2. Torque on a loop: A square loop of side lies in the horizontal z-y plane as shown in the figure. A current i flows counterclockwise around the loop. The z-axis points vertically out of the page. (a) Suppose that a vertical magnetic field B = B.k is applied. (i) Determine the magnetic force on each segment of the loop. (ii) Assuming the force on each segment acts at the center of the segment, compute the magnetic torque on each segment about the center of the square loop. (Recall that the torque T due to a force F applied at position r is given by 7 = r x F). (iii) What is the net torque on the loop? (b) Suppose a horizontal magnetic field B = B,i is applied instead. Repeat parts (i), (ii) and (iii) above for this circumstance. (c) Define the magnetic moment of the loop as a vector m that has magnitude ia where i is the current and a is the area of the loop. The direction of m is perpendicular to the loop and out of the page; it is determined by a right hand rule (point fingers along current, then thumb points along m). Verify that the results of parts (a) and (b) are consistent with the general formula Tt =M X B (4) where Ty is the net torque on the loop and B is the applied magnetic field. y-axis x-axis Figure 3: Square loop lies in a horizontal plane. The current flows counterclockwise. The definition of magnetic moment m and the formula for the torque given here are generally valid for loops of arbitrary shape, not only square. 6