Electromagnetic Induction

(a) Magnetic Flux & Flux Linkage

Magnetic Flux (Φ): The total number of magnetic field lines passing through a given area. Defined as:

Φ = ABcos Θ

where:

• B = Magnetic flux density (T)

• A = Area perpendicular to the field (m²)

• Θ = Angle between the field and normal to the surface

• Flux Linkage (NΦ): The total flux linked with a coil of N turns:

NΦ = NABcos Θ

(b) Faraday’s & Lenz’s Laws

Faraday’s Law: The magnitude of the induced electromotive force (emf) in a circuit is proportional to the rate of change of magnetic flux linkage:

emf = − (NΦ) / t

Lenz’s Law: The induced emf always acts in a direction to oppose the change in flux that caused it. This ensures energy conservation.

This is where the - sign comes from

(c) Applying Faraday’s & Lenz’s Laws

If magnetic flux linkage changes over time, an emf is induced:

emf = − d(NΦ) / dt

The negative sign follows Lenz’s law, indicating the induced emf opposes the flux change.

Applications:

• Transformers,

• electric generators, and

• induction heating.

(d) Induced emf in a Linear Conductor Moving Perpendicularly to a Magnetic Field

When a conductor moves at right angles to a uniform magnetic field, an emf is induced:

emf = Blv

where:

• B = Magnetic flux density (T)

• l = Length of conductor in the field (m)

• v = Velocity of conductor perpendicular to field (m/s)

This is the basis for how dynamos work.

(e) Instantaneous emf in a Rotating Coil

• When a coil rotates in a magnetic field, flux linkage changes sinusoidally.

• Instantaneous emf is given by:

emf = NAB ω sin⁡ (ωt)

where:

• N = Number of turns

• B = Flux density (T)

• A = Coil area (m²)

• ω = Angular velocity (rad/s)

• t = Time (s)

The maximum emf occurs when the coil is parallel to the field lines (maximum rate of change of flux).

The emf is zero when the coil is perpendicular to the field (no flux change).

This explains AC generator operation, where emf follows a sinusoidal pattern.