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Maxwell's Equations (Ampere)

Currents and changing E-fields create B-fields.

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\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

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Core idea

Overview

The Ampere-Maxwell Law relates the circulation of a magnetic field to the density of electric current and the rate of change of the electric field. It extends Ampere's original circuital law by introducing the displacement current term, which is essential for describing electromagnetic wave propagation.

When to use: Apply this equation when calculating the magnetic field produced by steady electric currents or in regions where a time-varying electric field exists, such as between the plates of a charging capacitor. It is critical for electromagnetic scenarios where the charge density is not stationary.

Why it matters: This law demonstrates that a changing electric field induces a magnetic field, completing the symmetry with Faraday's law. This interaction is the mechanism that allows light and other electromagnetic radiation to travel through a vacuum without a medium.

Symbols

Variables

\text{Not directly solvable here} = Note

Not directly solvable here

Note

V a r iab l e

Walkthrough

Derivation

Understanding Maxwell's 4th Equation: Ampere-Maxwell Law

Magnetic fields are produced by electric currents and by changing electric fields (displacement current).

Maxwell’s displacement current term is included to ensure charge conservation.

Start with the Magnetostatic Form:

For steady currents, currents $J$ produce a curling magnetic field.

\nabla \times B = μ_{0} J

Note the Continuity Requirement:

For time-varying charge density, $\nabla \cdot J \neq = 0$ , so an additional term is needed for consistency.

\nabla \cdot (\nabla \times B) = 0 but \nabla \cdot J = - \frac{\partial ρ}{\partial t}

State the Full Differential Form:

Both conduction current and a changing electric field contribute to the curl of $B$ .

\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

Result

\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

Source: OpenStax University Physics Vol. 2 — Electromagnetism

Free formulas

Rearrangements

Solve for $\oint B \cdot d l$

Make $\oint$ $B$ $\cdot$ d $l$ the subject

\oint B \cdot d l = μ_{0} I + μ_{0} ϵ_{0} \frac{d Φ _{E}}{d t}

This process transforms the differential form of Ampere's Law with Maxwell's addition into its integral form, primarily by applying Stokes' Theorem and defining total current and electric flux.

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 because I appears as a first-degree term multiplied by the constant mu_0. It has a y-intercept equal to the displacement current term and a slope of mu_0, meaning the output increases steadily as I increases.

Graph type: linear

Why it behaves this way

Intuition

Magnetic field lines form closed loops that 'swirl' around two types of sources: actual moving electric charges (currents) and regions where the electric field is changing over time.

\nabla \times B

Curl of the magnetic field

Measures the 'swirliness' or circulation of the magnetic field lines around a point. A non-zero curl indicates that magnetic field lines are looping around a source.

μ_{0}

Permeability of free space

A fundamental constant that quantifies how strongly a magnetic field is generated by electric currents or changing electric fields in a vacuum. It acts as a scaling factor.

J

Electric current density

The amount and direction of electric charge flow per unit area. It represents the conventional flow of charges that produces a magnetic field.

ϵ_{0}

Permittivity of free space

A fundamental constant that quantifies how strongly an electric field is generated by electric charges or changing magnetic fields in a vacuum. It acts as a scaling factor.

\frac{\partial E}{\partial t}

Time rate of change of the electric field

Describes how quickly the electric field is strengthening or weakening in a region. This changing electric field acts as an effective current (displacement current) that generates a magnetic field.

Signs and relationships

\mu_0\mathbf{J}: The positive sign indicates that the magnetic field curls around the direction of conventional current density according to the right-hand rule.
\mu_0\epsilon_0\frac{∂ \mathbf{E}}{∂ t}: The positive sign indicates that a changing electric field generates a magnetic field that curls around the direction of the electric field's change, analogous to how a conventional current generates a magnetic field.

Free study cues

Insight

Canonical usage

This equation is primarily used with SI units, ensuring dimensional consistency across all terms.

Common confusion

Forgetting the displacement current term ( $μ_{0}$ $ϵ_{0}$ \frac{∂ \mathbf{E}}{ $\partial$ t}) when dealing with time-varying electric fields, or incorrectly mixing SI and Gaussian (CGS) unit conventions.

Unit systems

$B$ Tesla (T) · Magnetic field strength.

$J$ Ampere per square meter (A/m2) · Electric current density.

$E$ Volt per meter (V/m) · Electric field strength.

$t$ second (s) · Time.

$μ_{0}$ Henry per meter (H/m) · Permeability of free space.

$ϵ_{0}$ Farad per meter (F/m) · Permittivity of free space.

One free problem

Practice Problem

A steady current density J = 5.0 A/m² flows through a conductor. If the electric field is static (meaning its derivative with respect to time is 0), calculate the magnitude of the curl of the magnetic field (∇ × B). Use μ₀ = 1.256637 × 10⁻⁶ T·m/A.

mu00.000001256637

eps08.854187817e-12

dE_dt0

Solve for: $o u t$

Hint: Since the electric field is not changing, the second term of the equation becomes zero.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Magnetic field of a wire.

Study smarter

Tips

The displacement current term (μ₀ε₀ ∂E/∂t) is often negligible in low-frequency circuits but becomes dominant at high frequencies.
In magnetostatics (where fields are constant over time), the equation simplifies to the classic Ampere's Law: ∇ × B = μ₀J.
Remember that the product μ₀ε₀ is equal to 1/c², the reciprocal of the speed of light squared.

Avoid these traps

Common Mistakes

Forgetting displacement current term.
Right hand rule direction.

Common questions

Frequently Asked Questions

Magnetic fields are produced by electric currents and by changing electric fields (displacement current).

Apply this equation when calculating the magnetic field produced by steady electric currents or in regions where a time-varying electric field exists, such as between the plates of a charging capacitor. It is critical for electromagnetic scenarios where the charge density is not stationary.

This law demonstrates that a changing electric field induces a magnetic field, completing the symmetry with Faraday's law. This interaction is the mechanism that allows light and other electromagnetic radiation to travel through a vacuum without a medium.

Forgetting displacement current term. Right hand rule direction.

Magnetic field of a wire.

The displacement current term (μ₀ε₀ ∂E/∂t) is often negligible in low-frequency circuits but becomes dominant at high frequencies. In magnetostatics (where fields are constant over time), the equation simplifies to the classic Ampere's Law: ∇ × B = μ₀J. Remember that the product μ₀ε₀ is equal to 1/c², the reciprocal of the speed of light squared.

References

Sources

Griffiths, David J. Introduction to Electrodynamics. 4th ed. Pearson, 2013.
Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics. 10th ed. Wiley, 2014.
Wikipedia: Maxwell's equations
IUPAC Gold Book: Permittivity of vacuum
IUPAC Gold Book: Permeability of vacuum
NIST CODATA
David J. Griffiths, Introduction to Electrodynamics, 4th ed.
David J. Griffiths, Introduction to Electrodynamics, 4th ed., Pearson, 2013

Maxwell's Equations (Ampere)

Overview

Variables

Derivation

Start with the Magnetostatic Form:

Note the Continuity Requirement:

State the Full Differential Form:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Maxwell''s Equations (Faraday)

Frequently Asked Questions

Sources