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A short conductor of length L carrying current I is placed at radius r from the axis of a long cylinder. The magnetic field is B = B0 * r_hat, where r_hat is the unit vector radially outward from the cylinder axis. Find the work done and power required to move this conductor one complete revolution in the positive (angular) direction at a rotational frequency of N revolutions per minute.
- Work = 4*pi*r*B0*I*L
- Work = 2*pi*r*B0*I*L
- Power = -2*pi*r*B0*I*L*N/60
- Power = -4*pi*r*B0*I*L*N/60
Correct answer: Work = 2*pi*r*B0*I*L
Solution
The conductor of length L carries current I along the z-axis (parallel to cylinder axis) at radius r. The field B = B0*r_hat is radial. The force on the conductor is F = I*L*(z_hat cross r_hat)*B0 = -I*L*B0*phi_hat (opposing tangential motion). Work done by external agent per revolution = I*L*B0 * 2*pi*r = 2*pi*r*B0*I*L. Power = Work * frequency = Work * N/60 = 2*pi*r*B0*I*L*N/60. The negative sign indicates the electric field does negative work (field opposes motion), so power BY the field = -2*pi*r*B0*I*L*N/60.
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