Faraday's Law of Induction:

Faraday's Law of Induction: Definition and Mathematical Representation

Introduction

Faraday’s Law of Electromagnetic Induction is a fundamental principle of electromagnetism discovered by Michael Faraday. It describes how a changing magnetic field within a closed loop induces an electromotive force (EMF) in the conductor. This principle is the working mechanism behind electric generators, transformers, and inductors.

Statement of Faraday’s Law

The induced electromotive force (EMF) in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.

Mathematical Formulation

Integral Form

${ℰ}=-\frac{d{\mathrm{\Phi }}_B}{dt}$

Where:

• ${ℰ}$ is the induced EMF (in volts),
• ${\mathrm{\Phi }}_B$ is the magnetic flux through the circuit,
• $\frac{d{\mathrm{\Phi }}_B}{dt}$ is the time derivative of the magnetic flux.

The magnetic flux is defined as:

${\mathrm{\Phi }}_B=\underset{S}{\int }\overrightarrow{B}\cdot d\overrightarrow{A}$

Where:

• $\overrightarrow{B}$ is the magnetic field (in tesla),
• $d\overrightarrow{A}$ is an infinitesimal area vector perpendicular to the surface $S$.

Differential Form

Using Maxwell's equations, the differential form of Faraday's Law is:

$\mathrm{\nabla }\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$

This expresses that a time-varying magnetic field produces a circulating electric field.

Physical Interpretation

• A changing magnetic field induces an electric field.
• The induced EMF drives a current if the circuit is closed.
• The negative sign indicates the direction of the induced EMF opposes the change in flux (as per Lenz's Law).

Lenz’s Law

Lenz’s Law gives the direction of the induced EMF:

${ℰ}=-\frac{d{\mathrm{\Phi }}_B}{dt}$

The negative sign shows that the induced current will create a magnetic field opposing the change in the original magnetic flux.

Applications

• Electric generators – convert mechanical energy to electrical energy using induction.
• Transformers – transfer electric power between circuits via changing magnetic flux.
• Inductive sensors – detect position or motion using electromagnetic principles.
• Electromagnetic brakes – generate resistance via induction in metallic conductors.
• Induction cooking – use changing magnetic fields to generate heat directly in cookware.

Examples

Example 1: Rotating Loop in a Magnetic Field

A loop rotating in a magnetic field $\overrightarrow{B}$ with angular velocity $\omega$ has a time-dependent flux:

${\mathrm{\Phi }}_B(t)=BA\cos (\omega t)$

Then the induced EMF is:

${ℰ}=-\frac{d{\mathrm{\Phi }}_B}{dt}=BA\omega \sin (\omega t)$

See Also

• Electromagnetic Induction
• Lenz's Law
• Magnetic Flux
• Maxwell's Equations
• Electromagnetism
• Electric Generator
• Transformer