3.10 Rectangular coil in a magnetic field

As the current carrying conductor experiences a force when placed in a magnetic field, each side of a current carrying rectangular coil experiences a force in a magnetic field. In the present section we shall see in what way the rectangular loop carrying current is influenced by a magnetic field.
Consider a rectangular coil of length l and breadth b carrying a current I placed in a uniform magnetic field B.
The magnitude of experienced by each side of loop is given below.
Force acting on side ab= IbB in the upward direction.
Force acting on side bc= IlB along PQ.
Force acting on side cd= IbB in the downward direction
Force acting on side da= IlB along RS.
As the force acting on the upper and lower sides are equal and opposite along the same line of action, they cancel each other. As the force acting on the sides bc and da are equal and opposite along different lines of action they constitute a couple. Hence the rectangular coil experiences a torque.
The perpendicular distance between the lines of action is b sin theta
Therefore the magnitude of torque acting on the coil is
T= b sin theta IlB
T= IlB sin theta
T= IAB sin theta, where A=lb the area
If there are N turns for the coil then T= NIAB sin theta
We know NIA = m the magnetic moment. Hence in vector form the expression for torque can be written as
T= mxB
Thus the torque acting on a coil in a magnetic field depends on the number of turns, area of current loop, strength of current and magnetic field.

Force Acting  on a current
carrying conductor

Moving Coil Galvanometer