Question

Derive the expression for the torque on a current-carrying coil in a magnetic field.

Answer 1

Consider a rectangular loop PQRS carrying current I is placed in a uniform magnetic field B. Let a and b be the length and breadth rectangular loop respectively. The unit vector n̂ normal to the plane of the loop makes an angle θ with the magnetic field.

Rectangular coil placed in a magnetic field

1. The magnitude of the magnetic force acting on the current-carrying arm PQ is F_PQ = IaBsin (π/2) = IaB.
2. The magnitude of the force on the arm QR is F_QR =IbBsin `(pi/2 - theta)` = lbB cos θ and its direction.
3. The magnitude of the force on the arm RS is F_RS =IaBsin (π/2) = IaB and its direction is downwards.
4. The magnitude of the force acting on the arm SP is F_SP = lbB`(pi/2 - theta)` = lbB cos θ and its direction.
5. Since the forces F_QR and F_SP are equal, opposite, and collinear, they cancel each other. But the forces F_PQ and F_RS.
   
   Side view of current loop
6. The magnitude of torque acting on the arm PQ about AB is τ_PQ = (1/2 sinθ) and points in the direction of AB. The magnitude of the torque acting on the arm. RS about AB is τ_RS = (b/2 sinθ) IaB and points also in the same direction AB.
7. The total torque acting on the entire loop about an axis AB is given by
   `tau = ("b"/2 sin theta) "F"_"PQ" + ("b"/2 sin theta) "F"_"RS"`
   = Ia(bsinθ)B
   τ = IABsinθ along the direction B.
   In vector form,
   `vec tau= (vec"IA") xx vec"B"`.
8. The above equation can also be written in terms of magnetic dipole moment.
   `vectau` = \[\overrightarrow{\mathrm{pm}}\] × \[\overrightarrow{\mathrm{B}}\] where \[\overrightarrow{\mathrm{pm}}\] = \[\overrightarrow{\mathrm{IA}}\]
9. If there are N turns in the rectangular loop, the torque is given by
   τ = NIAB sinθ

Special cases:

1. When θ =90° or the plane of the loop is parallel to the magnetic field, the torque on the current loop is maximum.
   τ_max = IAB
2. When θ = 0°/180° or the plane of the loop is perpendicular to the magnetic field, the torque on the current loop is zero.