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Physical Sciences
Electromagnetism
1865
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Maxwell's Equations

∇⋅E=ρε0,∇×E=−∂B∂t\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

Four equations unifying electricity and magnetism—predicting electromagnetic waves and light.

By James Clerk Maxwell

Physical Sciences
Maxwell's Equations
1865 · James Clerk Maxwell
Expert Reviewed
89%

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Story PortalWho discovered this?Discover how James Clerk Maxwell and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — Wi-Fi routers and electric motors.Math PortalWhat does it derive from?Trace the mathematical lineage from Coulomb and Ampère laws.Future PortalWhere is it going?Photonics and metamaterials
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Why it matters: Enabled radio, radar, Wi-Fi, optics, and modern physics.

Discoverers: James Clerk Maxwell (1865)

What does it mean?

Four equations unifying electricity and magnetism—predicting electromagnetic waves and light.

Why should I care?

Enabled radio, radar, Wi-Fi, optics, and modern physics.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
EEElectric fieldV/mElectric field vector
BBMagnetic fieldTMagnetic flux density
ρρCharge densityC/m³Free charge density

Worked Example

Speed of light c = 1/√(μ₀ε₀) derived from wave equation.

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Wi-Fi RouterElectric MotorSmartphoneTelecommunications

Pictures & video

Photograph of James Clerk Maxwell
James Clerk Maxwell, who unified electricity, magnetism and light (1865).Wikimedia Commons · Public domain

Sources & further reading

Maxwell's equations — WikipediaSearch WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Maxwell's Equations

∇⋅E=ρε0,∇×E=−∂B∂t\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

Real-world impact

Wi‑Fi, MRI & fiber optics

Unified electromagnetism powers radio, radar, lasers, and medical imaging.

Photo: Unsplash — MRI scanner

Four equations unifying electricity and magnetism—predicting electromagnetic waves and light.

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