Faraday's Law of Induction

Overview of Faraday's Law

Faraday's Law of Induction is one of the most fundamental principles in electromagnetism. Discovered by Michael Faraday in 1831, it states that a changing magnetic field induces an electromotive force (emf) in a conductor. This law is the foundation for understanding how generators work and how electromagnetic induction operates.

The law connects the rate of change of magnetic flux through a surface to the induced emf around the boundary of that surface. This relationship is crucial for understanding time-varying electromagnetic phenomena and is a cornerstone of Maxwell's equations.

Faraday's Law Statement

Faraday's Law of Induction

The induced emf is equal to the negative rate of change of magnetic flux:

\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]

Where:

\(\mathcal{E}\) is the induced electromotive force (emf) in volts
\(\Phi_B\) is the magnetic flux through the surface in webers
\(\frac{d\Phi_B}{dt}\) is the rate of change of magnetic flux
The negative sign indicates Lenz's Law (direction of induced current)

Formula Derivations

Derivation 1: Motional EMF Formula

Derive: \(\mathcal{E} = BLv\) for a conductor moving through a magnetic field

Step-by-Step Derivation:

Consider a conductor of length L moving at velocity v perpendicular to magnetic field B
The magnetic flux through the loop changes as the conductor moves
Flux change: \(\Delta\Phi_B = B \cdot \Delta A = B \cdot L \cdot \Delta x\)
Rate of change: \(\frac{d\Phi_B}{dt} = B \cdot L \cdot \frac{dx}{dt} = BLv\)
By Faraday's Law: \(\mathcal{E} = -\frac{d\Phi_B}{dt} = -BLv\)
For a complete circuit, the negative sign indicates direction, so: \(\mathcal{E} = BLv\)

Result: \(\mathcal{E} = BLv\) (This formula is NOT on the equation sheet)

Derivation 2: Rotating Loop EMF

Derive: \(\mathcal{E} = BA\omega\sin(\omega t)\) for a rotating loop

Step-by-Step Derivation:

Magnetic flux through a rotating loop: \(\Phi_B = BA\cos(\theta)\)
For rotation at angular velocity ω: \(\theta = \omega t\)
Flux becomes: \(\Phi_B = BA\cos(\omega t)\)
Rate of change: \(\frac{d\Phi_B}{dt} = -BA\omega\sin(\omega t)\)
By Faraday's Law: \(\mathcal{E} = -\frac{d\Phi_B}{dt} = BA\omega\sin(\omega t)\)

Result: \(\mathcal{E} = BA\omega\sin(\omega t)\) (This formula is NOT on the equation sheet)

Derivation 3: Magnetic Force on Moving Rod

Derive: \(F = ILB\) for the magnetic force on a moving rod

Step-by-Step Derivation:

When a rod moves through a magnetic field, it experiences motional EMF: \(\mathcal{E} = BLv\)
If the rod has resistance R, the induced current is: \(I = \frac{\mathcal{E}}{R} = \frac{BLv}{R}\)
The magnetic force on a current-carrying conductor is: \(F = ILB\) (from magnetic force on wire)
Substitute the induced current: \(F = \frac{BLv}{R} \cdot LB = \frac{B^2L^2v}{R}\)
This force opposes the motion (Lenz's Law)

Result: \(F = ILB\) where I is the induced current (This formula is NOT on the equation sheet)

Key Concepts

Magnetic Flux

Magnetic flux (\(\Phi_B\)) is the product of magnetic field strength, area, and the cosine of the angle between the field and the normal to the surface:

\[ \Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta \]

The flux can change due to:

Changing magnetic field strength
Changing area of the loop
Changing orientation (angle)

Rate of Change

The induced emf depends on how quickly the flux changes, not the absolute value of the flux. This means:

Faster changes produce larger emf
Constant flux produces zero emf
The direction of change affects the sign

This is why generators need to rotate - to create a changing flux.

