# Electromagnetic wave

In physics, an **electromagnetic wave** is a change, periodic in space and time, of an electric field **E**(**r**,*t*) and a magnetic field **B**(**r**,*t*). A stream of electromagnetic waves is referred to as **electromagnetic radiation**.
Because an electric as well as a magnetic field is involved, the term *electromagnetic* (EM) is used, a contamination of electric and magnetic. Examples of EM waves in increasing wavelength are: gamma rays, X-rays,
ultraviolet light, visible light, infrared, microwaves, and radio waves. All these waves propagate in vacuum with the same speed *c*, the speed of light. The speed of light in air (at standard temperature and pressure) is very close to the speed of light in vacuum (the refractive index of air, *n*, is 1.0002926, meaning that the speed of electromagnetic waves in air is *c*/*n* ≈ *c*).

Classically (non-quantum mechanically), EM radiation is produced by accelerating charges, for instance, by charges oscillating in a radio antenna. Quantum mechanically, EM radiation is seen as a beam of energy parcels (photons). Photons are emitted whenever a system in an energetically high state (of energy *E*_{2}) makes a transition to a state of lower energy (*E*_{1}); during this transition a photon of energy proportional to the difference *E*_{2} - *E*_{1} > 0 is emitted. This is what happens in a fluorescent tube: mercury atoms are brought into an energetically high state by collisions with electrons, and upon subsequent falling down to their lowest energy state they emit photons.

Electromagnetic waves were predicted on theoretical grounds by James Clerk Maxwell in 1861 and first emitted and received in the laboratory by Heinrich Hertz a quarter century later. The first to see the applicability for communication purposes was the inventor of radiotelegraphy Guglielmo Marconi (around 1900). Later applications are radio, television, radar, cellular phones, gps, and all kinds of wireless applications, from remote control of television sets to internet access in public places.

## Properties

In figure 1 we see a snapshot (i.e., a picture at a certain point in time) of the magnetic and electric fields in adjacent points of space. In each point, the vector **E** is perpendicular to the vector **B**. The wave propagates to the right, along an axis which we conveniently refer to as *z*-axis. Both **E** and **B** are perpendicular to the propagation direction, which is expressed by stating that an electromagnetic wave is a transverse wave, in contrast to sound waves, which are longitudinal waves (i.e., air molecules vibrate parallel to the propagation direction of the sound).

Assume that the snapshot in figure 1 is taken at time *t*, then at a certain point *z* we see an arrow of certain length representing **E**(*z*,*t*) and also a vector **B**(*z*,*t*). At a point in time Δ*t* later, the same values of **E** and **B** (same arrows) are seen at *z* + *c* Δ*t*. The arrows seem to have propagated to the right with a speed *c*.

The time *t* is fixed and the position *z* varies in figure 1. Conversely, we can keep the position fixed and imagine what happens if time changes. Focus on a fixed point *z*, then in progressing time the two vectors **E**(*z*,*t*) and **B**(*z*,*t*) in the point *z*, grow to a maximum value, then shrink to zero, become negative, go to a minimum value, and grow again, passing through zero on their way to the same maximum value. This cycle is repeated indefinitely. When we now plot **E** and **B** in the fixed point *z* as a function of time *t*, we see the same type (sine-type) function as in figure 1. The number of times per second that the vectors go through a full cycle is the frequency of the electromagnetic wave.

Periodicity in space means that the EM wave is repeated after a certain distance. This distance, the wavelength is traditionally designated by λ, see figure 1. If we go at a fixed time a distance λ to the right or to the left we encounter the very same fields **E** and **B**.

Basically, the only property distinguishing different kinds of EM waves, is their wavelength, see figure 2. Note the enormous span in wavelengths, from one trillionth of a millimeter for gamma-rays (radioactive rays) up to the VLF (very low frequency) radio waves of about 100 kilometer.

### Frequency of electromagnetic waves

Often EM waves are characterized by their frequency ν, instead of their wavelength λ. If the EM field goes through ν full cycles in a second, where ν is a positive integral number, the field has a frequency of ν Hz (hertz). The speed of propagation of the EM waves being *c*, in 1/ν seconds the wave propagates a distance *c* × (1/ν) meter (according to the formula: distance traveled is speed times traveling time). The distance covered in 1/ν seconds is by definition the wavelength λ:

- <math>

\lambda = \frac{c}{\nu} \quad \Longrightarrow\quad \nu = \frac{c}{\lambda}
</math>
If we express *c* in m/s then λ is obtained in m. To convert quickly from wavelength to frequency we can approximate *c* by 3·10^{8} m/s.

