A 200-turn rectangular coil of area 3.00 × 10⁻³ m² rotates at 50.0 rad s⁻¹ in a uniform 0.400 T field. At t = 0 its area vector is parallel to the field. Derive flux linkage and emf as functions of time, calculate peak and rms emf, and find the first time emf reaches its positive maximum. If connected to a 60.0 Ω resistor, find average power and mechanical torque averaged over a cycle for constant speed.

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NΦ = 0.240cos(50t) Wb-turn; ε = 12.0sin(50t) V; ε_rms = 8.49 V; first positive maximum at 0.0314 s. Average power = 1.20 W and average opposing torque = 0.0240 N m. Flux per turn is Φ = BAcosωt. Flux linkage is NBAcosωt = 200(0.400)(3.00 × 10⁻³)cos(50t) = 0.240cos(50t) Wb-turn. Faraday's law gives ε = −d(NΦ)/dt = NBAωsinωt = 12.0sin(50t) V; the sign reflects Lenz's law for the chosen winding orientation. Peak emf is 12.0 V and sinusoidal rms emf is 12.0/√2 = 8.49 V. First positive maximum occurs when 50t = π/2, so t = 0.0314 s. With a pure 60.0 Ω resistance, average power is ε_rms²/R = 8.49²/60.0 = 1.20 W. At constant angular speed, average mechanical input equals electrical dissipation in the ideal model. Thus average torque magnitude is P/ω = 1.20/50.0 = 0.0240 N m and acts opposite rotation.

Induction in a rotating coil

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