Before Maxwell: Four Disconnected Laws
The four experimental laws that Maxwell unified were each discovered independently over roughly four decades: Each law was experimentally well-established. But they described different phenomena with different mathematics, and there was no reason to think they were related. Maxwell saw the connection.
- Gauss's Law for electricity (1835): Electric field lines originate at positive charges and terminate at negative charges. The total flux through any closed surface equals the enclosed charge divided by the permittivity of free space.
- Gauss's Law for magnetism: There are no magnetic monopoles. Magnetic field lines always form closed loops — they never begin or end. The total magnetic flux through any closed surface is always zero.
- Faraday's Law of induction (1831): A changing magnetic field induces an electric field. This is the principle behind every transformer and electric generator ever built.
- Ampère's Law (1826): Electric currents create magnetic fields circling around them. This is the principle behind every electromagnet and motor.
The Displacement Current: Maxwell's Key Insight
The fourth equation, as Ampère originally wrote it, lacked the ∂D/∂t term. It described the magnetic field created by electric currents. Maxwell noticed a problem: Ampère's original law was mathematically inconsistent when applied to a capacitor charging. Current flows into the capacitor through a wire, and this current creates a magnetic field — but inside the capacitor, between the plates, there is no current. The magnetic field seemed to appear and disappear at the capacitor boundary, violating the mathematics of the curl operator. Maxwell's fix was to add the displacement current density term ∂D/∂t. Even when there is no physical movement of charges, a changing electric field produces an effective current that generates a magnetic field. This completed the mathematical symmetry between equations 3 and 4: just as a changing B produces E (Faraday), a changing E produces B (displacement current). Once the displacement current term was added, Maxwell could manipulate his equations to show that E and B fields could sustain each other in a self-propagating wave — even in empty space with no charges or currents. The wave speed came out as 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s. Maxwell wrote: "This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself is an electromagnetic disturbance." He was right. He had unified electricity, magnetism, and optics.
Numerical Electromagnetics: When Equations Can't Be Solved by Hand
Maxwell's equations have exact analytical solutions only for highly symmetric geometries — plane waves in free space, fields inside a rectangular waveguide, radiation from a Hertzian dipole. Real engineering geometries (antennas near structures, PCB traces above ground planes, shielding enclosures with apertures) require numerical methods. The three dominant numerical approaches are the Finite Element Method (FEM), the Method of Moments (MoM), and the Finite-Difference Time-Domain method (FDTD). FEM discretises the volume into tetrahedra and finds field distributions that satisfy Maxwell's equations at each element; MoM works on surfaces and is highly efficient for antenna problems; FDTD marches Maxwell's equations forward in time on a regular grid, naturally capturing transient behaviour and broadband responses. These tools — Ansys HFSS, CST Studio, COMSOL Multiphysics — are the modern implementation of Maxwell's 1865 insight. They solve the same four equations, for geometries and material distributions that would have taken Maxwell's contemporaries lifetimes of hand calculation. EngForge's transmission line calculator computes characteristic impedance, wave speed, reflection coefficients, and VSWR for coaxial, microstrip, and stripline geometries — directly from the material properties that appear in Maxwell's equations.