A circular loop of radius `R=20` cm is placed in a uniform magnetic field `B=2T` in `xy`-plane as shown in figure. The loop carries a current `i=1.0 A` in the direction shown in figure. Find the magnitude of torque acting on the loop.

Explanation

Torque acting on the loop
`tau =MB sin theta`
`M = i A = i. pi R^(2), theta = 90^(@)`
`tau = MB = I pi R^(2) B = (1) (pi)(0.2)^(2)(2)`
`= 8pi xx 10^(-2) Nm`
Note: Here direction of magnetic moment is along `-z`- axis and `B` is `x-y` plane hence angle between `vec(M)` and `vec(B)` is `90^(@)`.
If we want torque in vector form
`vec(M) = -i A hat(k) = -i pi R^(2) hat(k)`
`vec(B) = B cos 45^(@) hat(i) + B sin 45^(@) hat(j) = (B)/(sqrt(2))(hat(i) + hat(j))`
`tau = vec(M) xx vec(B) = -(i pi R^(2) B)/(sqrt(2))[hat(k) xx (hat(i) + hat(j))]`
`= -(i pi R^(2) B)/(sqrt(2))[hat(j) - hat(i)]`
`= -(i pi R^(2) B)/(sqrt(2))[hat(i) - hat(j)]`
`|vec(tau)| = tau= (i pi R^(2) B)/(sqrt(2))sqrt((1)^(2) + (-1)^(2)) = i pi R^(2) B`
`= 8pi xx 10^(-2) Nm`