Maxwell’s Equations
Four equations. Two fields. The complete theory of classical electromagnetism.
$$\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} & \text{(Gauss — electricity)} \\ \nabla \cdot \mathbf{B} &= 0 & \text{(Gauss — magnetism)} \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} & \text{(Faraday)} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} & \text{(Ampère–Maxwell)} \end{aligned}$$
The structure
Each equation says one thing about one operation on one field:
| Divergence ($\nabla \cdot$) | Curl ($\nabla \times$) | |
|---|---|---|
| Electric field $\mathbf{E}$ | Charges create it | Changing B curls it |
| Magnetic field $\mathbf{B}$ | No monopoles | Currents + changing E curl it |
Two rows, two columns, four equations. Divergence tells you about sources. Curl tells you about circulation. Together, they completely determine the fields.
The prediction: electromagnetic waves
In empty space (no charges, no currents: $\rho = 0$, $\mathbf{J} = 0$), the equations simplify to:
$$\nabla \cdot \mathbf{E} = 0 \qquad \nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \qquad \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
The curl equations now say: changing $\mathbf{B}$ creates $\mathbf{E}$, and changing $\mathbf{E}$ creates $\mathbf{B}$. They sustain each other. The result is a self-propagating wave — an oscillation of electric and magnetic fields that travels through space without any charges or currents.
The speed of this wave is:
$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3 \times 10^8 \text{ m/s}$$
Maxwell calculated this in 1865, compared it to the measured speed of light, and concluded: “light is an electromagnetic disturbance.”
This was one of the great unifications in physics. Optics, electricity, and magnetism — three separate subjects — were revealed to be one thing.
An electromagnetic wave propagating through space. The electric field (red) and magnetic field (blue) oscillate perpendicular to each other and to the direction of travel.
The electromagnetic spectrum
All electromagnetic waves have the same structure — oscillating $\mathbf{E}$ and $\mathbf{B}$ fields propagating at $c$. The only difference is frequency:
| Frequency | Wavelength | What we call it |
|---|---|---|
| $\sim 10^4$ Hz | km | Radio waves |
| $\sim 10^{10}$ Hz | cm | Microwaves |
| $\sim 10^{13}$ Hz | μm | Infrared |
| $\sim 10^{14}$ Hz | 400–700 nm | Visible light |
| $\sim 10^{16}$ Hz | nm | Ultraviolet, X-rays |
| $\sim 10^{20}$ Hz | pm | Gamma rays |
All of this — from the radio signal carrying music to the gamma ray from a distant supernova — is described by the same four equations.
Why these equations matter
Maxwell’s equations are the prototype for all modern field theories in physics. The structure — fields described by divergence and curl, changing in time, obeying local equations — became the template for:
- General relativity (Einstein’s field equations for gravity)
- Quantum field theory (the Standard Model of particle physics)
- Gauge theories (the mathematical framework unifying forces)
They also directly underlie every piece of technology involving electromagnetic waves: radio, television, mobile phones, Wi-Fi, radar, medical imaging, fibre optics, lasers.
Four equations. Everything.