Consider a rectangular coil, PQRS is carrying current ‘I’ as shown in figure. Let PQ=RS=L and QR=SP=B
Suppose the coil is placed in a uniform magnetic field of strength ‘B’ (magnetic flux density) as shown in figure. Let θ be the angle between the magnetic field and plane of coil.
Force experienced by PQ, F1=BI.PQ.sin 90º
∴ F1 = BI.l [perpendicularly outward]
Force experienced by QR, F2=BI.QR.sinθ
∴ F2 = BIb.sinθ (downward)
Force experienced by RS, F3=BI.RS.sin 90º
∴ F3 = BI.l [perpendicularly inward]
Force experienced by SP, F_4=BI.SP.sin(π-θ)
∴ F4 =BI.b.sinθ (upward)
Here, F1 and F3 are equal, opposite and acting at a certain separation,thus they form couple. F2 and F4 are equal, opposite and acting in a straight line. Their effect will be cancelled by the rigidity of the coil.
Torque produced by the couple
τ = F1 or F3 × XZ
If there are ‘N’ turns of coil,
∴ τ =BINAcosθ
This is the required relation.