Final answer:
To find the instantaneous emf induced in a rotating coil with a certain number of turns, radius, and angular velocity in a magnetic field, use Faraday's law of induction considering the angular change of the coil relative to the magnetic field.
Step-by-step explanation:
To calculate the instantaneous emf in the coil at the moment when its normal is at a 30.0° angle to the magnetic field, we can use Faraday's law of induction. The emf, ε, can be given by the rate of change of magnetic flux, Φ, through the coil: ε = -dΦ/dt. In this situation, where the magnetic field (B) is uniform and the coil rotates, the change in flux is due to the change in area orientation relative to the field
The magnetic flux through the coil is defined as Φ = B · A · cos(θ), where θ is the angle between the normal to the area (A) of the coil and the magnetic field (B). At a 30° angle, the flux through one turn of the coil is:
Φ = 0.0500 T · 0.050 m² · cos(30°)
The rate of change of flux would then involve the angular velocity (\(ω\)), which relates to how quickly the angle θ changes with time, since θ = ωt. Given ω = 20.0 rad/s, the rate of change of flux through one turn of the coil when θ = 30° can be calculated as the derivative of \(Φ(t)\), considering the trigonometrical relation that \(d(cos(ωt))/dt = -ω · sin(ωt)\).
Given the instantaneous rate of change of flux as dΦ/dt = B · A · ω · sin(θ), and that the coil has 500 turns, the instantaneous emf (ε) is:
ε = -N dΦ/dt = -500 · 0.0500 T · 0.050 m² · 20.0 rad/s · sin(30°)
By calculating this, we will obtain the instantaneous emf value for the student's question.