A force F is applied to a rod so that the rod slides with constant speed v over

Question

A force F is applied to a rod so that the rod slides with constant speed v over a friction less pair of parallel conducting rails that are separated by a distance /. The rod and rails have negligible resistance, but the rails are connected by a resistance R. as shown. There is a uniform magnetic field B perpendicular to and directed out of the plane of the paper. On the diagram at right, indicate the direction of the educed current in the resistor. Determine expressions for the following in terms of v. B. . and R. The induced emf in the rod The electric field in. the rod. (Hint: uses an eq. from Electrostatics unit.) The magnitude of the induced current in resistor K. The power dissipated in the resistor as the rod moves in the magnetic field. The magnitude of the external force F applied to the rod to keep it moving with constant speed v.

Explanation / Answer

b)Let l be the seperation of the rails and x be the distance from the left end of the rails to the rod. The area of surface S is then lx, and the magnetic flux through s is                = B.A                   = BA = Blx Taking the time derivative of both sides gives          d/dt = Bl (dx/dt)                   = Blv Where v= dx/dt  is the speed of the rod.  The emf induced in this circuit is             e = -d/dt                 = - Blv where the negaive sign tells that the emf is in the negative tangential direction. a) The direction of induced current in the resistor is clockwise.      c) The electric field in the rod is            E =e/l              = Bv d)The magnitude of induced current is            I = e/R              = (Blv)/R e) The power dissipated in the rod is          P =eI            = (Blv)^2/R f) The magnitude of external force is             F = BIL

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