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A long straight wire carries a steady current I. In the same plane lies a U-shaped conductor closed by a sliding connector of length l and resistance R, which moves to the right with constant velocity v. The resistance of the U-frame and the loop's self-inductance are negligible. Find the induced current in the loop as a function of the distance r between the connector and the straight wire.

1. I_induced = mu0*I*v*l / (2*pi*r*R)
2. I_induced = mu0*I*v*l / (pi*r*R)
3. I_induced = mu0*I*v / (2*pi*R)
4. I_induced = mu0*I*v*l² / (2*pi*r²*R)

Correct answer: I_induced = mu0*I*v*l / (2*pi*r*R)

Solution

The straight wire's field at the connector (distance r) is B = mu0*I/(2*pi*r). The connector of length l moving with speed v has motional EMF = B*l*v = mu0*I*l*v/(2*pi*r). Since only the connector has resistance R, the induced current is I_induced = EMF/R = mu0*I*v*l/(2*pi*r*R).

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