Figure P11.5 depicts a closed conducting loop in the xy-plane (z-0) in a region

Question

Figure P11.5 depicts a closed conducting loop in the xy-plane (z-0) in a region where a constant magnetic field B = azBo is present. In particular, this loop includes a resistor R and a rotating semicircular-shaped wire portion of radius a. The semicircular-shaped wire rotates at angular frequency w (rad/sec) (a) Determine the magnetic flux (t) linking the closed loop at a given time instant t (b) Determine the emf vind (t) generated by induction across the terminals of resistor R and the corresponding induced current iind(t). Assume the collective resistance of the conductors forming the loop is negligible ind (t) Figure P11.5

Explanation / Answer

Answer:- From Faraday's law, EMF will induce only when the flux linked with conductor will change. Flux linked with a conductor is the vector dot product of magnetic field and the area of magnetic filed line on the conductor surface. So for a changing flux either magnetic field should be changing or the area of conductor.

Here magnetic field is not changing but the area is changing, so EMF will induce here. At any time t, the net flux linked with the system is-

a) Flux = (B0 x w x h) + (B0 x 0.5 x pi x a2)cos(w0 x t), note when (w0 x t) = pi, sign changes to negative.

b) Induced EMF, vind(t) = -d(Flux)/dt = (B0 x 0.5 x pi x a2 x w0)sin(w0 x t), first term is constant in Flux, so giving zero.

Induced current, iind(t) =  vind(t)/R = ((B0 x 0.5 x pi x a2 x w0)/R)*sin(w0 x t).

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