can anyone please help me on this one?
A circular coil (980 turns, radius = 0.078 m) is rotating in a uniform magnetic field. At t = 0 s, the normal to the coil is perpendicular to the magnetic field. At t = 0.027 s, the normal makes an angle of 45 degrees with the field because the coil has made one-eighth of a revolution. An average emf of magnitude 0.058 V is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.
To find the magnitude of the magnetic field at the location of the coil, we can use Faraday's Law of electromagnetic induction, which states that the induced electromotive force (EMF) in a circuit is equal to the rate of change of magnetic flux through the circuit.
The magnetic flux through a coil is given by the equation Φ = B * A * cos(θ), where B is the magnitude of the magnetic field, A is the area of the coil, and θ is the angle between the normal to the coil and the magnetic field.
In this problem, the normal to the coil is initially perpendicular to the magnetic field, and after a time of 0.027 s, it makes an angle of 45 degrees with the field due to one-eighth of a revolution. We are given that an average EMF of magnitude 0.058 V is induced in the coil.
To find the magnitude of the magnetic field, we need to rearrange the equation Φ = B * A * cos(θ) to solve for B.
Step 1: Calculate the change in flux (∆Φ)
∆Φ = Φ_final - Φ_initial
We know that the EMF induced is given by the rate of change of magnetic flux, so we can write:
EMF = -∆Φ/∆t
Given that the average EMF induced is 0.058 V and the time interval ∆t = 0.027 s, we can substitute these values into the equation:
0.058 V = -∆Φ/0.027 s
Step 2: Calculate the change in angle (∆θ)
Given that the coil makes one-eighth of a revolution in 0.027 s, we can calculate the change in angle (∆θ) by multiplying the number of revolutions (1/8) by 360 degrees per revolution:
∆θ = (1/8) * 360 degrees = 45 degrees
Step 3: Calculate the change in flux using the formula:
∆Φ = B * A * ∆θ
We know that the coil has 980 turns, and the radius is 0.078 m, so the area (A) can be calculated as follows:
A = π * r^2
= π * (0.078 m)^2
Step 4: Substitute the values into the equation and solve for B:
0.058 V = -B * (π * (0.078 m)^2) * 45 degrees / 0.027 s
Simplifying the equation and solving for B:
B = (-0.058 V * 0.027 s) / [(π * (0.078 m)^2) * 45 degrees]
B ≈ 0.0073 T
Therefore, the magnitude of the magnetic field at the location of the coil is approximately 0.0073 Tesla (T).