A closely wound rectangular coil of 80 turns has dimen-sions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 37.0° with a magnetic field of 1.70 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

Verified step by step guidance

1. First, understand that the problem involves electromagnetic induction, where a change in the orientation of a coil in a magnetic field induces an electromotive force (emf). The key formula to use here is Faraday's Law of Induction, which states that the induced emf (ε) is equal to the negative rate of change of magnetic flux (Φ) through the coil: ε = -dΦ/dt.
2. Calculate the initial magnetic flux (Φ_initial) through the coil. The magnetic flux is given by Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area of the coil, and θ is the angle between the magnetic field and the normal to the plane of the coil. Here, B = 1.70 T, A = 25.0 cm * 40.0 cm = 0.25 m * 0.40 m = 0.10 m², and θ = 37.0°. Use these values to find Φ_initial.
3. Determine the final magnetic flux (Φ_final) when the coil is perpendicular to the magnetic field. In this position, θ = 0°, so cos(θ) = 1. Therefore, Φ_final = B * A * cos(0°) = B * A.
4. Calculate the change in magnetic flux (ΔΦ) as the coil rotates from its initial position to the final position. ΔΦ = Φ_final - Φ_initial.
5. Use Faraday's Law to find the average induced emf. Since the rotation takes 0.0600 s, the average emf is given by ε_avg = -ΔΦ / Δt, where Δt is the time interval of the rotation. Substitute the values of ΔΦ and Δt to find ε_avg.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a coil induces an electromotive force (emf) in the coil. The average emf is calculated by the rate of change of magnetic flux, which depends on the number of turns in the coil, the magnetic field strength, and the angle between the field and the coil's plane.

Magnetic Flux

Magnetic flux is the measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as the product of the magnetic field strength, the area of the coil, and the cosine of the angle between the field and the normal to the coil's surface. Changes in this flux are crucial for determining the induced emf.

Angular Motion and Time

The rotation of the coil from an initial angle to a perpendicular position involves angular motion, which affects the rate of change of magnetic flux. The time taken for this rotation is essential for calculating the average emf, as it determines how quickly the flux changes, influencing the magnitude of the induced emf according to Faraday's Law.