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Maxwell's Equations (Faraday)

Changing magnetic field induces electric field.

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

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Core idea

Overview

Faraday's law of induction describes how a time⁻varying magnetic field creates a spatially varying electric field.

When to use: Use when analyzing electromagnetic induction where changing magnetic flux induces electric fields and electromotive forces, designing transformers and generators that convert mechanical to electrical energy, understanding induced currents in conductors moving through magnetic fields, calculating induced voltages in coils experiencing varying magnetic fields, explaining wireless charging and inductive coupling, and applying Lenz's law to determine the direction of induced currents opposing flux change producing them.

Why it matters: It is Faraday's fundamental law of electromagnetic induction that forms the basis for electrical power generation, explaining how changing magnetic fields create electric fields and currents, enabling transformers, generators, induction motors, magnetic braking, wireless power transfer, and electromagnetic compatibility, while providing the time-dependent coupling between electric and magnetic fields essential for electromagnetic wave propagation through Maxwell's unified theory.

Symbols

Variables

\text{Not directly solvable here} = Note

Not directly solvable here

Note

V a r iab l e

Walkthrough

Derivation

Understanding Maxwell's 3rd Equation: Faraday's Law

A changing magnetic flux induces an electric field and an electromotive force around a loop.

Magnetic flux through a surface can change with time (due to a changing field or moving circuit).

State the Integral Form:

The induced EMF around a closed loop equals the negative rate of change of magnetic flux through the loop.

\oint_{C} E \cdot d l = - \frac{d Φ _{B}}{d t}

Use Stokes’ Theorem:

Convert the line integral into a surface integral, relating circulation of $E$ to time variation of $B$ .

\int_{S} (\nabla \times E) \cdot d a = - \int_{S} \frac{\partial B}{\partial t} \cdot d a

State the Differential Form:

A time-varying magnetic field produces a circulating (curling) electric field.

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

Result

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

Source: OpenStax University Physics Vol. 2 — Electromagnetism

Free formulas

Rearrangements

Solve for $\oint E \cdot d l$

Make oint mathbf{E} dmathbf{l} the subject

\oint E \cdot d l = - \frac{d Φ _{B}}{d t}

This derivation transforms the differential form of Faraday's Law of Induction into its integral form. It involves integrating both sides over a surface, applying Stokes' Theorem, and defining magnetic flux to relate the electromotive force around a closed loop to the rate of change of magnetic flux through the surface bounded by that loop.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line passing through the origin with a negative slope. Because the output is directly proportional to the negative of this variable, increasing the rate of change of magnetic flux results in a proportional decrease in the output value.

Graph type: linear

Why it behaves this way

Intuition

A time-varying magnetic field threading through a region generates closed-loop electric field lines that circulate around the changing magnetic flux.

E

Electric field vector, representing the force per unit charge experienced by a test charge.

The 'push or pull' that a positive test charge would feel at a given point.

B

Magnetic field vector, describing the influence that exerts forces on moving charges and magnetic dipoles.

The 'strength and direction' of the magnetic influence, similar to that around a permanent magnet.

\nabla \times E

The curl of the electric field, which quantifies the circulation or 'swirl' of the electric field lines around a point.

Imagine tiny paddlewheels in space; if they spin due to the electric field, the curl is non-zero, indicating a circulating electric field.

\frac{\partial B}{\partial t}

The time rate of change of the magnetic field, indicating how quickly the magnetic field's strength or direction is changing at a specific location.

How fast the magnetic 'influence' is increasing, decreasing, or rotating at a particular point in space.

Signs and relationships

-: The negative sign embodies Lenz's Law, stating that the induced electric field (and any resulting induced current) creates a magnetic field that opposes the change in the magnetic flux that produced it.

Free study cues

Insight

Canonical usage

Primarily used with SI units, where the dimensional consistency between the curl of the electric field and the time derivative of the magnetic field is maintained.

Common confusion

Misinterpreting the curl operator's units or failing to ensure dimensional consistency between the electric field gradient and the rate of change of the magnetic field.

Unit systems

$E$ V/m · Electric field strength, representing force per unit charge.

$B$ T · Magnetic flux density, representing magnetic force on a moving charge or current.

$t$ s · Time, the independent variable for the rate of change.

One free problem

Practice Problem

Concept check: Faraday's law reference.

Solve for: $o u t$

Hint: Reference only.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Wireless charging.

Study smarter

Tips

Curl E = -dB/dt.
Lenz's law (minus sign).

Avoid these traps

Common Mistakes

Omitting negative sign.
Confusing flux with field B.

Common questions

Frequently Asked Questions

A changing magnetic flux induces an electric field and an electromotive force around a loop.

Use when analyzing electromagnetic induction where changing magnetic flux induces electric fields and electromotive forces, designing transformers and generators that convert mechanical to electrical energy, understanding induced currents in conductors moving through magnetic fields, calculating induced voltages in coils experiencing varying magnetic fields, explaining wireless charging and inductive coupling, and applying Lenz's law to determine the direction of induced currents opposing flux change producing them.

It is Faraday's fundamental law of electromagnetic induction that forms the basis for electrical power generation, explaining how changing magnetic fields create electric fields and currents, enabling transformers, generators, induction motors, magnetic braking, wireless power transfer, and electromagnetic compatibility, while providing the time-dependent coupling between electric and magnetic fields essential for electromagnetic wave propagation through Maxwell's unified theory.

Omitting negative sign. Confusing flux with field B.

Wireless charging.

Curl E = -dB/dt. Lenz's law (minus sign).

References

Sources

David J. Griffiths, Introduction to Electrodynamics
Robert Resnick, David Halliday, Jearl Walker, Fundamentals of Physics
Wikipedia: Faraday's law of induction
Wikipedia: Maxwell's equations
Fundamentals of Physics, 11th Edition by Halliday, Resnick, Walker
Introduction to Electrodynamics, 4th Edition by David J. Griffiths
OpenStax University Physics Vol. 2 — Electromagnetism

Maxwell's Equations (Faraday)

Overview

Variables

Derivation

State the Integral Form:

Use Stokes’ Theorem:

State the Differential Form:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Maxwell''s Equations (Ampere)

Frequently Asked Questions

Sources