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Maxwell's Equations (Ampere) Calculator

Currents and changing E-fields create B-fields.

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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.

Symbols

Variables

\text{Not directly solvable here} = Note

Note

Apply it well

When To Use

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.

Avoid these traps

Common Mistakes

  • Forgetting displacement current term.
  • Right hand rule direction.

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
J5
eps08.854187817e-12
dE_dt0

Solve for:

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.

References

Sources

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