Principia Synthesia
SearchTimelineOrganizations
Sign inGet started
© 2026 Principia Synthesia
ThemePricingSupport
Principia Admin

Maxwell's Equations

The four fundamental equations of classical electromagnetism.

Jun 12, 2026··Fork (sign in)Last verified: Jun 12, 2026

Gauss, Faraday, Ampere — the complete classical field theory of electromagnetism.

#physics#electromagnetism#waves

Maxwell's Equations

Differential Form

EquationExpression
Gauss (electric)∇⋅E=ρ/ε0\nabla \cdot \mathbf{E} = \rho/\varepsilon_0∇⋅E=ρ/ε0​
Gauss (magnetic)∇⋅B=0\nabla \cdot \mathbf{B} = 0∇⋅B=0
Faraday∇×
E=−∂B/∂t\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t
∇×E=−∂B/∂t
Ampere-Maxwell∇×B=μ0J+μ0ε0 ∂E/∂t\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t∇×B=μ0​J+μ0​ε0​∂E/∂t

Electromagnetic Waves

In vacuum (ρ=0\rho=0ρ=0, J=0\mathbf{J}=0J=0) the equations reduce to the wave equation:

∇2E−μ0ε0∂2E∂t2=0\nabla^2\mathbf{E} - \mu_0\varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0∇2E−μ0​ε0​∂t2∂2E​=0

with wave speed c=1/μ0ε0≈3×108 m/sc = 1/\sqrt{\mu_0\varepsilon_0} \approx 3\times 10^8\,\text{m/s}c=1/μ0​ε0​​≈3×108m/s.

Related curriculum: book-classical-physics.

Related events

  • Maxwell's Treatise on Electricity and Magnetism
    Jan 1, 1873Physics History