LEC 4: Electromagnetic Induction: Faraday’s Law, Lenz’s Law, and Motional EMF

Electromagnetic Induction

Introduction

Electromagnetic induction is a fundamental concept in physics describing how a changing magnetic field can induce an electric current in a conductor. This topic is central to understanding the relationship between electricity and magnetism, and is covered in Physics 201A under Electrodynamics & Relativity.

Overview of Electromagnetic Discoveries

Historical Context

• Early Experiments: In the 18th and 19th centuries, scientists such as Coulomb, Ørsted, and Ampère established links between electricity and magnetism.

• Faraday’s Contribution: Michael Faraday introduced the concept of electric and magnetic fields and demonstrated that changing one could produce effects in the other.

• Maxwell’s Unification: James Clerk Maxwell later unified these ideas into four equations describing electromagnetic phenomena.

Time-Dependent Fields

Static vs. Dynamic Fields

• Stationary Fields: Electric fields from fixed charges and magnetic fields from constant currents are static.

• Time-Dependent Fields: When fields change with time, new phenomena such as electromagnetic induction arise.

• Example: The magnetic strip on credit cards uses electromagnetic induction for data reading.

Faraday’s Experiments on Time-Varying Magnetic Fields (1831)

Experimental Setup and Observations

• Galvanometer: Used to measure electric current induced in a coil.

• Stationary Magnet: No current is observed when the magnet is stationary.

• Moving Magnet: Current is induced when the magnet moves into or out of the coil, demonstrating that a changing magnetic field induces an emf.

• Direction of Current: The direction of the induced current depends on the direction of the magnet’s motion.

Magnetic Flux

Definition and Units

• Magnetic Flux (φm): Measures the amount of magnetic field lines passing through a given surface area.

• Formula:

$\phi_m = \int \vec{B} \cdot d\vec{A}$

• Units: Weber (Wb), where $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$

Faraday’s Law of Induction

Statement and Mathematical Formulation

• Faraday’s Law: The induced emf ($\varepsilon$) is the negative rate of change of magnetic flux through a circuit.

• Formula:

$\varepsilon = -\frac{d\phi_m}{dt}$

• Expanded Formula (using flux definition):

$\varepsilon = -\frac{d}{dt} \int \vec{B} \cdot d\vec{A}$

• For N turns:

$\varepsilon = -N \frac{d\phi_m}{dt}$

• Physical Meaning: Any change in magnetic field strength, area, or orientation induces a voltage.

Lenz’s Law

Direction of Induced EMF

• Lenz’s Law: The induced emf always generates a current whose magnetic field opposes the change in magnetic flux that produced it.

• Consequence: This law ensures conservation of energy in electromagnetic systems.

• Example: If a magnet approaches a coil, the induced current creates a magnetic field opposing the magnet’s motion.

Motional EMF

Induction by Motion

• Motional EMF: An emf can be induced by moving a conductor through a magnetic field or by changing the area exposed to the field.

• Examples:

  • Moving a coil into a steady magnetic field

  • Rotating a loop in a steady magnetic field

• Formula for Rotating Loop:

$\phi_m = B A \cos(\theta(t))$

$\theta = \omega t,\ \omega = \text{const.}$

• Where $\omega$ is angular velocity and $t$ is time.

Applications and Experiments

Classroom Experiments

• Jumping Ring Experiment: Demonstrates electromagnetic induction and Lenz’s law by causing a metal ring to jump due to induced currents.

• Other Experiments: Involve moving magnets, coils, and measuring induced currents with galvanometers.

• Observation Questions:

  • Which parts are moving?

  • Which parts are magnetic/non-magnetic?

  • Which parts are conductors/insulators?

Summary Table: Key Laws and Concepts

Concept	Definition	Formula	Units
Magnetic Flux	Amount of magnetic field lines through a surface	$\phi_m = \int \vec{B} \cdot d\vec{A}$	Weber (Wb)
Faraday’s Law	Induced emf is negative rate of change of magnetic flux	$\varepsilon = -\frac{d\phi_m}{dt}$	Volt (V)
Lenz’s Law	Induced current opposes change in flux	Direction: Opposes change	—
Motional EMF	EMF induced by motion in a magnetic field	$\phi_m = B A \cos(\omega t)$	Weber (Wb)

Summary

• Magnetic flux quantifies the amount of magnetic field passing through a surface.

• Faraday’s law relates the induced emf to the rate of change of magnetic flux.

• Lenz’s law determines the direction of the induced current, always opposing the change in flux.

• Motional emf arises from the movement of conductors in magnetic fields.

Additional info: These principles are foundational for technologies such as electric generators, transformers, and induction cooktops.