How to derive electromagnetic waves from Maxwell’s equations?

Hello everyone, in the previous article, we reviewed the following Maxwell’s equations. This set of equations was rated as the most beautiful formula in 2020, comparable to Newton’s mechanics formula, Einstein’s mass-energy equation, and Euler’s formula.

Some students may ask, apart from those complex mathematical symbols, isn’t it just:

• Gauss’s law of electricity
• Gauss’s law of magnetism
• Faraday’s law of electromagnetic induction
• Ampere’s loop law?

Yes — and that’s where it gets exciting.

The Breakthrough: Displacement Current Predicts Electromagnetic Waves

This equation predicted the existence of electromagnetic waves and ushered in a wireless era.

The red part in the equation is the displacement current density, also represented by JD, which represents the displacement current passing through a unit area.

This displacement current is independent of the conduction current in Ampere’s loop theorem. It was proposed by Maxwell to solve the problem of applying Ampere’s law to non-steady currents, proving that:

• Not only constant currents can generate magnetic fields
• But also changing electric fields can generate magnetic fields

So for the changing electric field and the changing magnetic field, it is you in me, I in you, what is this? Isn’t this electromagnetic wave? Yes, the greatness of Maxwell’s equations is that through simple mathematical derivation, we can get the wave equation of the electromagnetic field.

Let’s derive it together:

For simplicity, we finally gave the electric field equation under one-dimensional passive conditions, a perfect electric field wave equation. At the same time, according to Maxwell’s equations ④, we can also derive the wave equation of the magnetic field. We organize the simplest one-dimensional passive electromagnetic field wave equation below,

And the wave speed v of this wave equation is the speed of light in a vacuum ≈ 3*10^8 m/s

Maxwell’s Prediction and Hertz’s Confirmation

When Maxwell first predicted the existence of electromagnetic waves and suggested that light itself is an electromagnetic wave, the scientific community had serious doubts.

It wasn’t until 1888, nine years after Maxwell’s death, that Hertz confirmed the existence of electromagnetic waves through experiments. He also measured their propagation speed using the standing wave method, and published the results in:

“On Electromagnetic Waves in the Air and Their Reflections”

But confirming the wave speed didn’t necessarily mean light waves are electromagnetic waves.

---

Hertz’s Further Proof

In subsequent experiments, Hertz verified the:

• Reflection
• Refraction
• Focusing
• Interference
• Diffraction
• Polarization

…of electromagnetic waves — and pointed out that:

Electromagnetic waves are transverse waves —
with exactly the same properties as light waves

Only then was Maxwell’s prediction fully accepted by the scientific community.

Feynman and Einstein on Maxwell’s Genius

Feynman once praised Maxwell in his Lectures on Physics:

“Light — this common and elusive thing — is so important and mysterious that when writing Genesis, God needed a special day to create it. But when Maxwell completed his discovery, he could say: As long as there is electricity and magnetism, there will be light.”

Einstein’s theory of relativity was also deeply influenced by Maxwell, especially in terms of:

Invariance of the speed of light

Electromagnetic symmetry

Spacetime symmetry

The Legacy of Maxwell and the Future of RF Engineering

The progress of history is full of twists and turns. Maxwell was not recognized as the greatest mathematical physicist after Newton until the success of Hertz’s experiment.

Einstein once sighed:

“Physicists took many years to understand the full significance of Maxwell’s discovery. His genius was defeated only after Hertz confirmed the existence of electromagnetic waves by experiment.”

From Coulomb to RF Engineers

The theory of electromagnetism is a magnificent building.

• It took more than 110 years from Coulomb’s theorem (1785) to Hertz’s experiment (1888).
• Ampere, Gauss, and Faraday laid a solid foundation.
• Maxwell built the structure.
• Hertz brought it to life with experimental proof.

And today — our job as RF engineers and antenna developers is to:

Make this building more comfortable and convenient to live in.