Why the speed of light appears in Maxwell’s equations

Maxwell’s equations describe how electric and magnetic fields behave and interact. The reason the speed of light appears in them is because they predict that changing electric and magnetic fields propagate through space as waves — and the speed at which these waves travel is exactly what we call the speed of light.

Here’s how it works:

1. Faraday’s Law
   This says that a changing magnetic field creates an electric field.

2. Ampère-Maxwell Law
   Maxwell modified Ampère’s Law to include the “displacement current,” which means that a changing electric field creates a magnetic field.

When you combine these two, you see that a changing electric field creates a changing magnetic field, which in turn creates a changing electric field, and so on. This self-sustaining process forms an electromagnetic wave.

When you solve Maxwell’s equations in empty space (vacuum), you get a wave equation for both the electric and magnetic fields. The solution to this wave equation shows that the waves travel at a speed given by:

where:

• $mu_0$ is the permeability of free space (describes how a magnetic field interacts with the vacuum)

• $varepsilon_0$ is the permittivity of free space (describes how an electric field interacts with the vacuum)

Plugging in the measured values of $mu_0$ and $varepsilon_0$ gives you about $3 times 10^8$ meters per second — the speed of light.

Before Maxwell, light was known to travel at this speed, but it was not clear why. Maxwell’s unification of electricity and magnetism showed that light is an electromagnetic wave, so its speed naturally emerges from the properties of space encoded in $varepsilon_0$ and $mu_0$. This was one of the greatest triumphs of classical physics — unifying optics with electromagnetism.

In summary: the speed of light appears in Maxwell’s equations because they predict that electric and magnetic fields propagate as waves at a speed determined by the permittivity and permeability of free space — and that speed matches the speed of light.