Question 2: A current carrying loop is placed in a uniform magnetic field. Derive an equation for the torque acting on it.

Consider a rectangular coil carrying current I as shown in the figure. The coil is placed in a uniform magnetic field B and the sides PQ and RS are perpendicular and PS and QR is along the direction of the field.

Now we know that a current carrying conductor of length L when placed in a magnetic field, experiences a force = I (LxB). Now the sides QR and PS are parallel (angle=0) to the field so they don’t contribute to the force on the loop. Sides PQ and SR are perpendicular to the direction of the field, therefore, a force of magnitude ILB (angle = 90) is exerted on these sides. Let they are F1on SR and F2 on PQ. Now F1 is out of the page and F2 is into the page. (Use Fleming’s Left Hand Rule). This means two equal forces acting in opposite directions, not in the same straight line, on the same body. Hence they will produce a torque (couple) tending to rotate the coil along XX in clockwise direction. The torque of the couple is given by;

Ʈ = one of the force x perpendicular distance between them

= IBL x b                               (b is the breadth of the coil = perpendicular distance between the forces).

As Lb = area of the coil = A, hence,

Ʈ = IAB

This equation is derived for a single loop of a wire. If there are N turns of the wire in the loop, then N currents are flowing to produce the torque and hence;   Ʈ = NIAB

Now if the B-field is making some other angle α with the plane of the coil, then, it can be resolved in two components parallel and perpendicular to the plane of the coil. Then Bcosα is parallel to the plane and Bsinα is perpendicular to the plane. Therefore, the above equation can be written as;

Ʈ = IAB cosα                                        and for N turns of the coil,

Ʈ =N IAB cosα

A coil will rotate until its plane is normal to the field. At this position the coil links maximum number of lines of B. This equation can be applied to all flat coils placed in a uniform magnetic field.

Unit of torque:  The SI unit of torque is newton-meter (N m).