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Encyclopedia > 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 Nonhomogeneous electromagnetic wave equation with sources. The addition of sources to the wave equations makes the partial differential equations nonhomogeneous. In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...

Contents


SI units

Maxwell's equations in a vacuum with charge ρ and current sources can be written in terms of the vector and scalar potentials as

where

and

.
Compact fluorescent light bulb with E26 Edison screw base
Compact fluorescent light bulb with E26 Edison screw base

If we assume the Lorenz gauge Download high resolution version (502x730, 56 KB)A picture of a Compact fluorescent light bulb taken by Kevin Rector. ... Download high resolution version (502x730, 56 KB)A picture of a Compact fluorescent light bulb taken by Kevin Rector. ... The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludwig Lorenz. ...

then the nonhomogeneous wave equations become

.

CGS and Lorentz-Heaviside units

In cgs units these equations become

WMAP image of the CMB anisotropy,Cosmic microwave background radiation. (June 2003))
WMAP image of the CMB anisotropy,Cosmic microwave
background radiation
. (June 2003))

with Image File history File links Download high resolution version (2198x1274, 1278 KB)WMAP map of CMB anisotropy, from NASA.gov File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (2198x1274, 1278 KB)WMAP map of CMB anisotropy, from NASA.gov File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ...

and the Lorenz gauge

.

For Lorentz-Heaviside units, sometimes used in high dimensional relativistic calculations, the charge and current densities in cgs units translate as Lorentz-Heaviside units for Maxwells equations are often used in relativistic calculations. ...

.

Covariant form of the nonhomogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of relativity
Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of relativity

The relativistic Maxwell's equations can be written in covariant form as Image File history File links This animated GIF is meant to be used as an illustration for the time dilation article. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... This page does not deal with the statistical concept covariance of random variables, nor with the computer science concepts of covariance and contravariance. ...

where J is the four-current In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

and the electromagnetic four-potential is Electromagnetic potential is . ...

with the Lorenz gauge

.

Here

is the d'Alembertian operator.

In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

Solutions to the nonhomogeneous electromagnetic wave equation

Retarded solutions

Retarded spherical wave. The source of the wave occurs at time t'. The wavefront moves away from the source as time increases for t>t'. For advanced solutions, the wavefront moves backwards in time from the source t<t'.
Retarded spherical wave. The source of the wave occurs at time t'. The wavefront moves away from the source as time increases for t>t'. For advanced solutions, the wavefront moves backwards in time from the source t<t'.

In the case that there are no boundaries surrounding the sources, the solutions (cgs units) of the nonhomogeneous wave equations are Image File history File links Electromagnetic_wave_equation. ... Image File history File links Electromagnetic_wave_equation. ...

and

where

is a Dirac delta function. The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...


For SI units

.

For Lorentz-Heaviside units Lorentz-Heaviside units for Maxwells equations are often used in relativistic calculations. ...

.

These solutions are known as retarded solutions. They represent a superposition of spherical light waves traveling outward from the sources of the waves, from the present into the future. In physics, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...


Advanced solutions

There are also advanced solutions (cgs units)

and

.

These represent a superposition of spherical waves traveling from the future into the present.


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.)

A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ...

Undergraduate-level textbooks

  • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X.
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), W. H. Freeman. ISBN 0716708108.
  • 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

  • Jackson, John D. (1998). Classical Electrodynamics (3rd ed.), Wiley. ISBN 047130932X.
  • Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987).
  • Maxwell, James C. (1954). A Treatise on Electricity and Magnetism, Dover. ISBN 0486606376.
  • 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.)

Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... John Archibald Wheeler (born July 9, 1911) is an American theoretical physicist. ...

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.

 
 

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