Electromagnetic Induction

Module 6: Electromagnetism

Overview

Electromagnetic induction is the process of generating an electromotive force (EMF) by changing the magnetic flux through a circuit. This fundamental phenomenon underlies generators, transformers, and many electrical devices.

Key Syllabus Points:

  • Describe how magnetic flux can change
  • Analyse Faraday’s Law and Lenz’s Law qualitatively and quantitatively
  • Evaluate transformer operation and efficiency

Key Concepts

Magnetic Flux

Definition

Magnetic flux (\(\Phi\)) is a measure of the total magnetic field passing through a surface.

\[\Phi = B_\parallel A = BA\cos\theta\]

Where: - \(B\) = magnetic field strength (T) - \(A\) = area of surface (m²) - \(\theta\) = angle between field and surface normal

Unit: Weber (Wb), where 1 Wb = 1 T·m²

Faraday’s Law

Faraday’s Law of Electromagnetic Induction

The induced EMF equals the negative rate of change of magnetic flux through the circuit.

\[\varepsilon = -N\frac{\Delta\Phi}{\Delta t}\]

Where: - \(\varepsilon\) = induced EMF (V) - \(N\) = number of turns in coil - \(\Delta\Phi\) = change in magnetic flux (Wb) - \(\Delta t\) = time interval (s)

Ways to change flux: 1. Change the magnetic field strength (\(B\)) 2. Change the area (\(A\)) 3. Change the angle (\(\theta\)) 4. Move the conductor through the field

Lenz’s Law

Lenz’s Law

The direction of the induced current is such that it opposes the change in flux that produces it.

This is a consequence of conservation of energy - the induced current creates a magnetic field that opposes the change, requiring work to be done.

Application: - Flux increasing → induced current creates opposing field - Flux decreasing → induced current creates reinforcing field

Transformers

Ideal Transformer Equations

\[\frac{V_p}{V_s} = \frac{N_p}{N_s}\]

\[V_p I_p = V_s I_s\] (power conservation)

Type Turns Ratio Voltage Current
Step-up \(N_s > N_p\) Increases Decreases
Step-down \(N_s < N_p\) Decreases Increases

Real Transformer Losses

  1. Resistive losses: \(P = I^2R\) heating in windings
  2. Eddy current losses: Circulating currents in iron core
  3. Hysteresis losses: Energy lost reversing magnetic domains
  4. Flux leakage: Incomplete flux linkage between coils

Efficiency improvements: - Use laminated iron core (reduces eddy currents) - Use soft iron core (reduces hysteresis) - Use thick copper windings (reduces resistance)

Worked Examples

Example 1: Faraday’s Law Calculation

A coil with 200 turns and area 0.05 m² is perpendicular to a magnetic field that changes from 0.4 T to 0.1 T in 0.2 s. Calculate the induced EMF.

Solution:

Initial flux: \(\Phi_1 = BA = 0.4 \times 0.05 = 0.02\) Wb Final flux: \(\Phi_2 = BA = 0.1 \times 0.05 = 0.005\) Wb Change: \(\Delta\Phi = 0.005 - 0.02 = -0.015\) Wb

\[\varepsilon = -N\frac{\Delta\Phi}{\Delta t} = -200 \times \frac{-0.015}{0.2} = 15 \text{ V}\]

Example 2: Transformer Calculation

A step-down transformer has 1000 primary turns and 50 secondary turns. If the primary voltage is 240 V and the secondary current is 8 A, calculate: (a) Secondary voltage (b) Primary current (assuming 100% efficiency)

Solution:

  1. Using turns ratio: \[V_s = V_p \times \frac{N_s}{N_p} = 240 \times \frac{50}{1000} = 12 \text{ V}\]

  2. Using power conservation: \[I_p = \frac{V_s I_s}{V_p} = \frac{12 \times 8}{240} = 0.4 \text{ A}\]

Common Misconceptions

Avoid These Mistakes
  1. Confusing flux and field - Flux depends on area AND field, not just field strength
  2. Forgetting the negative sign - The negative in Faraday’s Law represents Lenz’s Law
  3. Assuming DC transformers work - Transformers require changing flux, so only work with AC
  4. Ignoring angle - Remember \(\Phi = BA\cos\theta\), flux depends on orientation
  5. Power gain misconception - Transformers conserve power (ideally), they don’t create energy

HSC Exam Analysis

Question Types

  1. Calculation questions (4-6 marks): Calculate EMF, flux change, transformer values
  2. Explanation questions (4-5 marks): Explain Lenz’s Law, transformer losses
  3. Application questions (5-7 marks): Power transmission, generator operation

Recent HSC Questions

  • 2024 Q25: Transformer efficiency and power transmission
  • 2023 Q27: Faraday’s Law application to moving conductor
  • 2022 Q26: Lenz’s Law and conservation of energy

Practice Problems

  1. A 500-turn coil rotates in a 0.2 T field. If the flux through it changes from maximum to zero in 0.05 s, calculate the average induced EMF.

  2. A transformer steps up 240 V to 12,000 V for transmission. If the transmission line current is 5 A, calculate the current in the primary coil (assuming 95% efficiency).

  3. Explain why the core of a transformer is made from laminated sheets rather than solid iron.

  4. A magnet is dropped through a copper tube. Using Lenz’s Law, explain why it falls slower than in free fall.