Maxwell's Equations

physics electromagnetism

Definition

Maxwell's Equations

Maxwell’s equations are a set of four coupled partial differential equations that describe how electric charges and currents produce electric and magnetic fields, and how those fields evolve in time. Together with the Lorentz force law, they form the foundation of classical electromagnetism.

In differential form, in vacuum:

$$\begin{array}{rlr}\nabla \cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0}& \text{(Gauss’s law)}\\ \nabla \cdot \mathbf{B}& =0& \text{(Gauss’s law for magnetism)}\\ \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}& \text{(Faraday’s law)}\\ \nabla \times \mathbf{B}& ={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}& \text{(Ampeˋre–Maxwell law)}\end{array}$$

where $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, $\rho$ is the charge density, $\mathbf{J}$ is the current density, ${\epsilon }_0$ is the vacuum permittivity, and ${\mu }_0$ is the vacuum permeability.

Physical Meaning

The four equations encode two kinds of physical content: two constrain the shape of the fields (divergence equations), and two govern their coupled time evolution (curl equations).

Gauss’s Law

Electric charges are the sources of the electric field. The flux of $\mathbf{E}$ through a closed surface is proportional to the enclosed charge. Positive charges emit field lines; negative charges absorb them. Field lines begin and end on charges.

Gauss’s Law for Magnetism

There are no magnetic monopoles. The flux of $\mathbf{B}$ through any closed surface is zero. Magnetic field lines form closed loops — they have no beginning and no end.

Faraday’s Law

A changing magnetic field induces a circulating electric field. This is the principle behind electric generators and transformers. The minus sign encodes Lenz’s law: the induced field opposes the change that produced it.

Ampère–Maxwell Law

A steady electric current produces a circulating magnetic field (Ampère’s original law). Maxwell added the displacement current term ${\mu }_0{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}$, which states that a changing electric field also produces a magnetic field. This term is the crucial addition that makes the equations consistent with charge conservation and that predicts electromagnetic waves.