Physical interpretation of conductivity with electromagnetic waves?

In summary: This is the physical meaning of the conductivity with respect to currents- it causes an exponential decay of the electric field strength as the wave travels through the material.
  • #1
randomafk
23
0
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.

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

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 [itex]J=\sigma E[/itex] ?

Thanks!
 
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  • #2
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, [itex]\epsilon(\omega)[/itex], 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 [itex]\omega=0[/itex]. 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).
 
  • #3
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.
 

1. How does conductivity affect the propagation of electromagnetic waves?

Conductivity is a measure of a material's ability to conduct electricity. In the context of electromagnetic waves, conductivity affects how easily the waves can travel through a material. Materials with high conductivity, such as metals, allow electromagnetic waves to pass through easily, while materials with low conductivity, such as insulators, impede the movement of the waves.

2. What is the relationship between conductivity and the speed of electromagnetic waves?

There is an inverse relationship between conductivity and the speed of electromagnetic waves. This means that as conductivity increases, the speed of the waves decreases. This is because highly conductive materials absorb and dissipate more energy from the waves, causing them to slow down.

3. How does the frequency of electromagnetic waves affect their interaction with conductive materials?

The frequency of electromagnetic waves plays a significant role in their interaction with conductive materials. At high frequencies, the waves have a shorter wavelength and are able to penetrate deeper into the material, causing more energy to be absorbed and dissipated. At low frequencies, the waves have a longer wavelength and are more likely to be reflected or transmitted through the material.

4. What is skin depth and how does it relate to conductivity?

Skin depth is a measure of how deeply electromagnetic waves can penetrate into a conductive material. It is directly related to the conductivity of the material, with higher conductivity resulting in a shallower skin depth. This is because highly conductive materials absorb and dissipate more energy, causing the waves to penetrate less deeply.

5. How does the shape and size of a conductive material affect its conductivity and interaction with electromagnetic waves?

The shape and size of a conductive material can greatly affect its conductivity and interaction with electromagnetic waves. Generally, larger and more massive materials have higher conductivity due to a larger number of free electrons available for conduction. Additionally, the shape of a material can influence how easily the waves can travel through it, with materials that are long and thin allowing for better wave propagation compared to materials that are short and wide.

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