Maxwell's equations

Maxwell's equations are the four coupled laws that govern the electric and magnetic fields in classical electromagnetism, relating each field to its sources, electric charge and current, and to changes in the other field. Together with the Lorentz force law, they encompass all the laws of electricity and magnetism.¹

The four equations in integral form

In vacuum, the integral forms state the following.¹

1. Gauss's law. The electric flux through any closed surface equals the enclosed charge divided by $\epsilon_0$:

$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{in}}}{\epsilon_0}.$$

1. Gauss's law for magnetism. The magnetic flux through any closed surface is zero, which is equivalent to saying magnetic field lines are continuous, with no beginning or end; no magnetic monopoles are known to exist.¹
2. Faraday's law. A changing magnetic flux induces an electromotive force, and hence an electric field, whose direction opposes the change (Lenz's law):

$$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_m}{dt}.$$

1. Ampère-Maxwell law. Magnetic fields are generated by moving charges or by changing electric fields:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I + \epsilon_0 \mu_0 \frac{d\Phi_E}{dt}.$$

In differential form, the same laws read, in vacuum and in MKS units,⁷

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, \qquad \nabla \cdot \vec{B} = 0, \qquad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \qquad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t},$$

where $\rho$ is the charge density and $\vec{J}$ the current density. Once the fields are found, the Lorentz force law $\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$ gives the force on a particle of charge $q$ moving with velocity $\vec{v}$.¹

The displacement current

The problem. Ampère's original law, $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$, fails for time-dependent situations. For a charging parallel-plate capacitor, one surface bounded by the same loop cuts the wire and gives $\mu_0 I$, while another surface passing between the plates encloses no conduction current and gives zero; the law's answer depends on the choice of surface, an internal contradiction.¹ The inconsistency is deeper still: taking the divergence of the quasi-static Ampère's law gives $\nabla \cdot \vec{J} = 0$, which cannot hold when charge is accumulating, since charge conservation requires $\nabla \cdot \vec{J} = -\partial\rho/\partial t$.⁸

Maxwell's fix. Adding the term $\epsilon_0 \, \partial\vec{E}/\partial t$, called the displacement current density, restores consistency, because $\nabla \cdot (\vec{J} + \epsilon_0 \, \partial\vec{E}/\partial t) = 0$ follows from Gauss's law.⁸ At the time there was no experimental evidence for this term; the equations demanded it.⁸ With the correction, the line integral of $\vec{B}$ around a loop is the same whether the spanning surface cuts the wire or passes between the capacitor plates, because the displacement current $\epsilon_0 \, d\Phi_E/dt$ through the second surface equals the conduction current $I$ through the first.¹ In a charging capacitor the displacement current between the plates equals the real current in the wires at every instant.¹ The term $\partial\vec{D}/\partial t$ is often called the displacement current density because its units are the same as current density, A/m².⁹

Electromagnetic waves

The displacement current makes the source-free equations symmetric: a changing $\vec{B}$ field induces an $\vec{E}$ field (induction), and a changing $\vec{E}$ field induces a $\vec{B}$ field.⁸ This symmetry is essential to the existence of combined electric and magnetic waves propagating through space; the extra term is responsible for electromagnetic waves.^{1 • 8} The wave speed that follows from the equations is $c = 1/\sqrt{\epsilon_0 \mu_0}$, independent of frequency.² Maxwell first realized the link between electromagnetism and light in 1862, and the prediction was not detected unequivocally for another 23 years.^{10 • 8}

History

James Clerk Maxwell (1831–1879) combined the existing laws of electricity and magnetism with insights of his own into a complete electromagnetic theory.¹ In 1864 he published "A Dynamical Theory of the Electromagnetic Field," presenting eight equations of the electromagnetic field and an experimentally testable electromagnetic theory of light; one of the most original aspects of the theory was the introduction of electrical displacement and the displacement current as another source of field.³ In his own mechanical picture, grounded in a belief in the ether, the displacement current was always associated with the motion of bound charges in a medium.³ The two-volume Treatise on Electricity and Magnetism followed in 1873, and Maxwell died in 1879 while beginning a radically revised edition.¹⁰

