Classical electromagnetism can be summarized by four coupled field laws. Together, Maxwell's equations describe how electric charge and current produce fields and how changing electric and magnetic fields generate one another.

Gauss's law for electricity

Electric charge produces net electric flux through a closed surface:

$$\oint_S\mathbf E\cdot d\mathbf A=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}.$$

Positive enclosed charge produces net outward electric flux, while negative enclosed charge produces net inward flux.

Gauss's law for magnetism

The net magnetic flux through every closed surface is zero:

$$\oint_S\mathbf B\cdot d\mathbf A=0.$$

Magnetic field lines therefore do not begin or end on isolated magnetic charges in classical electromagnetism.

Faraday's law

A changing magnetic flux produces circulating electric field:

$$\oint_C\mathbf E\cdot d\boldsymbol\ell =-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf A.$$

The minus sign expresses Lenz's law: the induced effect opposes the change that produces it.

Ampère–Maxwell law

Electric current and changing electric flux produce circulating magnetic field:

$$\oint_C\mathbf B\cdot d\boldsymbol\ell =\mu_0I_{\mathrm{enc}}+\mu_0\varepsilon_0\frac{d}{dt}\int_S\mathbf E\cdot d\mathbf A.$$

Maxwell's displacement-current term extends Ampère's law to time-varying electric fields.

One coupled theory

The equations are not four unrelated rules. Charge is a source of electric field; magnetic field has no corresponding monopole source; changing magnetic field creates circulating electric field; and current together with changing electric field creates circulating magnetic field.

This coupling allows electric and magnetic disturbances to sustain one another and leads naturally to electromagnetic waves.

These equations can also be written as local differential relations. That formulation requires divergence, curl and the integral theorems of vector calculus and is developed separately.