Electromagnetic wave equation

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The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field H, takes the form:

[ \nabla^2 \mathbf \ - \ \over \partial t^2} \ \ = \ \ 0]

[ \nabla^2 \mathbf \ - \ \over \partial t^2} \ \ = \ \ 0]

where c is the speed of light in the medium. In a vacuum, c = 2.998 x 10⁸ meters per second, which is the speed of light in free space.

The electromagnetic wave equation derives from Maxwell's equations.

In a linear, isotropic, non-dispersive medium, the magnetic flux density B is related to the magnetic field H by

[\mathbf = \mu \mathbf]

where μ is the magnetic permeability of the medium.

It should also be noted that in most modern literature, B is called the "magnetic field," and H is called either the "auxiliary magnetic field," or "the H vector."

In this article, it is most appropriate to use SI units through the motivation and derivation of the homogeneous wave equation. Once the marriage between electromagnetism and light has been made, and the relationship between the permitivity/permeability and the speed of light has been derived, it is often useful to use other units, such as cgs or Lorentz-Heaviside. At that point, we display results in all three sets of units.

Speed of propagation

In vacuum

If the wave propagation is in vacuum, then

[c = c_o = } = 2.998 \times 10^8 ] meters per second

is the speed of light in free space. The magnetic permeability [\ \mu_o] and the electric permittivity [\ \varepsilon_o] are important physical constants that play a key role in electromagnetic theory.

Symbol	Name	Numerical Value	SI Unit of Measure	Type
[ c \ ]	Speed of light	[ 2.998 \times 10^ ]	meters per second	defined
[ \ \varepsilon_0 ]	Permittivity	[ 8.854 \times 10^ ]	farads per meter	derived
[\ \mu_0 \ ]	Permeability	[ 4 \pi \times 10^ ]	henries per meter	defined

In a material medium

For the purposes of this article, we will assume that all materials are linear, isotropic, and non-dispersive. In that case, the speed of light in a material medium is

[c = = } ]

where

[ n = \sqrt ]

is the refractive index of the medium, [\mu \,] is the magnetic permeability of the medium, and [\varepsilon \,] is the electric permittivity of the medium.

The origin of the electromagnetic wave equation

Conservation of charge

Conservation of charge requires that the time rate of change of the total charge enclosed within a volume V must equal the net current flowing into the surface S enclosing the volume:

[ \oint_S \mathbf \cdot d \mathbf = - \int_V \rho \cdot dV]

where J is the current density (in amperes per square meter) flowing through the surface and ρ is the charge density (in coulombs per cubic meter) at each point in the volume.

From the divergence theorem, we can convert this relationship from integral form to differential form:

[ \nabla \cdot \mathbf = - ]

Ampere's Law prior to Maxwell's correction

In its original form, Ampere's Law (SI units) relates the magnetic field H to its source, the current density J:

[ \oint_C \mathbf \cdot d \mathbf = \int_S \mathbf \cdot d \mathbf]

Again, we can convert to differential form, this time using Stokes' theorem:

[ \nabla \times \mathbf = \mathbf ]

Inconsistency between Ampere's Law and Conservation of Charge

James Clerk Maxwell, who unified the laws of electricity and magnetism, discovered an important inconsistency between Ampere's Law and the Conservation of Charge.

If we take the divergence of both sides of Ampere's Law, we find

[ \nabla \cdot ( \nabla \times \mathbf ) = \nabla \cdot \mathbf ]

The divergence of the curl of any vector field – in this case, the magnetic field H – is always equal to zero:

[ \nabla \cdot ( \nabla \times \mathbf ) = 0 ]

Combining these two equations implies that

[\nabla \cdot \mathbf = 0 ]

From the conservation of charge, we know that

[ \nabla \cdot \mathbf = - ]

[ = 0]

This last result suggests that the net charge density at any point in space is a fixed constant that cannot ever change, which is of course absurd. Not only is this outcome contrary to all physical intuition, it also directly contradicts the empirical results of thousands of laboratory experiments. It requires not only that electrical charge is conserved, but that it cannot be re-distributed from one place to another. But we know that electrical currents can and do re-distribute electrical charge. As long as the total amount of charge remains constant, conservation of charge allows for the movement of charge from one place to another. So this last result is incorrect.

