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Concept
Inhomogeneous electromagnetic wave equation
Summary
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.
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:
{| class="wikitable" style="text-align: center;"
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! scope="col" style="width: 15em;" | Name
! scope="col" | SI units
! scope="col" | Gaussian units
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! scope="row" | Gauss's law
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! scope="row" | Gauss's law for magnetism
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! scope="row" | Maxwell–Faraday equation (Faraday's law of induction)
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! scope="row" | Ampère's circuital law (with Maxwell's addition)
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|}
where ε0 is the vacuum permittivity and μ0 is the vacuum permeability. Throughout, the relation
is also used.
Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. Substituting Gauss' law for electricity and Ampère's Law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇2X (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:
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:
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.
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