Maxwell's equations are the four partial differential equations that completely describe classical electromagnetism. They were formulated by James Clerk Maxwell in 1865 and unified electricity, magnetism, and optics into a single theoretical framework.
The Four Equations
\(\nabla \cdot \mathbf{E} = \dfrac{\rho}{\varepsilon_0}\) — Gauss's Law: electric field diverges from charges.
\(\nabla \cdot \mathbf{B} = 0\) — Gauss's Law for Magnetism: no magnetic monopoles.
\(\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}}{\partial t}\) — Faraday's Law: a changing magnetic field induces an electric field.
\(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\dfrac{\partial \mathbf{E}}{\partial t}\) — Ampère–Maxwell Law: currents and changing electric fields create magnetic fields.
Displacement Current
Maxwell's key insight was adding the displacement current term \(\varepsilon_0 \partial\mathbf{E}/\partial t\) to Ampère's law. This predicts that a time-varying electric field produces a magnetic field even in the absence of physical current — allowing electromagnetic waves to propagate through a vacuum.
Maxwell's equations in integral form. Together they are the complete foundation of classical electromagnetism.
Electromagnetic Wave Speed
Transverse EM wave: E (red) and B (blue dashed) oscillate perpendicular to each other and to the propagation direction z.
Taking the curl of Faraday's law and substituting Ampère–Maxwell gives a wave equation. The wave speed is:
Maxwell recognised this as the speed of light, demonstrating that light is an electromagnetic wave.
In Materials
In a dielectric medium with relative permittivity \(\varepsilon_r\) and relative permeability \(\mu_r\), the wave speed becomes \(v = c/\sqrt{\varepsilon_r \mu_r}\). This slowing is why wavelengths in PCB substrates are shorter than in free space.