Inhomogeneous electromagnetic wave equation explained

In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

Maxwell's equations

For reference, Maxwell's equations are summarized below in SI units and Gaussian units. They govern the electric field E and magnetic field B due to a source charge density ρ and current density J:

Name	SI units	Gaussian units
Gauss's law	\nabla ⋅ E = (\rho)/(\varepsilon _{0})	\nabla ⋅ E = 4 \pi\rho
Gauss's law for magnetism	\nabla ⋅ B = 0	\nabla ⋅ B = 0
Maxwell–Faraday equation (Faraday's law of induction)	\nabla x E =- (\partial B)/(\partial t)	\nabla x E =- (1)/(c) (\partial B)/(\partial t)
Ampère's circuital law (with Maxwell's addition)	\nabla x B = \mu _{0\left(J} + \varepsilon _{0} (\partial E)/(\partial t) \right)	\nabla x B = (1)/(c) \left( 4 \pi J + (\partial E)/(\partial t) \right)

where ε₀ is the vacuum permittivity and μ₀ is the vacuum permeability. Throughout, the relation$$\backslash varepsilon\_0\; \backslash mu\_0\; =\; \backslash dfrac$$is also used.

SI units

E and B fields

Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity (The last term in the right side is the vector Laplacian, not Laplacian applied on scalar functions.) gives the wave equation for the electric field E:$$\backslash dfrac\backslash dfrac-\backslash nabla^2\backslash mathbf\; =\; -\backslash left(\backslash dfrac\; \backslash nabla\backslash rho+\backslash mu\_0\backslash dfrac\backslash right)\backslash ,.$$

Similarly substituting Gauss's law for magnetism into the curl of Ampère's circuital law (with Maxwell's additional time-dependent term), and using the curl of the curl identity, gives the wave equation for the magnetic field B:$$\backslash dfrac\backslash dfrac-\backslash nabla^2\backslash mathbf\; =\; \backslash mu\_0\; \backslash nabla\backslash times\backslash mathbf\backslash ,.$$

The left hand sides of each equation correspond to wave motion (the D'Alembert operator acting on the fields), while the right hand sides are the wave sources. The equations imply that EM waves are generated if there are gradients in charge density ρ, circulations in current density J, time-varying current density, or any mixture of these.

These forms of the wave equations are not often used in practice, as the source terms are inconveniently complicated. A simpler formulation more commonly encountered in the literature and used in theory use the electromagnetic potential formulation, presented next.

A and φ potential fields

Introducing the electric potential φ (a scalar potential) and the magnetic potential A (a vector potential) defined from the E and B fields by:$$\backslash mathbf\; =\; -\; \backslash nabla\; \backslash varphi\; -\; \backslash frac\; \backslash ,,\backslash quad\; \backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\; \backslash ,.$$

The four Maxwell's equations in a vacuum with charge ρ and current J sources reduce to two equations, Gauss's law for electricity is:$$\backslash nabla^2\; \backslash varphi\; +\; \backslash frac\; \backslash left\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; \backslash right)\; =\; -\; \backslash frac\; \backslash rho\; \backslash ,,$$where

\nabla ^{2} here is the Laplacian applied on scalar functions, and the Ampère-Maxwell law is:$$\backslash nabla^2\; \backslash mathbf\; -\; \backslash frac\; \backslash frac\; -\; \backslash nabla\; \backslash left\; (\backslash frac\; \backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; \backslash right)\; =\; -\; \backslash mu\_0\; \backslash mathbf\; \backslash ,$$where \nabla ^{2} here is the vector Laplacian applied on vector fields. The source terms are now much simpler, but the wave terms are less obvious. Since the potentials are not unique, but have gauge freedom, these equations can be simplified by gauge fixing. A common choice is the Lorenz gauge condition:$$\backslash frac\; \backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0$$

Then the nonhomogeneous wave equations become uncoupled and symmetric in the potentials:

For reference, in cgs units these equations arewith the Lorenz gauge condition$$\backslash frac\; \backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0\backslash ,.$$

Covariant form of the inhomogeneous wave equation

See also: Covariant formulation of classical electromagnetism.

