A magnetic field B = Bo sin ωt k covers a large region

Question:
A magnetic field B = Bo sin ωt k covers a large region where a wire AB slides smoothly over two parallel conductors separated by a distance d. 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?

Solution:

Let wire AB at t = 0 move with velocity v.

At t, x(t) = vt

Motional emf across AB = e1 = Blv

e1 = (Bo sin ꞷt)vd(-j)

e2 = d(ϕB)/dt

e2 = -B0 ꞷ cos ꞷtx (t)d

Total emf in the circuit = emf due to change in field + motional emf across AB

Electric current in the clockwise direction is given as = Bod/R (ꞷx cos ꞷt + v sin ꞷt)

Therefore, external force is given as, Fext = Bo2d2/R (ꞷx cos ꞷt + v sin ꞷt)(sin ꞷt (i)

A magnetic field B = Bo sin ωt k covers a large region

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