Something was clearly missing from Ampere's Law, and Maxwell figured out what it was.

Maxwell's correction to Ampere's Law

To understand Maxwell's correction to Ampere's Law, we need to look at another of Maxwell's Equations, namely, Gauss's Law (SI units) in integral form:

[ \oint_S \varepsilon_o \mathbf \cdot d \mathbf = \int_V \rho \cdot dV]

Again, using the divergence theorem, we can convert this equation to differential form:

[ \nabla \cdot \varepsilon_o \mathbf = \rho ]

Taking the derivative with respect to time of both sides, we find:

[ ( \nabla \cdot \varepsilon_o \mathbf ) = ]

Reversing the order of differentiation on the left-hand side, we obtain

[ \nabla \cdot \varepsilon_o \over \partial t } = ]

This last result, along with Ampere's Law and the conservation of charge equation, suggests that there are actually two sources of the magnetic field: the current density J, as Ampere had already established, and the so-called displacement current:

[ \over \partial t } = \varepsilon_o \over \partial t } ]

So the corrected form of Ampere's Law, which Maxwell discovered, becomes:

[ \nabla \times \mathbf = \mathbf + \varepsilon_o \over \partial t } ]

Maxwell - First to propose that light is an electromagnetic wave

Maxwell's correction of Ampere's Law set the stage for an even more important and, at the time, startling discovery made by Heinrich Rudolph Hertz. Maxwell realized that the equations of electromagnetism suggest that electric and magnetic fields can propagate through free space – in other words, in the absence of matter – as electromagnetic waves, and further, that the speed of propagation of these waves is exactly the same as the speed of light. Reflecting on his discovery in 1865, Maxwell wrote:

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself . . . is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

To obtain electromagnetic waves in a vacuum note that Maxwell's equations (SI units) in a vacuum are

[ \nabla \cdot \mathbf = 0]

[ \nabla \times \mathbf = -\mu_o \frac} ]

[ \nabla \cdot \mathbf = 0]

[ \nabla \times \mathbf =\varepsilon_o \frac} ]

If we take the curl of the curl equations we obtain

[ \nabla \times \nabla \times \mathbf = -\mu_o \frac \nabla \times \mathbf = -\mu_o \varepsilon_o \frac } ]

[ \nabla \times \nabla \times \mathbf = \varepsilon_o \frac \nabla \times \mathbf = -\mu_o \varepsilon_o \frac } ]

If we note the vector identity

[\nabla \times \left( \nabla \times \mathbf \right) = \nabla \left( \nabla \cdot \mathbf \right) - \nabla^2 \mathbf]

where [ \mathbf ] is any vector function of space, we recover the wave equations

[ \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf \ \ = \ \ 0]

[ \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf \ \ = \ \ 0]

where

[c = } = 2.998 \times 10^8 ] meters per second

is the speed of light in free space.

Covariant form of the homogeneous wave equation

These relativistic equations can be written in covariant form as

[ \Box A^ = 0 ] [ \left( SI \right)]

[\Box A^ = 0 ] [\left( cgs \right) ]

where J is the four-current

[J^ = \left(c \rho, \mathbf \right)]

and the electromagnetic four-potential is

[A^=(\varphi, \mathbf c)] [\left( SI \right)]

[ A^=(\varphi, \mathbf ) ] [\left( cgs \right)]

with the Lorenz gauge

[\partial_ A^ = 0\,].

Here

[\Box = \nabla^2 - \frac ] is the d'Alembertian operator. The square box is not a typographical error; it is the correct symbol for this 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.