The relativistic Maxwell's equations can be written in covariant form aswhere$$\backslash Box\; =\; \backslash partial\_\; \backslash partial^\; =\; \backslash nabla^2\; -\; \backslash frac\; \backslash frac$$is the d'Alembert operator,$$J^\; =\; \backslash left(c\; \backslash rho,\; \backslash mathbf\; \backslash right)$$is the four-current,$$\backslash frac\; \backslash \; \backslash stackrel\backslash \; \backslash partial\_a\; \backslash \; \backslash stackrel\backslash \; \_\; \backslash \; \backslash stackrel\backslash \; (\backslash partial/\backslash partial\; ct,\; \backslash nabla)$$is the 4-gradient, andis the electromagnetic four-potential with the Lorenz gauge condition$$\backslash partial\_\; A^\; =\; 0\backslash ,.$$

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).$$-\; \_\; +\; \_\; A^\; =\; \backslash mu\_0\; J^$$where$$\_$$is the Ricci curvature tensor. Here the semicolon indicates covariant differentiation. To obtain the equation in cgs units, replace the permeability with 4π/c.

The Lorenz gauge condition in curved spacetime is assumed:$$\_\; =\; 0\; \backslash ,.$$

Solutions to the inhomogeneous electromagnetic wave equation

In the case that there are no boundaries surrounding the sources, the solutions (cgs units) of the nonhomogeneous wave equations are$$\backslash varphi\; (\backslash mathbf,\; t)\; =\; \backslash int\; \backslash frac\; \backslash rho\; (\backslash mathbf\text{'},\; t\text{'})\; \backslash ,\; d^3\backslash mathbf\text{'}\; dt\text{'}$$and$$\backslash mathbf\; (\backslash mathbf,\; t)\; =\; \backslash int\; \backslash frac\; \backslash frac\; \backslash ,\; d^3\backslash mathbf\text{'}\; dt\text{'}$$where$$\backslash delta\; \backslash left\; (t\text{'}\; +\; \backslash tfrac\; -\; t\; \backslash right)$$is a Dirac delta function.

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)$$\backslash varphi\; (\backslash mathbf,\; t)\; =\; \backslash int\; \backslash frac\; \backslash rho\; (\backslash mathbf\text{'},\; t\text{'})\; \backslash ,\; d^3\backslash mathbf\text{'}\; dt\text{'}$$and$$\backslash mathbf\; (\backslash mathbf,\; t)\; =\; \backslash int\; \backslash frac\; \backslash ,\; d^3\backslash mathbf\text{'}\; dt\text{'}\; \backslash ,.$$

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
• Covariant formulation of classical electromagnetism
• Maxwell's equations in curved spacetime
• Abraham–Lorentz force
• Green's function

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

• Book: Griffiths, David J.. Introduction to Electrodynamics . 3rd . Prentice Hall. 1998. 0-13-805326-X. registration.
• Book: Tipler, Paul . Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics . 5th . W. H. Freeman . 2004 . 0-7167-0810-8.
• Book: Purcell, Edward M. . Electricity and Magnetism . McGraw-Hill . New York . 1985 .
• Book: Hermann A. . Haus . James R. . Melcher . Electromagnetic Fields and Energy . Prentice-Hall . 1989 . 0-13-249020-X .
• Book: Banesh Hoffman . Relativity and Its Roots . Freeman . New York . 1983 .
• Book: David H. Staelin . Ann W. Morgenthaler . Jin Au Kong . Electromagnetic Waves . Prentice-Hall . 1994 . 0-13-225871-4 .
• Book: Stevens, Charles F. . The Six Core Theories of Modern Physics . MIT Press . 1995 . 0-262-69188-4. .

Graduate-level textbooks

• Robert Wald, Advanced Classical Electromagnetism, (2022).
• Book: Jackson, John D.. Classical Electrodynamics . 3rd . Wiley . 1998 . 0-471-30932-X.
• Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987).
• Book: Maxwell, James C. . A Treatise on Electricity and Magnetism . Dover . 1954 . 0-486-60637-6.
• Book: Charles W. . Misner . Kip Thorne . Kip S. . Thorne . John Archibald Wheeler . Gravitation . 1970 . W.H. Freeman . New York . 0-7167-0344-0. . (Provides a treatment of Maxwell's equations in terms of differential forms.)

Vector Calculus & Further Topics

• Book: Schey, Harry Moritz . Div, Grad, Curl, and all that: An informal text on vector calculus . 2005 . Norton . 978-0-393-92516-6 . 4th .
• Arfken et al., Mathematical Methods for Physicists, 6th edition (2005). Chapters 1 & 2 cover vector calculus and tensor calculus respectively.
• David Tong, Lectures on Vector Calculus. Freely available lecture notes that can be found here: http://www.damtp.cam.ac.uk/user/tong/vc.html