A magnetic field ( ) sinob= t b kcovers a large region where a wire ab slides smoothly over two parallel conductors separated by a distance d (fig. 6.10). The wires are in the x-y plane. The wire ab (of length d) has resistance r and the parallel wires have negligible resistance. If ab is moving with velocity v, what is the current in the circuit. What is the force needed to keep the wire moving at constant velocity?
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Answer:
Given :
Length of the wire AB = d
Resistance = R
Velocity = υ
Moving along the positive directions of x- axis over two parallel wires OX and CD at y=0 and y=d
Magnetic field covering the region is:
B→=B∘Sin(ωt)k^
At t = 0, AB is at x = 0
Velocity = υi^
Motional emf in AB = (B∘Sinωt)d.υ(j^)
The direction of the emf i aloy negative y - axis
Total emf induced = - B∘Sinωt.d.υ−B∘ωcm(ωt).x.d
This field is also along negative y - axis
Total emf induced = - B∘Sinωt.d.υ−B∘ωcm(ωt).x.d
= -B∘d[υsinωt+ωxcmωt]
Hence current induced I=BodR[vsinwt+wxcoswt]
∴ Force needed alongi^isFF∴F=BIl=Bosinwt+BodR[vsinwt+wxcoswt]d=B2od2R2[vsinwt+wxcoswt]sinwt
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