Inhomogeneous electromagnetic wave equation

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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, due to nonzero source charges and currents. The source terms in the wave equations makes the partial differential equations inhomogeneous. Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. The equations follow from Maxwell's equations.

Maxwell's equations[edit]

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 \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0} \nabla \cdot \mathbf{E} = 4\pi\rho
Gauss's law for magnetism \nabla \cdot \mathbf{B} = 0 \nabla \cdot \mathbf{B} = 0
Maxwell–Faraday equation (Faraday's law of induction) \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}
Ampère's circuital law (with Maxwell's addition) \nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) \nabla \times \mathbf{B} = \frac{1}{c} \left(4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t} \right)

where ε0 is the vacuum permittivity and μ0 is the vacuum permeability. Throughout, we also use the relation between ε0 and μ0 and the speed of light c, namely:

\varepsilon_0\mu_0 = \dfrac{1}{c^2}\,.

SI units[edit]

E and B fields[edit]

Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B.[1] Substituting Gauss' law for electricity into the curl of Faraday's law of induction, and using the curl of the curl identity ∇×(∇×X) = ∇(∇·X) − ∇2X gives the wave equation for the electric field E:

\dfrac{1}{c^{2}}\dfrac{\partial^{2}\mathbf{ E}}{\partial t^{2}}-\nabla^{2}\mathbf{ E} = -\left(\dfrac{1}{\varepsilon_{0}}\nabla\rho+\mu_{0}\dfrac{\partial\mathbf{ J}}{\partial t}\right)\,.

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:

\dfrac{1}{c^{2}}\dfrac{\partial^2 \mathbf{B}}{\partial t^2}-\nabla^{2}\mathbf{ B} = \mu_0\nabla\times\mathbf{J}\,.

The left hand sides of each equation are the wave terms, while the right hand sides are the wave sources. The equations imply that spatially and time-varying current densities J and gradients in charge density ρ will generate EM waves.

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[edit]

Introducing the electric potential φ (a scalar potential) and the magnetic potential A (a vector potential) defined from the E and B fields by:

  \mathbf{E} = - \nabla \varphi  - {\partial \mathbf{A} \over \partial t} \,,\quad \mathbf{B} = \nabla \times  \mathbf{A} \,,

the four Maxwell's equations in a vacuum with charge ρ and current J sources reduce to two equations, Gauss' law for electricity is:

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

and the Ampère-Maxwell law is:

  
   \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} \,.

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:

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

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

  
   \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} \,.

These equations can be further simplified using four vectors, as shown next.

CGS and Lorentz–Heaviside units[edit]

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[edit]

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} \quad \text{SI}
\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}\quad \text{cgs}

where

\Box = \partial_{\beta} \partial^{\beta} = \nabla^2 - {1 \over c^2} \frac{   \partial^2} { \partial t^2}

is the d'Alembert operator,

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

is the four-current,

  { \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

A^{\mu}=(\varphi, \mathbf{A} c)\quad \text{SI}
 A^{\mu}=(\varphi, \mathbf{A} ) \quad \text{cgs}

is the electromagnetic four-potential with the Lorenz gauge condition

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

Curved spacetime[edit]

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.

The Lorenz gauge condition in curved spacetime is assumed:

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

Solutions to the inhomogeneous electromagnetic wave equation[edit]

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    }   }\,,\quad\mathbf{J} \rightarrow {  \mu_0  \over {4 \pi     }   } \mathbf{J}
\,.

For Lorentz–Heaviside units,

 
\rho \rightarrow { \rho \over {4 \pi    }   }\,,\quad\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[edit]

References[edit]

  1. ^ Classical electrodynamics, Jackson, 3rd edition, p. 246

Electromagnetics[edit]

Journal articles[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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.