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Speed of electromagnetic wave in a medium (for example air) is different for different frequency (for exmaple red light is faster in air than blue light).

So how can we calculate the speed of light for different frequency?
Are there two different frequency for which speed of light is same?

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So how can we calculate the speed of light for different frequency?

It depends on how much prior information you want to require of your calculation. In the simplest case you may just look up an approximate formula for the speed of light in your medium of choice as a function of wavelength or (less common) frequency. Of course, this relies on somebody having measured this function. Or you could find tabulated values, which you calculate an interpolation for at arbitrary wavelengths/frequencies.

By calculating you could possibly also mean that you are not so much interested in specific, accurate values, but you rather want to roughly understand how dispersion (that is how the wavelength dependence of the speed of light is called) relates to microscopic material properties. For this you might be interested in Lorentz' oscillator model of dispersion/dielectrics. In short, this model assumes that electrons in a medium behave as if they were elastically bound to their equilibrium positions, and any displacement requires a force, which comes from the light (aka. electromagnetic wave). Just as a swing you are pushing with a certain force has a kind of delay before you see a significant displacement, the electrons in the medium show a delayed displacement (called dielectric polarization) upon the pushes of the incident light (electric field strength). The electrons are considered to be in a harmonic oscillator, and the delay depends on the (spectral) distance between the incident light frequency and the natural frequency of the harmonic oscillator.

However, this is an entirely classical picture, and although it can explain the general shape of dispersion graphs very well, it is not possible to calculate anything unless you know in advance the natural frequencies of all possible oscillators of the medium. And the latter cannot be calculated classically (only semi-classically sometimes, see Bohr's model of the atom).

If you really wanted to calculate dispersion, you would have to solve quantum mechanical problems, because these actually represent the harmonic oscillators implicitly used in Lorentz' model. However, quantum mechanical problems can hardly ever be solved analytically, but only by using numerical methods (the hydrogen atom being the most important exception). Moreover, the oscillators are usually not single atoms, but rather molecules, crystals or even amorphous matter, liquids and so on, which complicates things even more. Specifically for the solid state, the crystal is described by band theory, where the oscillators are (continuously distributed) transitions between energy bands.

Hence, you will often resort to some tabulated values or formulae, if you really want to do real world calculations and the effort of doing quantum mechanics is not justified.

Are there two different frequency for which speed of light is same?

Of course there are usually many frequencies which have the same speed of light, because the speed of light kind of tries to return to "normal" for any frequency that is far from the natural frequencies of an oscillator. In other words, the dispersion relations go up and down, up and down, and if you draw a horizontal line (representing a specific speed of light/index of refraction), you will often find many intersections with the respective graph of speed of light/index of refraction/dielectric coefficient.

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This question can be reformulated as:

“Does the index of refraction of a given medium depend on wavelength” (inverse of frequency), the answer is given by Cauchy’s formula:

https://en.wikipedia.org/wiki/Refractive_index

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  • $\begingroup$ What is the physical reason for Cauchy's formula $n(\lambda)$ being an even function of $\lambda$? $\endgroup$ – hyportnex Apr 14 at 18:37
  • $\begingroup$ Absorption by atoms of the medium as a function of wavelength: en.m.wikipedia.org/wiki/Sellmeier_equation (Cauchy’s formula is just an approximation). $\endgroup$ – Serge Hulne Apr 14 at 18:44
  • $\begingroup$ OK, so why Sellmeier's formula is an even function of $\lambda$? $\endgroup$ – hyportnex Apr 14 at 18:52
  • $\begingroup$ Here are some explanations as to the underlying physics : quora.com/… $\endgroup$ – Serge Hulne Apr 14 at 19:32

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