The electromagnetic wave equation is a second-order
partial differential equation that describes the propagation of
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
s through a
medium
Medium may refer to:
Science and technology
Aviation
* Medium bomber, a class of war plane
* Tecma Medium, a French hang glider design
Communication
* Media (communication), tools used to store and deliver information or data
* Medium ...
or in a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
. It is a
three-dimensional form of the wave equation. The
homogeneous form of the equation, written in terms of either the
electric field or the
magnetic field , takes the form:
where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
(i.e.
phase velocity) in a medium with
permeability , and
permittivity , and is the
Laplace operator. In a vacuum, , a fundamental
physical constant. The electromagnetic wave equation derives from
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
. In most older literature, is called the ''magnetic flux density'' or ''magnetic induction''. The following equations
predicate that any electromagnetic wave must be a
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example o ...
, where the electric field and the magnetic field are both perpendicular to the direction of wave propagation.
The origin of the electromagnetic wave equation
In his 1865 paper titled
A Dynamical Theory of the Electromagnetic Field,
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper
On Physical Lines of Force. In ''Part VI'' of his 1864 paper titled ''Electromagnetic Theory of Light'', Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:
The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.[See /upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf Maxwell 1864 page 499.]
Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with
Faraday's law of induction.
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:
These are the general Maxwell's equations specialized to the case with charge and current both set to zero.
Taking the
curl of the curl equations gives:
We can use the
vector identity
where is any vector function of space. And
where is a
dyadic which when operated on by the divergence operator yields a vector. Since
then the first term on the right in the identity vanishes and we obtain the wave equations:
where
is the speed of light in free space.
Covariant form of the homogeneous wave equation
These
relativistic equations can be written in
contravariant form as
where the
electromagnetic four-potential is
with the
Lorenz gauge condition
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
:
and where
is the
d'Alembert operator.
Homogeneous wave equation in curved spacetime
The electromagnetic wave equation is modified in two ways, the derivative is replaced with the
covariant derivative and a new term that depends on the curvature appears.
where
is the
Ricci curvature tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
and the semicolon indicates covariant differentiation.
The generalization of the
Lorenz gauge condition
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
in curved spacetime is assumed:
Inhomogeneous electromagnetic wave equation
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
inhomogeneous.
Solutions to the homogeneous electromagnetic wave equation
The general solution to the electromagnetic wave equation is a
linear superposition
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
of waves of the form
for virtually well-behaved function of dimensionless argument , where is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
(in radians per second), and is the
wave vector (in radians per meter).
Although the function can be and often is a monochromatic
sine wave, it does not have to be sinusoidal, or even periodic. In practice, cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of
Fourier decomposition
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.
In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the
dispersion relation:
where is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
and is the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
. The variable can only be used in this equation when the electromagnetic wave is in a vacuum.
Monochromatic, sinusoidal steady-state
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:
where
* is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
,
* is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
in
radians per second
The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega). ...
,
* is the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
in
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
, and
*
is
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
.
Plane wave solutions
Consider a plane defined by a unit normal vector
Then planar traveling wave solutions of the wave equations are
where is the position vector (in meters).
These solutions represent planar waves traveling in the direction of the normal vector . If we define the direction as the direction of , and the direction as the direction of , then by Faraday's Law the magnetic field lies in the direction and is related to the electric field by the relation
Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.
This solution is the linearly
polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.
Spectral decomposition
Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of
sinusoids
A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter. Capillaries are composed of only the tunica intima, consisting of a thin wall of simple squamous endothelial cells. They are the smallest blood vessels in the body: ...
. This is the basis for the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form
where
* is time (in seconds),
* is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
(in radians per second),
* is the
wave vector (in radians per meter), and
*
is the
phase angle (in radians).
The wave vector is related to the angular frequency by
where is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
and is the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
.
The
electromagnetic spectrum
The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies.
The electromagnetic spectrum covers electromagnetic waves with frequencies ranging fro ...
is a plot of the field magnitudes (or energies) as a function of wavelength.
