Survival guide to different Maxwell equations notations
Feynman's notation:
(1)
Pozar's notation:
(Time-domain) (Frequency-domain, Harmonic notation)
(2)
Collin's notation: pretty much the same plus the equations for the vector and scalar potentials (in free space):
(3)
where k0 is the wavenumber in empty space. In a general medium, loss mechanisms as dielectric damping and ohmic dissipation through conduction have to be considered. Magnetic polarization also results in loss through damping forces (while there is no magnetic conductivity). Complex permittivity, and permeability are already assumed in (2). However to explicitly outline contributions from damping and finite conductivity the following notation is convenient:
(4)
where χm , χe are the complex magnetic and electric susceptibility and the last term (tan δ) represents the real to imaginary part of the displacement current. In a medium with finite conductivity a conduction current will exist resulting in energy loss through Joule heating. This non-zero conductivity affects the imaginary part of permittivity in (4) such that the contribution due to Joule dissipation is practically indistinguishable from dielectric loss due to damping effects. Considering the combined damping and finite conductivity effects the wave equations for lossy media in absence of sources become:
(5)
If loss is removed k is real, the propagation constant γ=jk is imaginary, its real part (the attenuation constant) α=0, and its imaginary part (the phase constant) ß=k. If only the dielectric damping is present (σ=0), still ε=ε'-jε'' and γ=jk=jω√µε(1-jtanδ) with tanδ=ε''/ε' as derived from (4). In the case of a good (but not perfect) conductor σ>>ωε or, equivalently ε''>>ε' then:
(6)
Barzilai's notation:
(7)
notice the different general medium formalism leading to different signs in the wave equations.
Referring to Pozar's formalism, if n points (upward) along the axis to a planar discontinuity interface, n0 points toward the reader, t points left along the integration loop,
then the boundary conditions are:
(8)
where the vector equality: A∙BxC = C∙AxB=B∙CxA is also used.