Maxwells Equations and Constitutive Relations

In this section, we formulate the Maxwell equations that govern the behavior of the electromagnetic fields. We present the fundamental laws of electromagnetism, derive the boundary conditions and describe the properties of isotropic, anisotropic and chiral media by constitutive relations. Our presentation follows the treatment of Kong [122] and Mishchenko et al. [169], Other excellent textbooks on classical electrodynamics and optics have been given by Stratton [215], Tsang et al. [228], Jackson [110], van de Hulst [105], Kerker [115], Bohren and Huffman [17], and Born and Wolf [19]. 

The behavior of the macroscopic field at interior points in material media is governed by Maxwell s equations  

In our analysis we will assume that all fields and sources are time harmonic. With u) being the angular frequency and j = /, we write 

Taking into account the continuity equation in the frequency domain V J —j ujp = 0, we may express the Maxwell-Ampere law and the Gauss electric field law as 

Across the interface separating two different media the fields may be discontinuous and a boundary condition is associated with each of Maxwell s equations. To derive the boundary conditions, we consider a regular domain D enclosed by a surface S with outward normal unit vector n, and use the curl theorem 

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