrm{\tau }=\mathrm{NI}(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}})=\overrightarrow{\mathrm{M}}\times \overrightarrow{\mathrm{B}}$
Where, $\overrightarrow{\mathrm{M}}=\mathrm{NI}\overrightarrow{\mathrm{A}}$ is magnetic moment of loop.
A magnetic field, in which the plane of the coil in all positions remains parallel to the
direction of magnetic field is called radial magnetic field.
In a radial magnetic field, magnetic torque remains maximum for all the positions of the coil.