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

Question

The figure shows a rod of length L = 10.0 cm that is forced to move at constant speed u = 5.00 m/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 = 100 A through the long straight wire at distance a = 10.0 mm from the loop sets up a (nonuniform) magnetic field through the loop. Find the magnitude of the induced emf. Find the current induced in the loop. What is the magnitude of the force that must be applied to the rod to make it move at constant speed? At what rate does this force do work on the rod?

Explanation / Answer

B(r)=µ0*I/2pr

first lets find the flux

f=A*B, B is changing with r. So we will break the area into small strips with width dr and length x. dA=x*dr

df=B(r)*dA = (µ0*I/2pr)*x*r, then lets integrate it from a to L+a

--> f=(µ0*I/2p)*x*ln((L+a)/a)

then, form faraday law

e=-df/dt=(µ0*I/2p)*ln((L+a)/a)*dx/dt

dx/dt=v ---> e=(µ0*I/2p)*ln((L+a)/a)*v=2.4e-4 V



(2)

i=e/R=2.38e-4/0.4=6e-4 A

(3)

P=I2R=1.44e-7 Watt

(4)

to get the force we need to integrate

dF=i*dr*B =i*dr*µ0*I/2pr = (µ0i*I/2p)*dr/r (pay attention there are two currents here i and I, one is in the bar, second is in the wire)

F=integral(dF,dr)=(µ0i*I/2p)*ln((L+a)/a)=2.88e-8 N

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