An emf is induced by rotating a - turn, diameter coil in the Earth's magnetic field. What average emf is induced, given the plane of the coil is originally perpendicular to the Earth's field and is rotated to be parallel to the field in

The average emf induced by rotating a turn of coil in the earth's magnetic field is given by Faraday's law of induction. This law states that the induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.

Explanation:

When a coil is placed in a magnetic field, it experiences a flux equal to magnetic field strength multiplied by the area enclosed by the loop times the cosine of the angle between the normal to the area and the magnetic field. Mathematically

The flux Ф = BACosθ

When the loop is placed in the earth's magnetic field without any motion, this flux value is constant and no emf is induced. The moment rotation begins in the magnetic field, the angle between normal to the plane of the area and the magnetic field lines starts to change/vary though the area and the strength of the magnetic field remain constant. This causes the net magnetic flux through the loop to change as the magnetic flux is proportional to the angle θ depending on how fast the rotation is made that is the time interval dt an average emf is induced in the coil.

E = - dФ/dt = - d (BACosθ) / dt

Since B and A are constant

E = - BAd (Cosθ) / dt

Where θ is a function of time.

θ = ωt

Where ω = angular frequency measured in rad/s

E = - BAd (Cosωt) / dt

E = - (-ω) BASinωt = ωBASinωt

E = ωBASinωt = - d (BACosθ) / dt

Depending on the case presented both formulas work out well.

Now given that the plane becomes parallel with the field, the normal now becomes perpendicular to the field and θ = 90° so

E = ωBASin90° = ωBA (1) = ωBA