Inhomogeneous Electromagnetic Wave Equation - Solutions To The Inhomogeneous Electromagnetic Wave Equation

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

and

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)

and

.

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

Read more about this topic:  Inhomogeneous Electromagnetic Wave Equation

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