There is a related question (Why glass is transparent?) but I am coming at it only from Maxwell's equations. One can determine the skin depth $δ$ for poor conductors like (pure) water and glass using (see Wikipedia)
$$δ =2ρ \sqrt{\frac{ϵ}{μ_0}}$$
If I ignore the frequency dependence of the permittivity (only to get a board range for the skin depth of glass), using appropriate values for the resistivity ρ (water = $2.5×10^5$ Ω∙m and glass = $10^{10}−10^{14}$ Ω∙m), electric permittivity ($ϵ=ϵ_0ϵ_r$) and magnetic permeability ($μ ≈ μ_0$), I calculate that
$$δ(water) =10^4m$$ $$δ(glass) =10^8-10^{12} m$$
Maxwell’s equations determine the behavior of electromagnetic waves in conductors (as well as poor conductors), so if glass and water have such larger skin depths, then this is the reason why light is transparent for these two mediums – right? If so, I then have two related questions:
-
Mathematically, it’s fairly straight forward to show that the skin depth is independent of frequency. However, is there a physical explanation why the skin depth is independent of frequency for poor conductors but not for good conductors?
-
At least at optical frequencies, the skin depth is mainly dependent on the resistivity of the material. Since glass has a higher resistivity (is a poorer conductor) than water, electromagnetic waves penetrate farther through glass. So the key to understanding why glass is more transparent than water is physically understanding why δ ∝ ρ?
I have looked through the books of Griffiths and Jackson for help on this, and found nothing. Thank you in advance for any help on these questions.
Correction and edit due to Johannes’s comment below for question 2
Best Answer
I disagree with the premise of this question. Using DC permittivity and DC resistivity is an awful starting point if you want to understand anything about visible-light response. [Update: I should say that it's not that bad a starting point for metals specifically. Much worse for other materials.] When electrons move back and forth at 60 Hz, they usually move in a totally different way than when they move back and forth at 1 quadrillion Hz.
For example, in an n-type semiconductor, at 60 Hz, the conductance comes from electrons in the conduction band getting shifted within the band and traveling and sometimes bumping into defects. The conductance at 1 quadrillion Hz comes from electrons in the valence band being pulled into a quantum superposition state between valence and conduction band states. The superposition state happens to jiggle back and forth (by atomic-scale distances) at 1 quadrillion Hz, because of the energy difference between the two states and the laws of quantum mechanics. Soon the superposition is disturbed and you get an electron-hole pair.
For example, rubber has a very high resistivity but is not transparent. Indium-tin-oxide has a low resistivity but is transparent.
To understand visible absorption, you need to be thinking about energy levels and modes, not DC resistivity.
Water absorbs visible light because of various weak (harmonic) vibrational modes. Normally, vibration modes are only in the infrared, but water has unusually high-frequency vibration modes that reach just a bit into the visible. (Because hydrogen is light and bonds very tightly to oxygen. Just like a taut thin string on a guitar will vibrate at a higher frequency than a loose thick string.) Glass does not have that property.
Glass can be much more transparent than water: For example, fiber optics are glass strands through which light can travel many kilometers with negligible absorption. Fiber optics are manufactured very carefully to reduce absorption; if you made ordinary window glass that was 1km thick, it would certainly be opaque.