As was pointed out above, the wavelengths of the various parts of the EM spectrum differ many orders of magnitude. Furthermore, the sources of the radiations, the interactions with matter, and the detectors employed differ widely, too. So, it is not surprising that in the past different parts of the spectrum were discovered at different times and that electromagnetic radiation of different wavelengths is called by different names. In the table and in figure 2 some illustrative values are given for several kinds of EM waves, together with their names.

| |||||

Some typical values of: | |||||

| |||||

EM wave | λ (m) | ν (1/s) | hν (J) |
T (s) |
1/λ (cm^{−1}) |

| |||||

γ-rays | 1.00⋅10^{−14} | 3.00⋅10^{22} | 1.98⋅10^{−11} | 3.33⋅10^{−23} | 1.00⋅10^{12 } |

X-rays | 5.00⋅10^{−10} | 6.00⋅10^{17} | 3.96⋅10^{−16} | 1.67⋅10^{−18} | 2.00⋅10^{7 } |

Ultraviolet | 2.00⋅10^{−7 } | 1.50⋅10^{15} | 9.90⋅10^{−19} | 6.67⋅10^{−16} | 5.00⋅10^{4 } |

Visible | 6.00⋅10^{−7 } | 5.00⋅10^{14} | 3.30⋅10^{−19} | 2.00⋅10^{−15} | 1.67⋅10^{4 } |

Infrared | 5.00⋅10^{−6 } | 6.00⋅10^{13} | 3.96⋅10^{−20} | 1.67⋅10^{−14} | 2.00⋅10^{3 } |

Microwave | 1.00⋅10^{−2 } | 3.00⋅10^{10} | 1.98⋅10^{−23} | 3.33⋅10^{−11} | 1.00 |

Radio | 1.00⋅10^{2 } | 3.00⋅10^{6 } | 1.98⋅10^{−27} | 3.33⋅10^{−7 } | 1.00⋅10^{−4} |

### Monochromatic linearly polarized waves

The wave depicted in figure 1 is *monochromatic*, i.e., it is characterized by a single wavelength (monochromatic means "of one color". In the visible region, different wavelengths correspond to light of different colors).
It is known that EM waves can be linearly superimposed, which is due to the fact that they are solutions of a linear partial differential equation, the *wave equation* (see next section). A linear superposition of waves is a solution of the same wave equation as the waves themselves. Such a superposition is also an electromagnetic wave (a propagating periodic EM field). If waves of different wavelengths are superimposed, then a non-monochromatic wave is obtained (the term multi-chromatic wave would be apt, but is never used). By means of Fourier analysis a non-monochromatic wave can be decomposed into its monochromatic components.

The electric field vectors in figure 1 are all in one plane, this is the *plane of polarization*, and a wave with one fixed polarization plane, is called *linearly polarized.*

The radiation of many lasers is monochromatic and linearly polarized (at least to a very good approximation).

## Relation to Maxwell's equations

*In this section it will be shown that the electromagnetic wave depicted in figure 1 is a solution of the Maxwell equations in the vacuum. *

We assume that at some distance away from the source of EM waves (a radio transmitter, a laser, gamma radiating nuclei, etc.), there is no charge density ρ and no current density **J**. For that region of space, the microscopic (vacuum) Maxwell equations become (in SI units):

- <math>

\boldsymbol{\nabla} \cdot \mathbf{B} = 0, \qquad \boldsymbol{\nabla} \cdot \mathbf{E} = 0, </math> and

- <math>

\boldsymbol{\nabla} \times \mathbf{B}= \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}, \qquad \boldsymbol{\nabla} \times \mathbf{E}= -\frac{\partial \mathbf{B}}{\partial t}. </math> Apply to the last Maxwell equation the following relation, known from vector analysis and valid for any (differentiable) vector field,

- <math>

\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{E}) = \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{E}) - \boldsymbol{\nabla}^2 \mathbf{E}
</math>
and use that **∇** · **E** = 0, then **E** satisfies the wave equation,

- <math>

\boldsymbol{\nabla}^2 \mathbf{E} = \frac{\partial(\boldsymbol{\nabla}\times \mathbf{B})}{\partial t} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}.
</math>
Note that the displacement current (time derivative of **E**) is essential in this equation, if it were absent (zero), the field **E** would be a static, time-independent, electric field, and there would be no waves.

