Final answer:
The problem involves computing the induced electromotive force in a rotating circular coil within a magnetic field. To solve this, Faraday's law of electromagnetic induction is applied, with attention to the changing magnetic flux through the coil over time as it rotates at a given angular velocity.
Step-by-step explanation:
The question is about the induced electromotive force (emf) in a circular armature coil that is rotating in a magnetic field. Given that the coil has 280 loops and a diameter of 13.0 cm (radius of 0.065 m) and that it rotates at 120 revolutions per second in a uniform magnetic field with strength of 0.45 T, one can use Faraday's law of electromagnetic induction to calculate the emf. Faraday's law states that the emf induced in a coil is proportional to the rate of change of the magnetic flux through the coil.
The flux (Φ) through a single loop is given by the formula Φ = B × A × cos(θ), where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the plane of the loop. For a coil rotating in a magnetic field, θ changes with time as θ = ωt, with ω being the angular velocity of the coil. Therefore, the total change in flux for N loops is Φ_total = N × B × A × cos(ωt), and the induced emf (ε) is then the negative derivative of Φ_total with respect to time. The peak emf occurs when the time rate of change of flux is maximum. For this configuration, the angular velocity (ω) is 2π times the frequency (f), which is the number of revolutions per second.