KAHIBARO
Table of Contents
Light as an Electromagnetic Wave Speed
One of the most remarkable results in physics is that light has a definite speed in vacuum, and that this speed comes directly from the basic laws of electricity and magnetism. Maxwell discovered that electric and magnetic fields can form self-propagating waves, and the speed of those waves is
$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$
Here, $\varepsilon_0$ is the permittivity of free space and $\mu_0$ is the permeability of free space.
This means that the speed of light is not just an experimentally measured number. It is also a natural consequence of the structure of electromagnetism.
The speed of electromagnetic waves in vacuum is
$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$
This same speed is the speed of light in vacuum.
Numerical Value
Using the known values of $\mu_0$ and $\varepsilon_0$, we obtain
$$
c \approx 3.00 \times 10^8 \ \text{m/s}
$$
More precisely,
$$
c = 299{,}792{,}458 \ \text{m/s}
$$
This is an extremely large speed. Light can travel around Earth several times in one second.
| Quantity | Symbol | Approximate value |
|---|---|---|
| Speed of light in vacuum | $c$ | $3.00 \times 10^8 \ \text{m/s}$ |
| Permittivity of free space | $\varepsilon_0$ | $8.85 \times 10^{-12} \ \text{F/m}$ |
| Permeability of free space | $\mu_0$ | $4\pi \times 10^{-7} \ \text{H/m}$ |
Why This Speed Appears
A changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. Because of this mutual generation, a disturbance in the fields can move through space as a wave.
When Maxwell's equations are combined in empty space, they produce a wave equation of the form
$$
\frac{\partial^2 \mathbf{E}}{\partial x^2} = \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}
$$
and similarly for $\mathbf{B}$. A standard wave equation has the form
$$
\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}
$$
so by comparison,
$$
v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$
Maxwell realized that this predicted wave speed matched the known speed of light. This was the key insight that showed light is an electromagnetic wave.
Vacuum and Materials
The value $c$ refers specifically to light in vacuum. In materials such as air, water, or glass, light travels more slowly. The speed in a medium is
$$
v = \frac{1}{\sqrt{\mu \varepsilon}}
$$
For many materials, this is written in terms of the refractive index $n$:
$$
v = \frac{c}{n}
$$
Since most materials have $n > 1$, the speed in the material is less than $c$.
| Medium | Typical refractive index $n$ | Speed of light |
|---|---|---|
| Vacuum | $1.00$ | $c$ |
| Air | $\approx 1.0003$ | slightly less than $c$ |
| Water | $\approx 1.33$ | $\approx 0.75c$ |
| Glass | $\approx 1.5$ | $\approx 0.67c$ |
The symbol $c$ means the speed of light in vacuum, not in every substance.
In a material,
$$
v = \frac{c}{n}
$$
so light usually travels slower than $c$.
Relation to Wavelength and Frequency
For any wave, speed, wavelength, and frequency are related by
$$
v = f\lambda
$$
For electromagnetic waves in vacuum,
$$
c = f\lambda
$$
This means that if the frequency is high, the wavelength is short, and if the frequency is low, the wavelength is long. All electromagnetic waves in vacuum, radio waves, microwaves, visible light, X rays, travel at the same speed $c$.
This does not mean they are identical. They differ in frequency and wavelength, but not in their vacuum speed.
Visual Picture
The electric and magnetic fields oscillate while the wave moves forward. The fields are perpendicular to each other and also perpendicular to the direction of motion.
Travel Time and Distance
Because the speed of light is finite, light takes time to travel from one place to another. The basic relation is
$$
\text{distance} = \text{speed} \times \text{time}
$$
so for light in vacuum,
$$
d = ct
$$
and therefore
$$
t = \frac{d}{c}
$$
For example, if light travels $3.00 \times 10^8 \ \text{m}$, it takes about $1$ second.
This finite travel time is important in astronomy, communications, and measurement.
For light traveling in vacuum,
$$
d = ct
\quad \text{and} \quad
t = \frac{d}{c}
$$
Light does not travel instantaneously. It takes time to move across space.
Physical Meaning
The speed of light is a fundamental constant of nature. In electromagnetism, it is the natural speed at which electric and magnetic disturbances move through empty space. This gives light a special role in physics.
Later in physics, especially in relativity, the quantity $c$ becomes even more important. Here, the key point is simpler: Maxwell's equations predict a wave speed, and that speed is exactly the speed of light in vacuum.
Summary
The speed of light in vacuum is
$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3.00 \times 10^8 \ \text{m/s}
$$
This speed comes from the constants of electromagnetism. It is the speed of all electromagnetic waves in vacuum. In materials, light usually travels more slowly, with speed
$$
v = \frac{c}{n}
$$
and the wave relation in vacuum is
$$
c = f\lambda
$$
So the speed of light connects the electric and magnetic laws of Maxwell to the observable behavior of light itself.