[ - }_ + }_ A^ = 0 ]

where

[ }_ ]

is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

We have assumed the generalization of the Lorenz gauge in curved spacetime

[ }_ ].

Nonhomogeneous 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 nonhomogeneous.

Solutions to the homogeneous electromagnetic wave equation

The general solution to the electromagnetic wave equation takes the form

[ \mathbf( \mathbf, t ) = g(\phi( \mathbf, t )) = g( \omega t - \mathbf \cdot \mathbf ) ]

and

[ \mathbf( \mathbf, t ) = g(\phi( \mathbf, t )) = g( \omega t - \mathbf \cdot \mathbf ) ]

for virtually any well-behaved function g of dimensionless argument φ, where

[ \ \omega ] is the angular frequency (in radians per second), and

[ \mathbf = ( k_x, k_y, k_z) ] is the wave vector (in radians per meter).

Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g 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 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:

[ k = | \mathbf | = = ]

where k is the wavenumber and λ is the wavelength.

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:

[\mathbf ( \mathbf, t ) = \mathrm \ (\mathbf ) e^ \}]

where

• [j = \sqrt \, ] is the imaginary unit,
• [ \omega = 2 \pi f \, ] is the angular frequency in radians per second,
• [ f \, ] is the frequency in hertz, and
• [ e^ = \cos(\omega t) + j \sin(\omega t) \, ] is Euler's formula.

Plane wave solutions

Consider a plane defined by a unit normal vector

[ \mathbf = \over k } ].

Then planar traveling wave solutions of the wave equations are

[ \mathbf(\mathbf) = E_0 e^ \cdot \mathbf } ]

and

[ \mathbf(\mathbf) = H_0 e^ \cdot \mathbf } ]

where

[ \mathbf = (x, y, z) ] is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector [ \mathbf ]. If we define the z direction as the direction of [ \mathbf ] and the x direction as the direction of [ \mathbf ], then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation

[ \mp c \mu_o = ].

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. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

[ \mathbf ( \mathbf, t ) = \cos( \omega t - \mathbf \cdot \mathbf + \phi_0 ) ]

and

[ \mathbf ( \mathbf, t ) = \cos( \omega t - \mathbf \cdot \mathbf + \phi_0 ) ]

where

[ \ t ] is time (in seconds),

[ \ \omega ] is the angular frequency (in radians per second),

[ \mathbf = ( k_x, k_y, k_z) ] is the wave vector (in radians per meter), and

[ \phi_0 \,] is the phase angle (in radians).

The wave vector is related to the angular frequency by

[ k = | \mathbf | = = ]

where k is the wavenumber and λ is the wavelength.

The Electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.

Other solutions

Spherically symmetric and cylindrically symmetric analytic solutions to the electromagnetic wave equations are also possible.

References

Electromagnetics

Journal articles

• James Clerk Maxwell, "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) ISBN 0-13-249020-X
• Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).
• David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, Electromagnetic Waves (Prentice-Hall, 1994) ISBN 0-13-225871-4
• Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995) ISBN 0-262-69188-4.

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, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a treatment of Maxwell's equations in terms of differential forms.)

Vector calculus

• H. M. Schey, Div Grad Curl and all that: An informal text on vector calculus, 4th edition (W. W. Norton & Company, 2005) ISBN 0-393-925161.

See also

Theory and Experiment

• Maxwell's equations
• Electromagnetic radiation
• Charge conservation
• Light
• Electromagnetic spectrum
• Optics
• Special relativity
• General relativity
• Larmor power formula

Applications

• Rainbow
• Cosmic microwave background radiation
• Laser
• Laser fusion
• Photography
• X-ray
• X-ray crystallography
• RADAR
• Radio waves
• Optical computing
• Microwave
• Holography
• Microscope
• Telescope
• Gravitational lens
• Black body radiation

Biographies

• Albert Einstein
• James Clerk Maxwell
• Michael Faraday
• Heinrich Hertz

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