Maxwell’s equations are a set of four partial differential equations first combined by James Clerk Maxwell. Together, they completely describe classical electromagnetic phenomena, just as Newton’s laws completely describe classical mechanical phenomena. All four are named after persons other than Maxwell, but Maxwell was the first to add the displacement current term to Ampère’s Law, which led to the association of electromagnetic waves with Light and paved the way for the discovery of special relativity. All four equations can be written in both integral and differential forms, with both forms convenient for specific problems. Note that strictly speaking these are Maxwell’s equation in vacuo, with different forms for interaction with matter.

0.1 Notation

Throughout this article SI units are adopted for clarity, but the interesting mathematical aspects of the equations are independent of the constants μ₀ and 𝜖₀, and indeed of the physical meaning of the equations.

0.2 Gauss’ Law of Electrostatics

Differential form

Integral form

where q is the charge enclosed in the volume bounded by the surface S.

0.3 Gauss’ Law of Magnetostatics

This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date.

0.4 Faraday’s Law

Differential form

0.5 Ampère’s Law

Differential form

Integral form

0.6 Properties of Maxwell’s Equations

These four equations together have several interesting properties:

• Lorentz invariance
• Gauge invariance
• Invariance under the transformation B → , E → Bc