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 inhomogeneous electromagnetic wave equation (or often "nonhomogeneous electromagnetic wave equation") with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

SI units

Maxwell's equations in a vacuum with charge ρ and current $\mathbf{J}$ sources can be written in terms of the vector and scalar potentials as

$\nabla^2 \varphi + {{\partial } \over \partial t} \left ( \nabla \cdot \mathbf{A} \right ) = - {\rho \over \varepsilon_0}$

$\nabla^2 \mathbf{A} - {1 \over c^2} {\partial^2 \mathbf{A} \over \partial t^2} - \nabla \left ( {1 \over c^2} {{\partial \varphi } \over {\partial t }} + \nabla \cdot \mathbf{A} \right ) = - \mu_0 \mathbf{J}$

where

$\mathbf{E} = - \nabla \varphi - {\partial \mathbf{A} \over \partial t}$

and

$\mathbf{B} = \nabla \times \mathbf{A}$.

If the Lorenz gauge condition is assumed

${1 \over c^2} {{\partial \varphi } \over {\partial t }} + \nabla \cdot \mathbf{A} = 0$

then the nonhomogeneous wave equations become

$\nabla^2 \varphi - {1 \over c^2} {\partial^2 \varphi \over \partial t^2} = - {\rho \over \varepsilon_0}$

$\nabla^2 \mathbf{A} - {1 \over c^2} {\partial^2 \mathbf{A} \over \partial t^2} = - \mu_0 \mathbf{J}$ .

CGS and Lorentz–Heaviside units

In cgs units these equations become

$\nabla^2 \varphi - {1 \over c^2} {\partial^2 \varphi \over \partial t^2} = - {4 \pi \rho }$

$\nabla^2 \mathbf{A} - {1 \over c^2} {\partial^2 \mathbf{A} \over \partial t^2} = - {4 \pi \over c} \mathbf{J}$

with

$\mathbf{E} = - \nabla \varphi - {1 \over c} {\partial \mathbf{A} \over \partial t}$

$\mathbf{B} = \nabla \times \mathbf{A}$

and the Lorenz gauge condition

${1 \over c} {{\partial \varphi } \over {\partial t }} + \nabla \cdot \mathbf{A} = 0$.

For Lorentz–Heaviside units, sometimes used in high dimensional relativistic calculations, the charge and current densities in cgs units translate as

$\rho \rightarrow { \rho \over {4 \pi } }$

$\mathbf{J} \rightarrow { 1 \over {4 \pi } } \mathbf{J}$.

Covariant form of the inhomogeneous wave equation

See also: Covariant formulation of classical electromagnetism

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

$\Box A^{\mu} \ \stackrel{\mathrm{def}}{=}\ \partial_{\beta} \partial^{\beta} A^{\mu} \ \stackrel{\mathrm{def}}{=}\ {A^{\mu , \beta}}_{\beta} = - \mu_0 J^{\mu}$ $\left( SI \right)$

$\Box A^{\mu} \ \stackrel{\mathrm{def}}{=}\ \partial_{\beta} \partial^{\beta} A^{\mu} \ \stackrel{\mathrm{def}}{=}\ {A^{\mu , \beta}}_{\beta} = - \frac{4 \pi}{c} J^{\mu}$ $\left( cgs \right)$

where J is the four-current

$J^{\mu} = \left(c \rho, \mathbf{J} \right)$,

${ \partial \over { \partial x^a } } \ \stackrel{\mathrm{def}}{=}\ \partial_a \ \stackrel{\mathrm{def}}{=}\ {}_{,a} \ \stackrel{\mathrm{def}}{=}\ (\partial/\partial ct, \nabla)$

is the 4-gradient and the electromagnetic four-potential is

$A^{\mu}=(\varphi, \mathbf{A} c)$ $\left( SI \right)$

$A^{\mu}=(\varphi, \mathbf{A} )$ $\left( cgs \right)$

with the Lorenz gauge condition

$\partial_{\mu} A^{\mu} = 0$.

Here

$\Box = \partial_{\beta} \partial^{\beta} = \nabla^2 - {1 \over c^2} \frac{ \partial^2} { \partial t^2}$ is the d'Alembert operator.

Curved spacetime

The electromagnetic wave equation is modified in two ways in curved spacetime, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears (SI units).

$- {A^{\alpha ; \beta}}_{ \beta} + {R^{\alpha}}_{\beta} A^{\beta} = \mu_0 J^{\alpha}$

where

${R^{\alpha}}_{\beta}$

is the Ricci curvature tensor. Here the semicolon indicates covariant differentiation. To obtain the equation in cgs units, replace the permeability with 4π / c.

Generalization of the Lorenz gauge condition in curved spacetime is assumed

${A^{\mu}}_{ ; \mu} = 0$.

Solutions to the inhomogeneous electromagnetic wave equation

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

$\varphi (\mathbf{r}, t) = \int { { \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } \rho (\mathbf{r}', t') d^3r' dt'$

and

$\mathbf{A} (\mathbf{r}, t) = \int { { \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } { \mathbf{J} (\mathbf{r}', t')\over c} d^3r' dt'$

where

${ \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) }$

is a Dirac delta function.

For SI units

$\rho \rightarrow { \rho \over {4 \pi \varepsilon_0 } }$

$\mathbf{J} \rightarrow { \mu_0 \over {4 \pi } } \mathbf{J}$.

For Lorentz–Heaviside units,

$\rho \rightarrow { \rho \over {4 \pi } }$

$\mathbf{J} \rightarrow { 1 \over {4 \pi } } \mathbf{J}$.

These solutions are known as the retarded Lorenz gauge potentials. They represent a superposition of spherical light waves traveling outward from the sources of the waves, from the present into the future.

There are also advanced solutions (cgs units)

$\varphi (\mathbf{r}, t) = \int { { \delta \left ( t' - { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } \rho (\mathbf{r}', t') d^3r' dt'$

and

$\mathbf{A} (\mathbf{r}, t) = \int { { \delta \left ( t' - { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } { \mathbf{J} (\mathbf{r}', t') \over c } d^3r' dt'$.

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

See also

• Wave equation
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Larmor formula
• Formulation of Maxwell's equations in special relativity
• Maxwell's equations in curved spacetime
• Abraham–Lorentz force

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

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8. 
• 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 0-471-30932-X. 
• 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 0-486-60637-6. 
• 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 0-393-92516-1.