Maxwell’s Equations – The RF Foundations You Forgot

If you’ve ever wondered why antennas radiate or how electric and magnetic fields dance through space, it’s time to revisit the core: Maxwell’s Equations. Most hams recognize the names, but few realize how deeply they underpin everything from antenna design to common-mode current.

1) What Are Maxwell’s Equations?

Maxwell’s Equations describe how electric fields (E) and magnetic fields (B) behave and interact. They govern all classical electromagnetism and are the theoretical foundation of RF engineering.

There are four equations:

Gauss’s Law (Electric):
∇⋅E=ρε0

Electric charges create electric fields.

Gauss’s Law (Magnetic):
∇⋅B=0

There are no magnetic monopoles – magnetic field lines always loop.

Faraday’s Law:
∇×E=−∂B∂t

A changing magnetic field induces an electric field.

Ampère’s Law (with Maxwell's correction):
∇×B=μ0J+μ0ε0∂E∂t

Currents and changing electric fields create magnetic fields.

2) Why Hams Should Care

Maxwell’s Equations aren’t just academic—they explain:

  • How RF signals propagate: A time-varying E field creates a B field, and vice versa—this leads to electromagnetic wave propagation.
  • Why antennas radiate: Acceleration of charges (alternating current) leads to field changes per Maxwell.
  • The difference between conduction and radiation: A conductor can carry RF without radiating unless there’s a discontinuity, current imbalance, or acceleration—exactly what happens in poorly choked feedlines.
  • How common-mode arises: It’s not just “RF on the outside of the coax” — it’s an unintended mode of field symmetry that radiates because Maxwell demands it.

3) The Real Magic: Displacement Current

Maxwell’s “correction” to Ampère’s Law added the displacement current term—this subtle insight unified AC circuit theory with field theory. It’s why capacitors pass RF, and why antennas can radiate even when there's no “DC continuity.”

4) Visualizing the RF Field Dance

In free space, an RF wave has:

  • An E-field oscillating in one direction
  • A B-field oscillating orthogonally
  • The wave itself propagating perpendicular to both (right-hand rule)

This self-sustaining dance of E and B is why your antenna doesn't need a wire between it and your target—it is the wire, just made of fields.

5) So What? Practical Takeaways for Hams

  • Your antenna’s current distribution matters: Radiation comes from current change, not just current magnitude.
  • Common-mode chokes break field symmetry, not just “block current.”
  • Baluns exist to make the field equations work in your favor.
  • Displacement current = energy transfer through space, even without conductive paths.
  • RF doesn’t “leak” — it radiates when Maxwell says it must.

Maxwell Isn't Just Theory – It's The Manual

Whether you're tuning a dipole, diagnosing RFI, or designing an RF preamp—Maxwell’s Equations are silently at play. They don’t just describe the game; they are the game. Every signal you send is a ripple in the vast field described by these four elegant laws.

Next time you ask, “Why is this radiating?” — ask Maxwell.

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Written by Joeri Van Dooren, ON6URE – RF, electronics and software engineer, complex platform and antenna designer. Founder of RF.Guru. An expert in active and passive antennas, high-power RF transformers, and custom RF solutions, he has also engineered telecom and broadcast hardware, including set-top boxes, transcoders, and E1/T1 switchboards. His expertise spans high-power RF, embedded systems, digital signal processing, and complex software platforms, driving innovation in both amateur and professional communications industries.