Multiple Turns

For a coil with N turns, the total induced emf is:

\[ \mathcal{E} = -N\frac{d\Phi_B}{dt} \]

This is because each turn contributes to the total flux linkage, and the emf is proportional to the total flux linkage.

Direction of Induced Current

The negative sign in Faraday's Law indicates that the induced current creates a magnetic field that opposes the change in flux that produced it. This is Lenz's Law in action.

The direction can be determined using the right-hand rule for the induced current.

Interactive Simulation

Faraday's Law Simulation

Move the magnet to see how changing magnetic flux induces current in the coil.

Magnet Speed: Coil Turns:

Current Flux: 0 Wb

Rate of Change: 0 Wb/s

Induced EMF: 0 V

For Better Understanding:

Example Problems

Example 1: Simple Loop in Changing Field

Problem: A circular loop of radius 0.1 m is placed in a uniform magnetic field of 0.5 T. The field is increasing at a rate of 0.1 T/s. What is the induced emf in the loop?

Solution:

Calculate the area: \(A = \pi r^2 = \pi(0.1)^2 = 0.0314 \text{ m}^2\)
Calculate the flux: \(\Phi_B = BA = (0.5)(0.0314) = 0.0157 \text{ Wb}\)
Calculate the rate of change: \(\frac{d\Phi_B}{dt} = A\frac{dB}{dt} = (0.0314)(0.1) = 0.00314 \text{ Wb/s}\)
Apply Faraday's Law: \(\mathcal{E} = -\frac{d\Phi_B}{dt} = -0.00314 \text{ V}\)

Answer: The induced emf is -3.14 mV (negative sign indicates direction).

Example 2: Rotating Loop

Problem: A rectangular loop of area 0.02 m² rotates in a uniform magnetic field of 0.3 T at 60 rad/s. What is the maximum induced emf?

Solution:

The flux is: \(\Phi_B = BA\cos(\omega t)\)
The rate of change is: \(\frac{d\Phi_B}{dt} = -BA\omega\sin(\omega t)\)
The maximum emf occurs when \(\sin(\omega t) = \pm 1\)
Maximum emf: \(|\mathcal{E}| = BA\omega = (0.3)(0.02)(60) = 0.36 \text{ V}\)

Answer: The maximum induced emf is 0.36 V.

Applications

Electric Generators: Convert mechanical energy to electrical energy by rotating coils in magnetic fields
Transformers: Use changing magnetic fields to transfer energy between circuits
Induction Cooktops: Use changing magnetic fields to heat conductive materials
Magnetic Recording: Use changing magnetic fields to store and retrieve information
Wireless Charging: Transfer energy through changing magnetic fields without physical contact

Quick Quiz: Faraday's Law

1. What does Faraday's Law relate?

Rate of change of magnetic flux to induced emf

Electric field to magnetic field

Current to resistance

Voltage to power

2. If the magnetic flux through a loop is constant, what is the induced emf?

Maximum

Zero

Negative

Positive

3. For a coil with N turns, the induced emf is:

N times the single-turn emf

The same as single-turn emf

1/N times the single-turn emf

N² times the single-turn emf

4. What happens to the induced emf if the rate of flux change doubles?

It doubles

It halves

It stays the same

It quadruples

5. The negative sign in Faraday's Law represents:

Energy loss

Lenz's Law

Resistance

Power dissipation

Learning Objectives

State Faraday's Law: Express the relationship between changing magnetic flux and induced emf
Calculate Induced EMF: Use Faraday's Law to find induced emf in various situations
Understand Flux Changes: Identify the different ways magnetic flux can change
Apply to Multiple Turns: Calculate emf for coils with multiple turns
Connect to Applications: Relate Faraday's Law to real-world devices

Key Takeaways

Changing Flux: Only changing magnetic flux induces emf, not constant flux
Rate Dependence: The induced emf is proportional to the rate of change of flux
Multiple Turns: For N turns, the emf is N times the single-turn emf
Direction: The negative sign indicates the direction of the induced current
Fundamental Principle: Faraday's Law is the foundation for electromagnetic induction