Unit 18: Electromagnetic Induction

Faraday's Laws of Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field induces an electromotive force (EMF) in a conductor. Michael Faraday formulated two fundamental laws that quantify this phenomenon.

First Law

Faraday's first law states that whenever the magnetic flux linked with a closed circuit changes, an EMF is induced in the circuit. The induced EMF persists only as long as the flux is changing.

Second Law

The second law provides the magnitude of the induced EMF: ε = -dΦ/dt, where ε is the induced EMF, Φ is the magnetic flux through the circuit, and dΦ/dt is the rate of change of flux. The negative sign, explained by Lenz's law, indicates the direction of the induced EMF.

Definition of Magnetic Flux: Φ = B·A·cosθ, where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the field normal and the area vector.

Applications

• Electric generators and alternators
• Induction cooktops
• Magnetic flow meters
• Wireless charging systems

Lenz's Law and Motional EMF

While Faraday's laws give the magnitude of induced EMF, Lenz's law determines its direction, ensuring compliance with the law of conservation of energy.

Lenz's Law

Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is why Faraday's law includes a negative sign: ε = -dΦ/dt. The induced magnetic field created by the induced current works against the original change in flux.

Motional EMF

When a conductor moves through a magnetic field, an EMF is induced due to the magnetic Lorentz force on the charge carriers. For a straight rod of length l moving with velocity v perpendicular to a uniform magnetic field B, the motional EMF is given by:

ε = B l v

Here, B, l, and v must be mutually perpendicular. If any of these vectors are not perpendicular, the formula generalizes to ε = B l v sinθ, where θ is the angle between v and B (or between l and B depending on geometry).

Applications

• DC generators (split-ring commutator)
• Rail guns
• Induction brakes in trains
• Seismic sensors

AC Generators and Eddy Currents

Alternating current (AC) generators convert mechanical energy into electrical energy using electromagnetic induction. Eddy currents are loops of induced current that arise in bulk conductors exposed to changing magnetic fields.

AC Generator (Alternator)

A simple AC generator consists of a rectangular coil rotating with constant angular speed ω in a uniform magnetic field B. As the coil turns, the magnetic flux through it varies sinusoidally, producing an alternating EMF.

Flux through the coil: Φ = NBA cos(ωt), where N is the number of turns, A is the area of each turn, and θ = ωt.

Induced EMF (using Faraday's law):

ε = -dΦ/dt = NBA ω sin(ωt)

Thus the output is a sinusoidal voltage: ε = ε₀ sin(ωt) with peak value ε₀ = NBA ω. The frequency of the AC is f = ω/(2π).

Eddy Currents

When a bulk conductor (e.g., a metal plate) experiences a changing magnetic field, circulating currents called eddy currents are induced within the material. These currents flow in closed loops perpendicular to the magnetic field.

Effects:

• Joule Heating: Eddy currents dissipate energy as heat (P = I²R), useful in induction heating and cooking appliances.
• Magnetic Damping: The currents create opposing magnetic fields that resist motion, forming the basis of eddy‑current brakes in roller coasters and trains.
• Energy Loss: In transformers and motors, eddy currents cause unwanted losses; they are minimized by laminating the core.

Applications

• Induction furnaces for metal melting
• Non‑destructive testing (eddy‑current testing)
• Speedometers in vehicles
• Electromagnetic shielding

Self and Mutual Inductance

Inductance quantifies the ability of a conductor to induce EMF in itself or in a neighboring conductor due to a change in current.

Self‑Inductance

Consider a coil of N turns carrying a current I. The magnetic flux linked with the coil is Φ = L I, where L is the self‑inductance. A changing current induces an EMF in the same coil:

ε = -L (dI/dt)

The unit of inductance is the henry (H): 1 H = 1 V·s/A. Self‑inductance depends on geometry (number of turns, coil area, length, core material).

Mutual Inductance

Two coils placed near each other exhibit mutual inductance: a changing current in the primary coil (I₁) induces an EMF in the secondary coil:

ε₂ = -M (dI₁/dt)

Similarly, a change in I₂ induces EMF in the primary: ε₁ = -M (dI₂/dt). The mutual inductance M (also in henrys) is symmetric: M₁₂ = M₂₁ = M. It depends on the relative orientation, distance, and number of turns of the coils.

Coefficient of Coupling

The coupling coefficient k relates self and mutual inductance: M = k √(L₁ L₂), where 0 ≤ k ≤ 1. Perfect coupling (k = 1) occurs when all flux from one coil links the other.

Applications

• Inductors in filters and chokes
• Transformers (see next section)
• Wireless power transfer
• Inductive sensors

Energy Stored in an Inductor

Just as a capacitor stores energy in its electric field, an inductor stores energy in its magnetic field. When a current I flows through an inductor of inductance L, the magnetic field energy is:

U = ½ L I²

Derivation: The instantaneous power supplied to the inductor is P = ε I = L (dI/dt) I. Integrating from 0 to I gives the stored energy.

Analogy: The expression mirrors the capacitor energy formula U = ½ C V², with inductance playing the role of capacitance and current analogous to voltage.

Applications

• Energy storage in switch‑mode power supplies (SMPS)
• Inductive kickback protection (flyback diodes)
• Tuned circuits in radios (LC oscillators)
• Superconducting magnetic energy storage (SMES)

Transformers

A transformer is a static device that transfers electrical energy between two or more circuits through electromagnetic induction, allowing AC voltage levels to be stepped up or stepped down.

Basic Principle

An alternating current in the primary winding (N₁ turns) creates a changing magnetic flux in the core. This flux links the secondary winding (N₂ turns), inducing an EMF according to Faraday's law. Assuming negligible leakage flux and core losses (ideal transformer):

V₁ / V₂ = N₁ / N₂

where V₁ and V₂ are the RMS voltages across primary and secondary.

Power Conservation (Ideal Transformer)

For an ideal transformer, input power equals output power:

V₁ I₁ = V₂ I₂

Consequently, the current ratio is the inverse of the turns ratio:

I₁ / I₂ = N₂ / N₁

Thus a step‑up transformer (N₂ > N₁) raises voltage and reduces current, while a step‑down transformer does the opposite.

Real Transformer Considerations

• Core Losses: Hysteresis and eddy‑current losses in the ferromagnetic core.
• Copper Losses: Resistive I²R losses in the windings.
• Leakage Inductance: Not all flux links both windings; modeled as series inductances.
• Magnetizing Current: Small current required to establish the core flux.