The modern four-equation form is much simpler than what Maxwell published, and it emerged through an analytical reinterpretation in which charge and current become the fundamental quantities and the field acts directly on matter at the point of interest; contributors include Lorentz and Heaviside, with Hertz succeeding in experimentally contrasting the prediction of electromagnetic waves.^{11 • 3} Later, applying Einstein's theory of relativity to the completed theory showed that electric and magnetic forces are not separate but different manifestations of the same electromagnetic force.¹

Maxwell's equations in matter

Inside materials, electromagnetic fields are described using the auxiliary fields $\vec{D}$ and $\vec{H}$, and the equations are rewritten in terms of free charge and free current:⁵

\ \nabla \cdot \vec{D} = \rho_{\text{free}}, \qquad \nabla \cdot \vec{H} = -\nabla \cdot \vec{M}, \qquad \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \qquad \nabla \times \vec{H} = \vec{J}_{\text{free}} + \frac{\partial \vec{D}}{\partial t}.^[14 \]

This formulation absorbs the bound charge and bound currents of the material into the definitions of $\vec{D}$ and $\vec{H}$, so that only the free charge and current, which are typically the known quantities, appear as sources.⁵ To apply the macroscopic equations, the relations between $\vec{D}$ and $\vec{E}$, and between $\vec{H}$ and $\vec{B}$, must be specified; in simple linear media these are the constitutive relations $\vec{D} = \epsilon \vec{E}$ and $\vec{B} = \mu \vec{H}$.¹²

Charge conservation and relativistic form

The equations are mutually consistent only because they encode charge conservation. Taking the divergence of the Ampère-Maxwell equation and using Gauss's law yields the continuity equation $\nabla \cdot \vec{J} = -\partial\rho/\partial t$; conversely, charge conservation can be taken as the reason Maxwell added the displacement term.^{7 • 4}

In the covariant language of special relativity, the electric and magnetic fields combine into the antisymmetric field-strength tensor $F^{\alpha\beta}$, and the two equations with physical sources, Gauss's law and the Ampère-Maxwell law, become the single equation $\partial_\beta F^{\alpha\beta} = \mu_0 j^\alpha$. The antisymmetry of $F^{\alpha\beta}$ then implies $\partial_\alpha j^\alpha = 0$, charge conservation, automatically.⁶ The two source-free equations are likewise summarized by the vanishing divergence of the dual field tensor.⁶ Written in terms of the scalar and vector potentials in the Lorenz gauge, the whole theory reduces to a single 4-vector wave equation for the potential 4-vector $\Phi^\mu = (c\vec{A}, \phi)$.¹³

References

1. 16.1 Maxwell's Equations and Electromagnetic Waves, University Physics Volume 2, OpenStax
2. Answer Key Chapter 16, University Physics Volume 2, OpenStax
3. Electromagnetic field presented in introductory physics textbooks and consequences for its teaching, Phys. Rev. Phys. Educ. Res. 19, 020113 (2023)
4. Lecture 1, Electromagnetic Fields, Forces, and Motion (6.641), MIT OpenCourseWare
5. Differential Forms and Maxwell's Equations, Electromagnetism, Fluids and Waves (UCL)
6. Notes on Electrodynamics, Southern Illinois University Carbondale
7. Maxwell's Equations, Geometry of Special Relativity, Oregon State University
8. Chapter 3, Electromagnetic Interactions (22.105), MIT OpenCourseWare
9. Chapter 3, Electromagnetics and Applications (6.013), MIT OpenCourseWare
10. The discovery of Maxwell's equations, APS March Meeting 2012
11. Maxwell's Consistent Equations, 18.013A, MIT OpenCourseWare
12. Lecture 2: Media and Boundary Conditions, Electromagnetics and Applications (6.013), MIT OpenCourseWare
13. The potential 4-vector, Electromagnetism lecture notes, University of Texas at Austin
14. 4.4: Static fields in inhomogeneous materials

Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Maxwell's equations and potentials

Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026

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