You are given a circular loop of wire of cross-sectional area A, resistivity rho

Question

You are given a circular loop of wire of cross-sectional area A, resistivity rho, and radius r. You place the loop in a uniform magnetic field of magnitude B_0 that makes an angle of 30 degree to the axis of the loop. During a time T you decrease the magnetic field to zero at a steady rate. Find the power dissipated in the loop while the magnetic field is changing. You measured the total energy dissipated in the loop while you changed the magnetic field, and called it U_1. You now repeat the experiment, this time turning off the field more slowly, in a time T_2 = 9T. In terms of U_1, find the total energy U_2 dissipated during this second experiment.

Explanation / Answer

(a) Magnitude of induced emf, V1 = d/dt = (r2)(B/t)cos30o = (r2Bo/T)cos30o

Resistance of wire, R = L/A = 2r/A

Power dissipated in the loop, P1 = V2/R = (r3Bo2A / 2T2)cos230o

(b) U1 = P1T = (r3Bo2A / 2T)cos230o

=> U2 = U1/9

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