In electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus. These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations. A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium. For an inhomogeneous medium it necessarily requires 8 × 8 matrices.
Introduction Maxwell's equations in the standard vector calculus formalism, in an inhomogeneous medium with sources, are:
∇ ⋅ D ( r , t ) = ρ ∇ × H ( r , t ) − ∂ ∂ t D ( r , t ) = J ∇ × E ( r , t ) + ∂ ∂ t B ( r , t ) = 0 ∇ ⋅ B ( r , t ) = 0 . {\displaystyle {\begin{aligned}&{\mathbf {\nabla } }\cdot {\mathbf {D} }\left({\mathbf {r} },t\right)=\rho \,\\&{\mathbf {\nabla } }\times {\mathbf {H} }\left({\mathbf {r} },t\right)-{\frac {\partial }{\partial t}}{\mathbf {D} }\left({\mathbf {r} },t\right)={\mathbf {J} }\,\\&{\mathbf {\nabla } }\times {\mathbf {E} }\left({\mathbf {r} },t\right)+{\frac {\partial }{\partial t}}{\mathbf {B} }\left({\mathbf {r} },t\right)=0\,\\&{\mathbf {\nabla } }\cdot {\mathbf {B} }\left({\mathbf {r} },t\right)=0\,.\end{aligned}}}
The media is assumed to be linear, that is
D = ε E , B = μ H {\displaystyle {\mathbf {D} }=\varepsilon \mathbf {E} \,,\quad \mathbf {B} =\mu \mathbf {H} } , where scalar ε = ε ( r , t ) {\displaystyle \varepsilon =\varepsilon (\mathbf {r} ,t)} is the permittivity of the medium and scalar μ = μ ( r , t ) {\displaystyle \mu =\mu (\mathbf {r} ,t)} the permeability of the medium (see constitutive equation). For a homogeneous medium ε {\displaystyle \varepsilon } and μ {\displaystyle \mu } are constants. The speed of light in the medium is given by
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