In the very same way we derive a wave equation for **B**,

- <math>

\boldsymbol{\nabla}^2 \mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}.
</math>
Observe that **E** and **B** are related by the third and fourth Maxwell equation, which express the fact that a displacement current causes a magnetic field, and a changing magnetic field causes an electric field (Faraday's law of induction), respectively. So a time-dependent electric field that is not associated with a time-dependent magnetic field cannot exist, and conversely. Indeed, in special relativity **E** and *c* **B** can be transformed into one another by a Lorentz transformation of the electromagnetic field tensor, which shows their close relationship.

The wave equation is without doubt the most widely studied differential equation in mathematical physics. In figure 1 the electric field depicts a particular solution, with special initial and boundary conditions. The snapshot that is depicted has the analytic form

- <math>

\mathbf{E}(\mathbf{r}, t) = \mathbf{e}_x E_0 \sin\big[k(z-ct)\big]\quad\hbox{with}\quad k \equiv \frac{2\pi}{\lambda}.
</math>
The snapshot is taken at <math> \, t = 2\pi n /(ck)</math> for some arbitrary integer *n*. We assumed here that the direction of **E** defines the direction of the *x*-axis with unit vector **e**_{x} along this axis. The quantity *E*_{0} is the amplitude of the wave. Insertion of this expression in in the left hand side of the wave equation for **E** gives

- <math>

\boldsymbol{\nabla}^2 \mathbf{E} = \mathbf{e}_x E_0 \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial z^2} = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big].
</math>
Insertion of this expression in in the right hand side of the wave equation for **E** gives

- <math>

\frac{\mathbf{e}_x E_0}{c^2} \frac{\partial^2 \sin\big[k(z-ct)\big]}{\partial t^2} = - \frac{c^2\,k^2}{c^2} \mathbf{e}_x E_0 \sin\big[k(z-ct)\big] = - k^2 \mathbf{e}_x E_0 \sin\big[k(z-ct)\big],
</math>
so that it follows that the special solution, depicted in figure 1, is indeed a solution of the wave equation for **E**.

We could now proceed in the very same way and solve the wave equation for **B**, but then we could easily overlook the relation between the two fields. So we rather substitute the solution for **E** into the fourth Maxwell equation and use the definition of curl as a determinant,

- <math>

\begin{vmatrix} \mathbf{e}_x &\quad \mathbf{e}_y & \mathbf{e}_z \\ \frac{\partial}{\partial x}&\quad \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ E_0 \sin\big[k(z-ct)\big] &\quad 0 & \quad 0 \\ \end{vmatrix} = k \mathbf{e}_y E_0 \cos\big[k(z-ct)\big] = - \frac{\partial \mathbf{B}}{\partial t}. </math> It is easy to see that

- <math>

\mathbf{B}(\mathbf{r},t) = \mathbf{e}_y B_0 \sin\big[k(z-ct)\big] \quad \hbox{with}\quad B_0 \equiv \frac{E_0}{c}
</math>
is a solution of this equation. It follows that **E** and **B** are perpendicular (along the *x*-axis and *y*-axis, respectively) and are in phase. That is, **E** and **B** are simultaneously zero and attain simultaneously their maximum and minimum. The fact that (in vacuum) the amplitude of **B** is a factor *c* smaller than that of **E** is due to the use of SI units, in which the amplitudes have different dimensions. In Gaussian units this is not the case and *E*_{0} = *B*_{0}.

## Energy and energy flow in electromagnetic field

### Relation between power densities

In this section the following balance of power (energy per unit time) densities will be discussed:

- <math>

\boldsymbol{\nabla}\cdot \mathbf{S} + \frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} = - \mathcal{E}_\textrm{Joule} \qquad\qquad\qquad(1) </math> where

- <math>

\begin{align}
\mathbf{S} &= \mathbf{E} \times \mathbf{H}\equiv \mathbf{E} \times \frac{1}{\mu_0}\mathbf{B} \qquad\mathrm{(the\,\, Poynting\,\,vector)} \\
\mathcal{E}_\textrm{Field} &= \frac{1}{2} ( \mathbf{E}\cdot\mathbf{D}+ \mathbf{B}\cdot\mathbf{H})
\equiv \frac{1}{2} ( \epsilon_0 \mathbf{E}\cdot\mathbf{E}+ \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B})
\\
\mathcal{E}_\textrm{Joule} &= \mathbf{E}\cdot \mathbf{J} \\
\end{align}
</math>
The terms in equation (1) have dimension W/m^{3} (watt per volume). The quantity <math>\scriptstyle -\mathcal{E}_\textrm{Joule}\, >\, 0</math> represents the rate at which energy is produced per unit volume by ordinary Joulean (resistance) heating. The quantity <math>\scriptstyle \mathcal{E}_\textrm{Field}</math> is the energy density of the EM field and the time derivative is the rate of increase (power density). The vector **S**, *Poynting's vector*, is the power flux, the amount of energy crossing unit area perpendicular to the vector, per unit time.

