Alternating Voltage and Current Developed in a Coil Rotating in Magnetic Field

In households and industries, the electricity supplied is alternating current (AC), not direct current (DC). The basic device that produces AC is the AC generator (alternator), which works on the principle of electromagnetic induction.

When a closed coil is rotated in a uniform magnetic field, the magnetic flux linked with the coil changes continuously. This change induces an emf and hence an alternating current in the coil.

A current whose magnitude varies continuously and whose direction reverses periodically.

Mathematically: I = I₀​ sin ωt

Setup and Assumptions

• Uniform magnetic field \[\vec{B}\].
• Coil of area A, N turns.
• Coil rotates with constant angular velocity ω\omegaω.
• At t = 0, the plane of the coil is perpendicular to \[\vec{B}\] (flux is maximum).

Derivation

1. Magnetic flux through one turn at time t: At time t, the coil has rotated through angle θ = ωt.
The component of \[\vec{B}\] perpendicular to the plane of the coil is B cos ⁡θ.

2. Express θ in terms of ω and t:

3. Rate of change of flux (for one turn):

4. Induced emf in one turn (Faraday’s Law):

5. For a coil of N turns:

6. Define peak voltage V₀: The maximum value of sin⁡(ωt) is 1.

Therefore: V = V₀ sin⁡(ωt)