# Physical interpretation of conductivity with electromagnetic waves?

## Main Question or Discussion Point

Hi all,

I'm trying to understand exactly what the physical meaning of conductivity/current is in relation to waves.

if we have a wave traveling through a conductor, we find that it decays exponentially, i.e.

$e^{-\alpha z}$
where $\alpha=imag(k)=\omega\sqrt{\frac{\epsilon\mu}{2}} \left[\sqrt{1+(\frac{\sigma}{\epsilon\omega})^2}+1 \right]^{1/2}$
and $k^2=\mu\epsilon\omega^2+i\mu\sigma\omega$

My question is, what is the physical interpretation of the conductivity(σ) with respect to currents? How does it cause an exponential decay of the field strength as the wave travels through the material? Does it absorb the electric field by creating a current since $J=\sigma E$ ?

Thanks!

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vanhees71
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In my opinion, the most clear treatment of these issues is given in the Feynman Lectures vol. II. There Feynman explains in very simple terms the classical theory of electromagnetic waves in media.

A insulating homogeneous and isotropic dielectric's properties can be described in a simplified model as a rigid lattice of ions (atomic nuclei) surrounded by electrons harmonically bound to this positive background, including a friction term. Fourier decomposition in time leads to the complex valued dielectric function, $\epsilon(\omega)$, with the usual analyticity constraints for causality (use of the retarded propagator for the response to an external perturbation like an incoming electromagnetic wave).

If you have a conductor, you have in addition (quasi-)free electrons which have no harmonic binding force but only a friction force, which leads to a pole of the dielectric function at $\omega=0$. That's the only formal difference between an insulator and a conductor.

Superconductivity must be treated as a special case, leading to the London or the Ginzburg-Landau phenomenological theories (the latter is particularly interesting since it can be understood as the Abelian Higgs Mechanism applied to the classical electromagnetic field; see Weinberg, Quantum Theory of Fields, Vol. II).

When an electromagnetic wave travels through a conducting medium, the electric field of the wave exerts a force on the free electrons. This force causes them to accelerate and thereby gain some velocity. Moving charge is known as current, so the electric field of the em wave has therefore created electric currents in the material. The electrons are actually not perfectly free and isolated, and therefore cannot be perfectly accelerated by the em wave. The bond of the electron to the solid as well as the bumping into other particles creates a net drag on the electron as it tries to accelerate. The average amount of drag on electron's when being accelerated in a certain material is known as the "electrical resistivity" ρ. The electrical conductivity σ of a material is just the inverse of its resistivity.

The creation of currents in non-perfect conductors has two effects. First, when an electron being accelerated by the em wave bumps into an atom, it gets knocked out of the oscillation, and looses some of its kinetic energy to the atom. Therefore, some of the energy in the wave gets transferred to the coherent kinetic energy of the electron it accelerates, which then gets transferred to the random kinetic energy of the atom it bumps into. As a result, the wave dies down and the material heats up. This is known as "Joule heating" or "resistive heating". The second effect is that the induced oscillating currents radiate new waves which also carry much of the energy away. As a result, a conductor tends to reflect much of the energy of an incident em wave instead of transmitting it. The wave inside the conductor spatially decays because much of its energy is reflected back at the conductor's surface.