Multiplying the terms in equation (1) by a volume Δ*V* and a time span Δ*t*, both small enough that the terms in the equation may assumed to be constant over the volume and the time span, the equation represents the conservation of energy: the energy generated in Δ*V*Δ*t* (the right hand side) is equal to the increase in energy in Δ*V* plus the net flow of energy leaving Δ*V*. Hence, when this equation is multiplied by Δ*V*Δ*t* it expresses the conservation of energy.

### Intensity

The intensity *I* of an electromagnetic wave is by definition the modulus of the Poynting vector,
the amount of energy carried by the wave across a unit surface in unit time [dimension: volt×ampere/meter^{2} = joule/(second×m^{2}) = W/m^{2}],

- <math>

I \equiv |\mathbf{E} \times \mathbf{H}| = \Big[ (\mathbf{E}\cdot\mathbf{E})\,(\mathbf{H}\cdot\mathbf{H})- (\mathbf{E}\cdot\mathbf{H})\,(\mathbf{E}\cdot\mathbf{H})\Big]^{1/2}.
</math>
Use **E** ⋅ **H** = 0 and that in vacuum and SI units

- <math>

\mathbf{H}(\mathbf{r},t) = c\,\epsilon_0\ \mathbf{E}(\mathbf{r},t), </math> then

- <math>

I = c\,\epsilon_0\, |\mathbf{E}(\mathbf{r},t)|^2.
</math>
Often *I* is time-averaged over a complete cycle. Since the integral of cos^{2} and sin^{2} over a cycle is 1/2, we get for the time-averaged intensity:

- <math>

\bar{I} = \frac{1}{2} c\,\epsilon_0\, E_0^2,
</math>
where *E*_{0} is the amplitude of the electric field.

Clearly, the cycle averaged intensity of a wave going through vacuum is constant, independent of the direction of propagation, *z*. This is because in vacuum the wave does not lose energy to the medium. In a medium this may be different, <math>\,\bar{I}\,</math> may decrease because of energy loss.
The Lambert-Beer law states in that case

- <math>

\frac{d\bar{I}(z)}{dz} = - k \bar{I}(z) \quad\Longrightarrow\quad \bar{I}(z) = \bar{I}_0 e^{-kz}.
</math>
Hence the electromagnetic wave is damped by the factor exp[− *kz*/2]. This damping gives rise to an imaginary component of the index of refraction.

### Derivation of relation between power densities

Equation (1), the balance of power, will be proved. Recall from elementary electricity theory the laws of Joule and Ohm. They state that the amount of energy *W* per unit time, produced by a conduction current *I*, is equal to

- <math>

W = I^2\, R = I\,V, \quad\hbox{with}\quad R,\, W > 0 ,
</math>
where *R* is the resistance and *V* a voltage difference.

Assuming that the current flows along *z*, one may introduce the current density *J*_{z}. Using

- <math>

V = - E_z \Delta z \quad\hbox{and}\quad I = J_z \Delta x \Delta y </math> one obtains

- <math>

W = - E_z J_z\, \Delta x\Delta y\Delta z .
</math>
One could continue discussing the system with the small volume <math> \Delta x\Delta y\Delta z</math>. However, because all terms in the equations would be multiplied by the same volume, it is more convenient to consider densities and to divide out the volume. Nevertheless, we still refer to *the* system. Thus, define

- <math>

\mathcal{E}_\textrm{Joule} \equiv -\frac{W}{ \Delta x\Delta y\Delta z} = \mathbf{E}\cdot \mathbf{J} < 0.
</math>
The negative quantity <math>\mathcal{E}_\textrm{Joule}</math> is the loss of energy of the system per unit time and per unit volume (according to Joule's and Ohm's laws). One may look upon the quantity <math> -\mathcal{E}_\textrm{Joule}</math> as the work (per unit time and unit volume) done by the Lorentz force *q*(**v**×**B** + **E**) on the moving particles constituting the current density **J**. Since **v**⋅(**v**×**B**) = 0, this work depends only on the electric field **E**.

