A rectangular loop PQRS made from a uniform wire has length a, width b and mass m. It is free to rotate about the arm PQ, which remains hinged along a horizontal line taken as the y-axis (see figure). Take the vertically upward direction as the z-axis. A uniform magnetic field `vec(B) = (3 hat(i) + 4 hat(k)) B_(0)` exists in the region. The loop is held in the x-y plane and a current I is passed through it. The loop is now released and is found to stay in the horizontal position in equilibrium

(a) What is the direction of the current I in PQ?
(b) Find the magnetic force on the arm RS.
(c) Find the expression for I in terms of `B_(0)`, a, b and m

Explanation

`(a, c)` If loop is in equilibrium
Torque due to weight of loop `=` Torque due to magnetic field
`tau_(1) = tau_(2)`
`vec(tau)_(1) + vec(tau)_(2) = 0`
`vec(tau)_(1) = ((a)/(b)hat(i)) xx (-mg hat(k)) = (mga)/(2)hat(j)`
Let direction of current in loop be clockwise
Magnetic moment of loop `vec(M) = i ab (-hat(k)) = -i ab hat(k)`
`vec(tau)_(2) = vec(M) xx vec(B) = [-i ab hat(k)] xx [(3 hat(i) + 4 hat(k))B_(0)]`
`= -3i abB_(0)(hat(k) xx hat(i)) = -3i abB_(0) hat(j)`
`vec(tau)_(1) + vec(tau)_(2) = 0`
`(mga)/(2) hat(j) - 3iabB_(0)hat(j) = 0 rArr i = (mg)/(6 B_(0)b)`
Since current `i` is positive, hence assumed direction is correct.
(b) `vec(F)_(RS) = i(-b hat(j)) xx [(3 hat(i) + 4 hat(k)) B_(0)]`
`= -i bB_(0)[3(hat(j) xx hat(i)) + 4(hat(j) xx hat(k))]`
`= -i bB_(0)[-3 hat(k) + 4hat(i)]`
`= i bB_(0)[-4 hat(i) + 3 hat(k)]`