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I'm curious about the refractive index changing in the atmosphere. As I understand it (and I may very well be wrong/naive) when a ray of light enters from one medium into another, the electric field component of the light perpendicular to the interface between the mediums will be altered as such:

$$E^{\;Medium 2}_{\perp} = \frac{\epsilon_{Medium 1}}{\epsilon_{Medium 2}} E^{\;Medium 1}_{\perp}$$

However wouldn't the dielectric constant of air in the atmosphere be constant, meaning that the light entering it would undergo the same amount of refraction? I know this is not the case and I'm wondering if anyone could explain how the dielectric constant changes (if it does) and how this relates to the refractive index changing, or could at least point me to good resources on the matter.

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The refractive index of air depends slightly on temperature, barometric pressure, and humidity.

A simplified formula given by NIST is

$$n = 1 + 7.86\cdot10^{-4}p/(273+t) - 1.5\cdot10^{-11}{\rm RH}(t^2+160)$$

where $p$ is pressure in kPa, $t$ is temperature in Celsius, and RH is relative humidity as a percentage. This formula is valid specifically for 633 nm light.

NIST also provides a calculator to get the refractive index from wavelength, temperature, pressure, humidity, and CO2 content.

The main reason for the refractive index to change is that refraction occurs when an EM wave interacts with molecules in the medium. So if there are more molecules in a given volume of air, the refractive index will increase. And the molecular density of the air depends on the temperature and pressure of the air.

Humidity changes the refractive index because water molecules interact more weakly with the EM wave than the nitrogen and oxygen that make up the bulk of the air, so increasing partial pressure of water vapor will lead to a reduction in the refractive index. A small effect, not included in the simplified formula, can also be measured for variations in the partial pressure of CO2.

how [changing dielectric constant] relates to the refractive index changing,

$$n=\sqrt{\varepsilon_R}$$

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  • $\begingroup$ Does the relative permittivity change with molecular density? If it does, is there a formula for that? $\endgroup$
    – Aidan
    Commented Aug 15, 2019 at 22:35
  • $\begingroup$ @Aidan, $\varepsilon_R = n^2$. $\endgroup$
    – The Photon
    Commented Aug 15, 2019 at 22:44
  • $\begingroup$ I’ve a followup on this and maybe I can’t see the forest for the trees, but why does the permittivity $\epsilon_r$ DECREASE with increasing humidity? Given that low frequency $\epsilon_{r,H2O}\approx 80$ I would assume that $\epsilon_r$ of humid air would increase from $\epsilon_r \approx 1.00059$ towards larger values as a logical consequence of an interpolation between dry air and water… $\endgroup$
    – Nikolaij
    Commented May 5 at 2:11
  • $\begingroup$ @Nikolaij, the formula I quoted is from Appendix B of the linked NIST site. It shows a negative trend for n vs RH. But this is a fitting formula that will work best near some specific point where the fit was made. You could play with the calculator to see if the trend holds for other conditions. $\endgroup$
    – The Photon
    Commented May 5 at 2:35
  • $\begingroup$ Also where did you get that figure $\epsilon_R=80$? Even in liquid state $n_{H_2O}\approx 1.33$, IIRC, and that is at a much higher density than in gas state. $\endgroup$
    – The Photon
    Commented May 5 at 2:38

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