What Is the Speed of an Electromagnetic Wave in a Medium?

The speed of an electromagnetic wave in a material is generally lower than the speed of light in vacuum, but there is not one universal value for every material or every wave. For an ordinary, uniform, transparent, non-dispersive medium, you can calculate its phase speed from the medium’s electric permittivity and magnetic permeability. In optics, the same result is usually expressed using the refractive index.

The basic speed formula

For a linear, homogeneous, isotropic, non-dispersive medium with no free charge or current and with positive, constant permittivity and permeability, Maxwell’s equations give the wave equation

$$\nabla^2 \mathbf{E} = \mu\epsilon\frac{\partial^2 \mathbf{E}}{\partial t^2},$$

where $\mathbf{E}$ is the electric field, $\epsilon$ is the permittivity, and $\mu$ is the permeability. Comparing this with the standard wave equation gives the wave’s phase speed:

$$v = \frac{1}{\sqrt{\mu\epsilon}}.$$

In vacuum, the corresponding constants are $\mu_0$ and $\epsilon_0$, so the vacuum speed of light is

$$c = \frac{1}{\sqrt{\mu_0\epsilon_0}}.$$

If the material’s properties are written as relative permittivity $\epsilon_r = \epsilon/\epsilon_0$ and relative permeability $\mu_r = \mu/\mu_0$, then

$$v = \frac{c}{\sqrt{\epsilon_r\mu_r}}.$$

For most ordinary optical materials, $\mu_r$ is very close to 1. In that common case, $v \approx c/\sqrt{\epsilon_r}$.

Using refractive index

The refractive index $n$ is defined, for a transparent medium at a specified frequency, as the ratio of the vacuum speed to the phase speed in the medium:

$$n = \frac{c}{v}, \qquad v = \frac{c}{n}.$$

For a simple non-dispersive medium with positive permittivity and permeability, this agrees with $n = \sqrt{\epsilon_r\mu_r}$. In a real optical material, however, the refractive index can depend on frequency, so it is more precise to write $n(\omega)$, where $\omega$ is angular frequency. Unless stated otherwise, the “speed of light in a material” in elementary optics usually means this phase speed.

Worked example: light in glass

Suppose a glass sample has refractive index $n = 1.50$ for the light being considered. Using $c \approx 3.00\times10^8\ \text{m/s}$,

$$\begin{aligned} v &= \frac{c}{n} \\ &= \frac{3.00\times10^8\ \text{m/s}}{1.50} \\ &= 2.00\times10^8\ \text{m/s}. \end{aligned}$$

So the phase speed is about $2.00\times10^8\ \text{m/s}$. If the light’s vacuum wavelength is $600\ \text{nm}$, its frequency is unchanged when it enters the glass. Its wavelength there is

$$\lambda_{\text{glass}} = \frac{\lambda_0}{n} = \frac{600\ \text{nm}}{1.50} = 400\ \text{nm}.$$

The shorter wavelength, together with the unchanged frequency, gives the lower speed through $v=f\lambda$.

What changes at a boundary?

When a wave crosses from one stationary medium into another, its frequency is set by the source and remains the same across the boundary. Its speed and wavelength change. For example, entering a material with a higher refractive index lowers the phase speed and shortens the wavelength. This is the wave description behind refraction; the change in direction follows from the wavefronts meeting the boundary at an angle.

The wave does not generally “slow down” because it travels a shorter distance between collisions with atoms. The material’s response to the electric and magnetic fields changes how the combined wave propagates. A microscopic account involves interactions with the material, but the macroscopic speed is captured by its electromagnetic properties or measured refractive index.

Phase speed, group speed, and signal speed

In a dispersive material, different frequencies have different phase speeds. A narrow-band pulse can also have a group velocity, which describes how its envelope travels when the pulse is not strongly distorted. Group velocity is not always the speed at which energy or information is transmitted, particularly in strongly absorbing or anomalously dispersive regions. The speed of a causal signal’s leading edge does not exceed $c$.

Thus, saying “light travels at $c/n$ in a material” is a useful standard approximation when the question concerns phase speed at a specified frequency. It should not be read as a claim that every measure of a pulse’s speed is always exactly $c/n$.

Common mistakes

• Using $c/n$ without specifying the material or wavelength. Since $n$ can depend on frequency, the speed can vary with the light’s color.
• Confusing vacuum wavelength with wavelength in the material. The frequency stays fixed at the boundary; the wavelength changes with speed.
• Treating $v=1/\sqrt{\mu\epsilon}$ as universal in every material. It assumes a simple, homogeneous, isotropic medium. Anisotropic, absorbing, or strongly dispersive media need a more complete description.
• Assuming every definition of wave speed gives the same answer. Phase, group, and signal speeds can differ in dispersive media.