Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 1 of 9 Electromagnetic wave equation From Wikipedia, the free encyclopedia 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 B, takes the form: 2 where is the speed of light in the medium, and is the Laplace operator. In a vacuum, c = c0 = 299,792,458 meters per second, which is the speed of light in free space.[1] The electromagnetic wave equation derives from Maxwell's equations. It should also be noted that in most older literature, B is called the magnetic flux density or magnetic induction . Contents ■ 1 The origin of the electromagnetic wave equation ■ 2 Covariant form of the homogeneous wave equation ■ 3 Homogeneous wave equation in curved spacetime ■ 4 Inhomogeneous electromagnetic wave equation ■ 5 Solutions to the homogeneous electromagnetic wave equation ■ 5.1 Monochromatic, sinusoidal steady-state ■ 5.2 Plane wave solutions ■ 5.3 Spectral decomposition ■ 5.4 Other solutions ■ 6 See also ■ 6.1 Theory and Experiment ■ 6.2 Applications ■ 7 Notes ■ 8 References ■ 9 Further reading ■ 9.1 Electromagnetism ■ 9.1.1 Journal articles ■ 9.1.2 Undergraduate-level textbooks ■ 9.1.3 Graduate-level textbooks ■ 9.2 Vector calculus ■ 9.3 Biographies http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 2 of 9 ■ 10 External links The origin of the electromagnetic wave equation In his 1864 paper titled A Dynamical Theory of the Electromagnetic Field, Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force A postcard from Maxwell to Peter Tait. (http://upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf) . In PART VI of his 1864 paper which is entitled 'ELECTROMAGNETIC THEORY OF LIGHT' [2], 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. [3] Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics 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: where ρ = 0 because there's no charge density in free space. Taking the curl of the curl equations gives: http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 3 of 9 By using the vector identity where is any vector function of space, it turns into the wave equations: where m/s 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: . Where 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 Time dilation in transversal motion. The curved spacetime requirement that the speed of light is constant in every inertial reference frame leads to the Main article: Maxwell's equations in curved theory of Special Relativity. spacetime http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 4 of 9 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 and the semicolon indicates covariant differentiation. The generalization of the Lorenz gauge condition in curved spacetime is assumed: . Inhomogeneous electromagnetic wave equation Main article: 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 inhomogeneous. Solutions to the homogeneous electromagnetic wave equation Main article: Wave equation The general solution to the electromagnetic wave equation is a linear superposition of waves of the form and for virtually any well -behaved function g of dimensionless argument φ, where is the angular frequency (in radians per second), and 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 : where k is the wavenumber and λ is the wavelength . http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 5 of 9 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, ■ is the angular frequency in radians per second, ■ is the frequency in hertz, and ■ is Euler's formula. Plane wave solutions Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation Consider a plane defined by a unit normal vector . Then planar traveling wave solutions of the wave equations are and where is the position vector (in meters). These solutions represent planar waves traveling in the direction of the normal vector . If we define the z direction as the direction of and the x direction as the direction of , then by Faraday's Law the magnetic field lies in the y 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. 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 http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 6 of 9 Electromagnetic spectrum illustration. and where is time (in seconds), is the angular frequency (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 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. In spherical coordinates the solutions to the wave equation can be written as follows: , http://en.wikipedia.org/wiki/Electromagnetic_wave_equation 5/31/2011 Electromagnetic wave equation - Wikipedia, the free encyclopedia Page 7 of 9 and , . These can be rewritten in terms of the spherical bessel function. In cylindrical coordinates, the solutions to the wave equation are the ordinary bessel function of integer order. See also Theory and Experiment ■ Maxwell's equations ■ Special relativity ■ Wave equation ■ General relativity ■ Electromagnetic modeling ■ Photon dynamics in the double-slit ■ Electromagnetic radiation experiment ■ Charge conservation ■ Photon polarization ■ Light ■ Larmor power formula ■ Electromagnetic spectrum ■ Theoretical and experimental justification ■ Optics for the Schrödinger equation Applications ■ Rainbow ■ Radio waves ■ Cosmic microwave background radiation ■ Optical computing ■ Laser ■ Microwave ■ Laser fusion ■ Holography ■ Photography ■ Microscope ■ X-ray ■ Telescope ■ X-ray crystallography ■ Gravitational lens ■ RADAR ■ Black body radiation Notes 1. ^ Current practice is to use c0 to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. See NIST Special Publication 330 , Appendix 2, p. 45 (http://physics.nist.gov/Pubs/SP330/sp330.pdf) 2. ^ Maxwell 1864 (http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf)