The figure below shows a rod of length L = 10.0 cm that is forced to move at con

Question

The figure below shows a rod of length L = 10.0 cm that is forced to move at constant v = 5.00m/s along horizontal rails. The rod, rails, and connecting strip at the right form a conducting loop. The rod has resistance 0.400 ohm. The rest of the loop has negligible resistance. A current, i = 100A, is passing through the long straight wire shown above the loop. The wire is a distance a = 10.0 mm from the loop and creates the magnetic field shown below. Find the (a) emf and (b) the current induced in the loop. c) At what rate is thermal energy generated in the rod? (d) What is the magnitude of the force that must be exerted on the rod to force it to move at a constant speed? (e) At what rate does this force do work on the rod?

Explanation / Answer

here,

length of rod, l = 10 cm = 0.1 m

velocity, v = 5 m/s

resistance in rod, R = 0.4 ohms

Current in wire, Iw = 100 A

distance to wire, d = 10 mm = 0.01 m

Part a:
induce emf = - uo*I/(2*pi) * ln ((d + l)/d) * v

induce emf = - (4*pi*10^-7*100)/(2*pi) * ln((0.01 + 0.1)/0.01) * 5

induce emf = - 2.39*10^-4 V

Part B:
induced current, I = induced emf / resistance

induced current, I = (2.39*10^-4) / 0.4

induced current, I = 5.95*10^-4 A

Part C:

Thermal energy = ( induced current )^2 * Resistance

Thermal energy = ( 5.95*10^4 )^2 * 0.4

Thermal energy = 1.416*10^9 watts

Part D:
Force, F = (uo * induced current * Iw)/(2*pi) * ln((d+l)/d)

Force, F = (4*pi*10^-7 * 5.95*10^-4 *100)/(2*pi) * ln((0.01+0.1)/0.01)

Force, F = 2.853 * 10^- 8 N

Part e:

Work done = Force * velocity

work done = 2.853 * 10^- 8 * 5

work done = 1.427*10^-7 watts

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