Question

Derive an expression for the net torque on a rectangular current carrying loop placed in a uniform magnetic field with its rotational axis perpendicular to the field.

Answer 1

1. Consider rectangular loop abcd placed in a uniform magnetic field `vec"B"` such that the sides ab and cd are perpendicular to the magnetic field `vec"B"` but the sides bc and da are not, as shown in figure (a) below:
   
   Fig (a): Loop abcd placed in a uniform
2. The force `vec"F"_4` on side 4 (bc) will be `vec"F"_4` = Il₂ B sin(90° - θ)
3. The force `vec"F"_2` on side 2 (da) will be equal and opposite to `vec"F"_4` and both act along the same line. Thus, `vec"F"_2` and `vec"F"_4` will cancel out each other.
4. The magnitudes of forces `vec"F"_1` and `vec"F"_3`  on sides 1 (cd) and 3 (ab) will be Il₁ B sin 90° i.e., Il₁ B. These two forces do not act along the same line and hence they produce a net torque. 
5. This torque results into rotation of the loop so that the loop is perpendicular to the direction of `vec"B"`, the magnetic field.
   
   Fig (b): Side view of the loop abcd at an angle θ
6. Now the moment arm is `1/2 (l_2 sintheta)` about the central axis of the loop. Hence, the torque τ due to forces `vec"F"_1` and `vec"F"_3` will be 
   τ = `("I l"_1 "B"1/2"l"_2 sintheta) + ("I l"_1 "B"1/2"l"_2 sintheta)`
   = `"I" l""_1 "l"_2 "B" sin theta`
7. If the current-carrying loop is made up of multiple turns N, in the form of a flat coil, the total torque will be 
   τ' = `"N"tau = "NI""l"_1"l"_2 "B" sintheta`
   τ' = (NIA)B sinθ; where A is the are enclosed by the coil = l₁l₂ 
   This is the required expression.