Supplement 1.3: The Speed of Electromagnetic Waves

The phase velocity of monochromatic waves

The speed of a monochromatic wave can be easily calculated. We consider the electric field $\overrightarrow{E}$ of a wave moving in the direction of the wave vector $\overrightarrow{k}$:

The speed of the wave is obtained by setting

i.e., looking at positions $\overrightarrow{r}$ at times t of a fixed value of the field. Since ${\overrightarrow{E}}_o=\text{const}\text{.}$ in the case of plane waves, this is equivalent with a constant argument of the sine function:

The velocity is calculated by differentiating $\left|\overrightarrow{r}\right|=r$ versus t:

This is the phase velocity c of the wave. In case of electromagnetic waves this is the speed of light. With $\omega =2\pi f$ and $k=2\pi /\lambda$ it follows:

We can learn more about the dependence of the speed of light on electric and magnetic parameters by solving the wave equation

with                                 $\overrightarrow{E}(\overrightarrow{r},t)={\overrightarrow{E}}_{o}sin\left(\overrightarrow{k}\cdot \overrightarrow{r}-\omega t\right)$

(using the equations for $\overrightarrow{B}$ leads to an identical result).

 

Second differentiation of $\overrightarrow{E}$ versus space and time leads to

and hence:                             $c_o=\frac{\omega }{k}=\frac{1}{\sqrt{{\epsilon }_o{\mu }_o}}$

This is the vacuum speed of light c_o, which is a fundamental constant in physics. With the vacuum permittivity ε_o=8.854·10^-12 A·s/(V·m) and the vacuum permeability μ_o=1.256·10^-6 V·s/(A·m) one obtains:

or 300 000 km/s, approximately.

The speed of light c in matter is smaller than the speed of light in vacuum:

with the relative permittivity ε_r and permeability μ_r of the material. This equation is called Maxwell's relation. For transparent materials, it is μ_r≈1. For water and glass at frequencies of visible light, it is ε_r≈1.8 and 2.25, which explains their lower speed of light as indicated in chapter 1, section electromagnetic waves on page 2.

The refractive index n of matter is given by $n=\sqrt{{\epsilon }_r{\mu }_r}$, and hence:

With this result the electromagnetic wave equations are:

The squared speed of light connects the second order spatial and temporal derivatives of the electric and the magnetic field quantities.