Maxwell's Equations — The Four Laws of Light

Gauss's Law

\nabla \cdot E = \frac{ρ}{ε _{0}}

// differential form

\oint_{\partial V} E \cdot d A = \frac{Q _{enc}}{ε _{0}}

// integral form

Electric charge is the source of the electric field. Field lines begin and end on charge.

\nabla \cdot B = 0

// differential form

\oint_{\partial V} B \cdot d A = 0

// integral form

There are no magnetic monopoles. Every field line that leaves must return — closed loops only.

\nabla \times E = - \frac{\partial B}{\partial t}

// differential form

\oint_{\partial S} E \cdot d ℓ = - \frac{d Φ _{B}}{d t}

// integral form

A changing magnetic field winds up an electric field. This is how every generator on Earth works.

\nabla \times B = μ_{0} J + μ_{0} ε_{0} \frac{\partial E}{\partial t}

// differential form

\oint_{\partial S} B \cdot d ℓ = μ_{0} I_{enc} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}

// integral form

Currents and changing electric fields both curl up a magnetic field. Maxwell's added term closed the loop.

// the consequence

\nabla^{2} E = μ_{0} ε_{0} \frac{\partial ^{2} E}{\partial t ^{2}}

The speed of that wave is fixed entirely by two constants of the vacuum — the permeability μ₀ and the permittivity ε₀:

c = \frac{1}{μ _{0} ε _{0}} \approx 2.998 \times 1 0^{8} m/s

Maxwell computed that number and found it equal to the measured speed of light. Electricity, magnetism, and optics were one subject all along.