Multipole expansion
Assuming monochromatic fields varying in time as
, if one uses Maxwell's Equations to eliminate , the electromagnetic wave equation reduces to the
Helmholtz Equation for :
with as given above. Alternatively, one can eliminate in favor of to obtain:
A generic electromagnetic field with frequency can be written as a sum of solutions to these two equations. The
three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
with coefficients proportional to the
spherical Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
. However, applying this expansion to each vector component of or will give solutions that are not generically divergence-free (), and therefore require additional restrictions on the coefficients.
The multipole expansion circumvents this difficulty by expanding not or , but or into spherical harmonics. These expansions still solve the original Helmholtz equations for and because for a divergence-free field , . The resulting expressions for a generic electromagnetic field are:
where
and
are the ''electric multipole fields of order (l, m)'', and
and
are the corresponding ''magnetic multipole fields'', and and are the coefficients of the expansion. The multipole fields are given by
where are the
spherical Hankel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
, and are determined by boundary conditions, and
are
vector spherical harmonics normalized so that
The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae
radiation patterns, or nuclear
gamma decay. In these applications, one is often interested in the power radiated in the
far-field. In this regions, the and fields asymptotically approach
The angular distribution of the time-averaged radiated power is then given by
See also
Theory and experiment
*
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
*
Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
*
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
*
Electromagnetic modeling
*
Electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
*
Charge conservation
In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is al ...
*
Light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
*
Electromagnetic spectrum
The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies.
The electromagnetic spectrum covers electromagnetic waves with frequencies ranging fro ...
*
Optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
*
Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
*
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
*
Inhomogeneous electromagnetic wave equation
*
Photon polarization
*
Larmor power formula
*
Applications
*
Rainbow
*
Cosmic microwave background radiation
*
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
*
Laser fusion
Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with thermonuclear fuel. In modern machines, the targets are small spherical pellets about the size of ...
*
Photography
Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is emplo ...
*
X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
*
X-ray crystallography
*
Radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
*
Radio waves
*
Optical computing
*
Microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
*
Holography
*
Microscope
A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisi ...
*
Telescope
A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to obse ...
*
Gravitational lens
*
Black-body radiation
Biographies
*
André-Marie Ampère
*
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
*
Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
*
Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The uni ...
*
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
*
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
*
Hendrik Lorentz
Notes
Further reading
Electromagnetism
Journal articles
* Maxwell, James Clerk, "''
/upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf A Dynamical Theory of the Electromagnetic Field'", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
Undergraduate-level textbooks
*
*
* Edward M. Purcell, ''Electricity and Magnetism'' (McGraw-Hill, New York, 1985). .
* Hermann A. Haus and James R. Melcher, ''Electromagnetic Fields and Energy'' (Prentice-Hall, 1989) .
* Banesh Hoffmann, ''Relativity and Its Roots'' (Freeman, New York, 1983). .
*
David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, ''Electromagnetic Waves'' (Prentice-Hall, 1994) .
* Charles F. Stevens, ''The Six Core Theories of Modern Physics'', (MIT Press, 1995) .
* Markus Zahn, ''Electromagnetic Field Theory: a problem solving approach'', (John Wiley & Sons, 1979)
Graduate-level textbooks
*
*
Landau, L. D., ''The Classical Theory of Fields'' (
Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987). .
*
* Charles W. Misner,
Kip S. Thorne
Kip Stephen Thorne (born June 1, 1940) is an American theoretical physicist known for his contributions in gravitational physics and astrophysics. A longtime friend and colleague of Stephen Hawking and Carl Sagan, he was the Richard P. F ...
,
John Archibald Wheeler, ''Gravitation'', (1970) W.H. Freeman, New York; . ''(Provides a treatment of Maxwell's equations in terms of differential forms.)''
Vector calculus
*P. C. Matthews ''Vector Calculus'', Springer 1998,
*H. M. Schey, ''Div Grad Curl and all that: An informal text on vector calculus'', 4th edition (W. W. Norton & Company, 2005) .
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