Question

A wire of length L is bent round into

1. a square coil having N turns and
2. a circular coil having N turns. 

The coil in both cases is free to turn about a vertical axis coinciding with the plane of the coil, in a uniform, horizontal magnetic field and carry the same currents. Find the ratio of the maximum value of the torque acting on the square coil to that on the circular coil.

Answer 1

Let the side of the square be a.

Total wire length for N turns:

L = N × 4a

a = `L/(4 N)`    ...(i)

Area of square:

A_s = a²

= `(L/(4 N))^2`    ...[From equation (i)]

= `L^2/(16 N^2)`

Let radius be r,

Total wire length (L) = N × 2πr

r = `L/(2 pi N)`    ...(ii)

Area of circle (A_c) = πr²

= `piL/(2 pi N)^2`    ...[From equation (ii)]

= `L^2/(4 pi N^2)`

Ratio of torques:

`tau_s/tau_c = A_s/A_c`

= `(L^2/(16 N^2))/(L^2/(4 pi N^2))`

= `1/16 xx 4 pi`

= `pi/4`

∴ `tau_"square"/tau_"circle" = pi/4`