The secret of the speed of light: electromagnetic runaway

The 4 Maxwell's equations

All electromagnetic phenomena reduce to a system of four fundamental relations: Maxwell's equations. On their own, these equations predict everything about the dynamics of electric and magnetic fields.

1. Gauss's law for electricity: \(\nabla \cdot \mathbf{E} = \dfrac{\rho}{\varepsilon_0}\) (The divergence of the electric field is proportional to the charge density \(\rho\) → an electric charge creates an electric field around it)
   
2. Gauss's law for magnetism: \(\nabla \cdot \mathbf{B} = 0\) (The net magnetic flux through a closed surface is zero → magnetic field lines form closed loops, there are no magnetic monopoles)
   
3. Faraday's law: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) (Any variation of the magnetic field in time creates a rotating electric field)
   
4. Ampère–Maxwell law: \(\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) (A time-varying electric field in turn creates a magnetic field)

The last two equations are the key to the mystery of runaway. A time-varying magnetic field (\( \partial \mathbf{B}/\partial t \)) creates a rotating electric field. Symmetrically, a time-varying electric field (\( \partial \mathbf{E}/\partial t \)) creates a rotating magnetic field. It is this dynamic coupling that sustains itself throughout their propagation.

Maxwell and the discovery of the speed of light

When James Clerk Maxwell (1831–1879) unified the laws of electricity and magnetism in the 19th century, he discovered a surprising phenomenon: a local disturbance of an electric or magnetic field creates a wave that propagates all by itself in vacuum, without the need for additional energy.

The wave equation and the prediction of the speed of light

Maxwell showed that this wave travels at a finite speed, determined by two properties of vacuum:

The formula \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \) gives this speed.

• μ₀: the vacuum's ability to carry a magnetic field
• ε₀: the vacuum's ability to let an electric field pass through

Light, an electromagnetic wave

By combining Faraday's and Ampère-Maxwell's laws, he obtained a wave equation. Inserting the values of μ₀ and ε₀, he found: \( c \approx 300\,000 \) km/s, the speed of light!

Maxwell's conclusion

In 1865, Maxwell concluded that light is an electromagnetic wave, just like radio waves or microwaves. A discovery that revolutionized physics.

N.B.:
In Maxwell's time (1865), the speed of light was not yet known with great precision. The experimental measurements available, made notably by Armand Fizeau (1819-1896) and Albert Michelson (1852-1931), made it possible to estimate \( v \) with a significant margin of error. Maxwell's theoretical calculation therefore gave a value very close to what would be confirmed later, which constituted an impressive validation of his theory.

Calculation of the propagation speed of the E and B fields

When Maxwell assembled the laws of electricity and magnetism, he discovered a structure that sustains itself as soon as one locally disturbs an electric or magnetic field. This "runaway" is stable, without amplification, and has no internal energy source. Vacuum carries the wave without feeding it.

The wave equation and the prediction of the speed of light

This dynamic loop between \( \mathbf{B} \) and \( \mathbf{E} \) produces a structure that propagates through vacuum at a finite speed.

The term \(\mu_0 \varepsilon_0\) of the fourth equation fixes the speed of light: \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. \)

• \( \mu_0 \) = vacuum permeability constant, or the vacuum's ability to carry a magnetic field
• \( \varepsilon_0 \) = vacuum electric permittivity constant, or the vacuum's ability to let an electric field "pass"

Obtaining the electromagnetic wave equation

By mathematically combining Faraday's and Ampère-Maxwell's laws (taking the curl of one and using the other), the coupling is eliminated to obtain an equation describing each field alone.

Wave equation for the electric field

For the electric field, we find: \( \nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \) This is the classical form of the wave equation. The term \( \mu_0 \varepsilon_0 \) appears in front of the second time derivative. In mathematical physics, the standard wave equation is written \( \nabla^2 \psi = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} \), where \( v \) is the propagation speed.

Identification with the speed of a wave

By immediate identification: \( \frac{1}{v^2} = \mu_0 \varepsilon_0 \quad \Rightarrow \quad v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \)

Result of the calculation: Light is an electromagnetic wave

Maxwell calculated this value with the constants known in his time:

• \( \mu_0 = 4\pi \times 10^{-7} \, \text{N} \cdot \text{A}^{-2} \)
• \( \varepsilon_0 \approx 8.854 \times 10^{-12} \, \text{C}^{2} \cdot \text{N}^{-1} \cdot \text{m}^{-2} \)

The result is astonishing: \( v \approx 2.998 \times 10^{8} \, \text{m/s} \), the speed of the waves corresponds to the speed of light measured experimentally!

N.B.:
This mathematical demonstration assumes vacuum, that is, the absence of charges and currents (\( \rho = 0, \, \mathbf{J} = 0 \)). The reasoning consists in taking the curl of Faraday's law and using the vector identity \( \nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} \), combining it with Gauss's law (\( \nabla \cdot \mathbf{E} = 0 \) in vacuum) and the Ampère-Maxwell law. One thus obtains the same wave equation for the magnetic field \( \mathbf{B} \). This theoretical prediction by James Clerk Maxwell (1831-1879) in 1865 was confirmed experimentally by Heinrich Hertz (1857-1894) in 1887, when he produced the first radio waves (electromagnetic waves).

Limits of Maxwell's equations

Maxwell found that electromagnetic waves propagate at a certain speed in vacuum, but the demonstration that this is a limiting speed. Maxwell did not yet know relativity. He therefore had no reason to think that this was a universal speed limit. The notion of a universal limit only appeared with special relativity, well after Maxwell.

It was only with special relativity (1905) by Albert Einstein (1879-1955) that the speed of light in vacuum \( c \) was recognized as the **maximum speed at which information, energy or any physical interaction can propagate**. In other words, \( c \) is a **fundamental limit imposed by the structure of spacetime**, and not a simple consequence of Maxwell's equations.

Key takeaways

Electromagnetic phenomena obey Maxwell's equations and can be summarized as follows:

• An electric charge (electric charge) creates an electric field around it.
• A time-varying magnetic field generates an electric field.
• A time-varying electric field in turn creates a magnetic field.
• These two fields sustain each other, propagating through space as electromagnetic waves at the speed of light.

Thus, light and all electromagnetic waves are nothing other than the manifestation of this phenomenon of self-sustaining electric and magnetic fields in vacuum.