In the figure is a U-shaped conducting rail that is oriented vertically and imme

Question

In the figure is a U-shaped conducting rail that is oriented vertically and immersed in a horizontal magnetic field (that points into the paper). The rail has no electrical resistance and does not move. A slide bar with mass m, length L, and resistance R can slide vertically without friction while maintaining electrical contact with the rail. The slide wire is released from rest. It accelerates downward because of its own weight and eventually reaches a terminal velocity v_term. a) Explain why the bar reaches a terminal velocity instead of accelerating uniformly under the force of gravity. b) Find an expression for the terminal velocity using only the given parameters and known constants. Then calculate the value of v_term if L = 20 cm, m = 10 g, R = 0.10 ohm, and B = 0.50 T

Explanation / Answer

At the terminal speed, the acceleration of the wire, and thus the net force, must be zero. There is a gravitational force downward on the wire, so gravitational force must be upward. In terms of magnitudes,

Newton’s Second Law is F = FG FB = ma = 0 mg IB sin = 0

The direction of current flow though the slide wire is necessarily perpendicular to the magnetic field,

so sin = 1. From the definition of resistance, V = IR = E. mg = IB = E R B From Faraday’s Law,

the magnitude of the emf is

E = d dt =

d /dt B . dA

= d dt [BA cos ]

= B cos dA dt

= B cos ds dt

= Bv cos

where s is the distance from the slide wire to the top of the “U”.

Note that the magnetic field and area vectors are parallel,

so cos = 1.

E = Bv mg = Bv (lB) / R

vterm = mgR / l2B2

here vterm is proportional to g

so terminal velocity instead of accelerating uniformly under the force of gravity

2.

here L = 20 Cm = 0.2 m m = 10 g = 0.01 Kg R = 0.01 ohm B = 0.50 T

so terminal velocity

vterm = 0.01 x 9.8 x 0.01 / ( 0.22 x 0.502 )

= 0.098 m/s Ans

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