5.9.5. Electromagnetic Waves

Table of Contents

Waves Made of Fields

Electromagnetic waves are traveling disturbances of the electric field and the magnetic field. Unlike mechanical waves, they do not need a material medium such as air or water. They can travel through empty space, which is why sunlight can reach Earth from the Sun.

In an electromagnetic wave, the electric field and magnetic field change with time and position, and they support each other as the wave moves forward. A changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. This idea is one of the great results of Maxwell's equations.

An electromagnetic wave is a self-propagating combination of changing electric and magnetic fields that can travel through vacuum.

How the Fields Are Oriented

A simple electromagnetic wave is transverse. This means the fields oscillate perpendicular to the direction in which the wave travels. The electric field is perpendicular to the magnetic field, and both are perpendicular to the direction of propagation.

If the wave moves in the $x$ direction, one possible arrangement is that the electric field oscillates in the $y$ direction and the magnetic field oscillates in the $z$ direction.

Rendering scientific sketch…

Orientation of fields in an electromagnetic wave

The direction of wave travel is given by the cross product of the fields,

$$
\vec E \times \vec B
$$

This points in the direction in which energy moves.

For an electromagnetic wave,
$$
\vec E \perp \vec B, \qquad \vec E \perp \text{direction of travel}, \qquad \vec B \perp \text{direction of travel}
$$
and
$$
\vec E \times \vec B
$$
points along the wave's direction of propagation.

A Simple Mathematical Form

A plane electromagnetic wave traveling in the positive $x$ direction can be written as

$$
\vec E(x,t) = E_0 \sin(kx - \omega t)\,\hat y
$$

$$
\vec B(x,t) = B_0 \sin(kx - \omega t)\,\hat z
$$

Here, $E_0$ and $B_0$ are the maximum values of the electric and magnetic fields, $k$ is the wave number, and $\omega$ is the angular frequency.

The two fields oscillate in phase. This means they reach maxima, minima, and zero at the same places and times.

Speed of Electromagnetic Waves

Maxwell's equations predict that electromagnetic waves travel at speed

$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$

where $\mu_0$ is the permeability of free space and $\varepsilon_0$ is the permittivity of free space. This speed is the speed of light in vacuum.

This was a remarkable discovery because it showed that light itself is an electromagnetic wave.

In vacuum, electromagnetic waves travel at
$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 3.00 \times 10^8 \,\text{m/s}
$$

Relation Between Electric and Magnetic Fields

The amplitudes of the electric and magnetic fields are not independent. In vacuum they satisfy

$$
E_0 = c B_0
$$

and more generally at every point in the wave,

$$
E = cB
$$

for a plane wave in vacuum.

This means the electric field is numerically much larger than the magnetic field because $c$ is a very large number.

For a plane electromagnetic wave in vacuum,
$$
E = cB
$$
and therefore
$$
E_0 = cB_0
$$

Wavelength, Frequency, and Speed

Like all waves, electromagnetic waves have wavelength $\lambda$, frequency $f$, and speed $v$. In vacuum,

$$
v = c
$$

and

$$
c = f\lambda
$$

This relation connects how fast the wave travels, how often it oscillates, and how far apart successive crests are.

Energy Transport

Electromagnetic waves carry energy. Sunlight warming your skin is a direct example. The flow of electromagnetic energy is described by the Poynting vector, which is treated separately in its own chapter, but the key idea here is simple. As the wave moves, energy is transported in the same direction as the wave.

The energy is shared between the electric field and the magnetic field. In a plane wave in vacuum, the average energy stored in the electric field equals the average energy stored in the magnetic field.

Why Electromagnetic Waves Exist

The existence of electromagnetic waves follows from Maxwell's equations in empty space. In vacuum, where there are no charges and no currents, the electric and magnetic fields satisfy wave equations. For the electric field,

$$
\nabla^2 \vec E = \mu_0 \varepsilon_0 \frac{\partial^2 \vec E}{\partial t^2}
$$

and similarly for the magnetic field,

$$
\nabla^2 \vec B = \mu_0 \varepsilon_0 \frac{\partial^2 \vec B}{\partial t^2}
$$

These are standard wave equations, showing that both fields can propagate through space as waves.

In empty space, both $\vec E$ and $\vec B$ satisfy wave equations, so changing fields can travel as waves even without matter.

Electromagnetic Waves and Light

Visible light is only one part of the electromagnetic family. Radio waves, microwaves, infrared, ultraviolet, X rays, and gamma rays are all electromagnetic waves. They differ in frequency and wavelength, but they are all the same kind of physical phenomenon.

A useful summary is shown below.

Type	Wavelength	Frequency	Typical example
Radio	Long	Low	Broadcasting
Microwave	Shorter than radio	Higher	Microwave ovens, radar
Infrared	Shorter than microwave	Higher	Heat radiation
Visible	About $400$ to $700$ nm	Moderate	Human vision
Ultraviolet	Shorter than visible	Higher	Sunburn
X rays	Very short	Very high	Medical imaging
Gamma rays	Shortest	Highest	Nuclear processes

Key Features to Remember

Electromagnetic waves are transverse waves of electric and magnetic fields. They can travel in vacuum. The fields are perpendicular to each other and to the direction of motion. They move at speed $c$ in vacuum, and their field magnitudes satisfy $E = cB$.

Main facts for electromagnetic waves in vacuum:
$$
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}, \qquad c = f\lambda, \qquad E = cB
$$
The wave is transverse, and $\vec E$, $\vec B$, and the direction of propagation are all mutually perpendicular.

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