Problem 2.1 If the total force on a current loop (or any system, for that matter) is zero (dF = 0), show that the total torque N = 7 x dF' is independent of the choice of origin. Hint: Change to a primed coordinate system where 7 = ' + 1y, with an arbitrary shift . Problem 2.4 a) Show that the angular momentum 7 x p of the circulating particle with respect to the center is mrvn where 7n is a unit vector perpendicular to the plane of the circle. Here, n points in a direction given by the right-hand rule applied to the particle's motion. Show that the magnetic moment associated with the motion of the point charge is qur/2 and thus that the gyromagnetic ratio is given by (2.19). Evaluate numerically the gyromagnetic ratio v (2.19), choosing the same mass (1.67 x 10727 kg) and charge (1.60 x 10~ C) as for a proton. The difference between vour answer and (2.17) is due to the more complicated motion of the proton constituents, the 'quarks.\" For related reasons, a neu- tron has a nonvanishing magnetic moment despite its zero overall charge. y = 2.675 x 108 rad/s/T (2.17) y (point charge in circular motion) = (2.19) 2m