physics conceptual ?

Question

Two pieces of the same wire have the same length. From one piece, a square coil containing a single loop is made. From the other, a circular coil containing a single loop is made. The coils carry different currents. When placed in the same magnetic field with the same orientation, they experience the same torque. What is the ratio Isquare / Icircle of the current in the square coil to that in the circular coil?

Answer 1

Torque on a coil (length L) wherein I current is flowing (say clockwise) in placed in magnetic field (B) at angle (p) is given by cross product of position vector & force vector.

Tq (vector) = r x F (1)
Here the force experienced by coil if given by
F (Vector) = I (L x B)

Tq (vector) = r x I (L x B) = r x I (L x B)
= I B x (r x L) by vector property
= I B x A (where A is cross-section area vector)
= I A B sin (p)

now L length of wire makes a “square coil” & a “circular coil”

then for “square coil (perimeter = L =4* side)
Area of square coil (As) = (L/4)*(L/4) = L^2 / 16
for “circular coil ” (perimeter L = 2*Pi* radius)
Area of circular coil (Ac) = Pi* (L/2*Pi)^2 = L^2 / 4*Pi

Coils are placed identically in Mag field but having Is and Ic currents in such a way that they experience same torque.
So, we can write

Is*As*B sin (p) = Ic*Ac*B sin (p) or (Is / Ic) = (Ac / As)
(Is / Ic) = (L^2 / 4*Pi) / (L^2 / 16) = (16/4*Pi)

(Isquare / Icircle) = 4/Pi = 4 : Pi