A circular ring with area 4.45 cm2 is carrying a current of 13.5 A. The ring, initially at rest, is immersed in a region of unif

a) (0.0015139 i^ + 0.0020185 j^ + 0.00060556 k^) N.m b) ΔU = -0.000747871 J c) w = 47.97 rad/s The relevant data includes the area of the circular ring as 4.45 cm², the current of 13.5 Amps, and the magnetic field strength as (1.05×10−2T)(12i^ + 3j^ - 4k^). The initial magnetic moment orientation is expressed as μi = μ(−0.8i^ + 0.6j^). The moment of inertia for the ring is given as 6.50×10−7 kg⋅m². To find the initial magnetic moment (μ), we apply μ = N*I*A, using N=1 for a single coil. Substituting the values yields μ = 0.0060075 A-m². The torque on the ring is determined using the cross product of the magnetic moment and the magnetic field. Calculating this gives the initial torque as (0.0015139 i^ + 0.0020185 j^ + 0.00060556 k^). The magnetic potential energy is then calculated using the dot product of the magnetic moment with the magnetic field. This leads to initial energy stored in the ring as Ui = 0.000495556 J after computation. Following a 90-degree rotation, the final potential energy is found to be Uf = -0.000252315 J. The decrease in potential energy is determined to be ΔU = -0.000747871 J. Thus, the potential energy stored in the ring decreased by ΔU = -0.000747871 J.