Maxwell's Equations

Four equations, and everything electrical, magnetic and optical follows — including the punchline hiding in the algebra: fields can chase each other through empty space at $1/\sqrt{\mu_0\varepsilon_0}$, which works out to the measured speed of light. Light is not like electromagnetism; it is electromagnetism.

Maxwell's four equations unify electricity, magnetism, and light.

$$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0} \qquad\text{(Gauss's law for E)}$$

$$\nabla\cdot\vec{B} = 0 \qquad\text{(No magnetic monopoles)}$$

$$\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t} \qquad\text{(Faraday's law)}$$

$$\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t} \qquad\text{(Ampere-Maxwell)}$$

Displacement current — Maxwell added $\varepsilon_0\dfrac{\partial\vec{E}}{\partial t}$ to Ampere's law. This term allows electromagnetic waves to exist in vacuum.

Electromagnetic waves — Maxwell's equations predict waves travelling at:

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00\times 10^8\;\text{m/s}$$

Key insight: Maxwell showed that light is an electromagnetic wave. This unified optics with electromagnetism — one of the greatest achievements in physics.

Common pitfall: An electromagnetic wave needs no medium — the fields sustain each other. The $E$ and $B$ fields are in phase, mutually perpendicular, and both perpendicular to the travel direction; a common error is drawing them out of phase like energy-trading oscillators.