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A perfectly conducting wire PQ of length 10 cm slides at a constant speed of 1 cm/s along two frictionless, zero-resistance horizontal rails. One end of the rails carries an inductor L = 1 mH in series with a resistor R = 1 ohm, and a switch S. A uniform magnetic field B = 1 T is directed perpendicular to the plane of the rails. The wire keeps moving at 1 cm/s even after S is closed. If the current 1 ms after closing S is x * 10⁻³ A, find x. [Take e⁻¹ = 0.37]

1. x = 0.63
2. x = 0.37
3. x = 1.00
4. x = 0.50

Correct answer: x = 0.63

Solution

The constant-velocity wire gives a steady motional EMF, so the circuit behaves like a battery (emf/R) charging through an inductor: i(t) = (emf/R)(1 - e^(-t/tau)).

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