Apply one of Maxwell's equations:

- <math>

\mathbf{E}\cdot \mathbf{J} = \mathbf{E}\cdot\left( - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} + \frac{1}{\mu_0} \boldsymbol{\nabla}\times \mathbf{B} \right) </math> Use a rule known from vector analysis and apply another one of Maxwell's equations,

- <math>

\begin{align} \mathbf{E}\cdot(\boldsymbol{\nabla}\times \mathbf{B}) =& \mathbf{B}\cdot(\boldsymbol{\nabla}\times \mathbf{E}) - \boldsymbol{\nabla}\cdot(\mathbf{E} \times \mathbf{B}) \\ =& -\mathbf{B}\cdot\left(\frac{\partial \mathbf{B}}{\partial t}\right) - \boldsymbol{\nabla}\cdot(\mathbf{E} \times \mathbf{B}) \end{align} </math> Define

- <math>

\mathcal{E}_\textrm{Field} \equiv \frac{1}{2}\left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) = \frac{1}{2}\left( \mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H} \right) </math> with

- <math>

\mathbf{D}\equiv \epsilon_0\mathbf{E}\quad\hbox{and}\quad\mathbf{B} \equiv \mu_0\mathbf{H},
</math>
where ε_{0} is the electric constant and μ_{0} the magnetic constant. Define also

- <math>

\mathbf{S} \equiv \frac{1}{\mu_0} \mathbf{E}\times\mathbf{B} = \mathbf{E}\times\mathbf{H}
</math>
where **S** is the Poynting vector called after John Henry Poynting. This vector is perpendicular to the plane of **E** and **B** and by the right-hand rule, it points in the direction of propagation of the EM wave. The divergence of the Poynting vector is the energy flow associated with the electromagnetic wave, i.e., with the pair **E**(**r**,*t*) and **B**(**r**,*t*). By definition **∇**·**S** gives the flow leaving the system and − **∇**·**S** gives the flow entering the system.
The total energy balance becomes

- <math>

-\mathcal{E}_\textrm{Joule} = \frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} + \boldsymbol{\nabla}\cdot \mathbf{S}. </math>

Here an example is found of the conservation of energy, known as *Poynting's theorem*: The energy produced per unit time according to Joule's law <math> -\mathcal{E}_\textrm{Joule}</math> is equal to the rate in increase of the electromagnetic energy of the system <math>\mathcal{E}_\textrm{Field}</math>, plus the flow of EM radiation **∇**·**S** leaving the system.

If there is no current, **J** = 0, then

- <math>

\frac{\partial \mathcal{E}_\textrm{Field}}{\partial t} = - \boldsymbol{\nabla}\cdot \mathbf{S}, </math> which is the continuity equation. The increase of field energy per unit time is the flow of radiation energy into the system.

### Example

As an example we give an order-of-magnitude-calculation of an electromagnetic energy density.
Consider to that end a radio station with a signal of strength *P* kW. We compute the energy density at a distance *R* from the station. First one must assume what the shape is of the waves emitted by the antenna, are they spherical or cylindrical? Choose the latter and call the cylinder height *z*. Further it is assumed that power density is homogeneous and that all power crosses the cylindrical walls, that is, power crossing the top and bottom of the cylinder is assumed to be zero. Also no absorption by the atmosphere or the Earth will occur. When a steady state is reached (some time after the beginning of the transmission), the time derivative of <math>\scriptstyle \mathcal{E}_\mathrm{Field}</math> vanishes. The energy density at a distance *R* becomes constant in time,

- <math>

\mathcal{E}_\mathrm{Field}(R) = \frac{P} {2\pi R z c} \quad \mathrm{J/m^3},
</math>
where *c* is the speed of propagation of the radio signal (is speed of ligth ≈ 3·10^{8} m/s).^{[1]}
To give a numerical example: *P* = 100 kW, *R* = 5 km (about 3 miles), *z* = 50 m, then
<math>\mathcal{E}_\textrm{Field}</math> = 2.1·10^{−10} J/m^{3}.

## Fourier expansions of the fields

*See Fourier expansion electromagnetic field for the expansion of the vector fields*.**A**(vector potential),**E**, and**B**together with the Fourier expressions for the EM energy and the EM momentum

## Photons

*See photons for an overview and see quantization of the electromagnetic field for a more detailed discussion.*

## Notes

- ↑
*P*is the integral of <math>\scriptstyle -\mathcal{E}_\textrm{Joule}</math> = −**E**⋅**J**over the volume of the cylinder; <math>\scriptstyle (2\pi R z c)\,\mathcal{E}_\mathrm{Field}</math> is the integral of the Poynting vector**S**over the surface of the cylinder.