# Differences in speed of light calculations in glass

I wanted to determine the speed of light in glass by using both of the following formulas : $$v = \frac{1}{\sqrt{\mu \epsilon}} \simeq \frac{1}{\sqrt{\mu_0 \epsilon_r \epsilon_0}} \;\;\; \text{and} \;\;\; v=\frac{c}{n}$$

For the first one, I got that $v = 1.383 \times 10^8$ m/s by taking $\epsilon_r = 4.7$ F/m (I think it's the smallest value that I can take for glass: in my book, they say its value varies from 5 to 10). For the second one, I got $v = 2 \times 10^8$ m/s by taking $n = 1.5$

I think the difference between the two values is not negligeable, so how to explain this difference? Is it only due to the fact that $\epsilon_r$ is a function of the frequency of the light propagating in the glass? Why would it make such a difference in the speed?

First, The relative permittivity is dimensionless - not units of F/m. It is defined as the ratio of the permeability of the material to that of the vacuum.

Second, by selecting a value of 4.7. you are defining the index of refraction to be $\sqrt{4.7}\approx{2.2}$. This is because for a dielectric such as glass, $\mu\approx\mu_0$ so $$n\equiv\sqrt{\frac{\mu\epsilon}{\mu_0\epsilon_0}}\approx\sqrt{\frac{\epsilon}{\epsilon_0}}\equiv\sqrt{\epsilon_r}$$

So your differences in velocity are just from choosing $n=2.2$ in one case and $n=1.5$ in the other.

As for the frequency dependance, both the permittivity and the index of refraction are in general, frequency dependant (leading to an effect known as dispersion) When you see a constant value of say 1.5, it is just an average approximation over some wavelength. It could be that the value you found (4.7) is for UV or IR light and you are comparing it to the value in the somewhere in the visible spectrum.

As it turns out when I looked up values for relative permeability of schott glass, I found much smaller values of around 2.4:

http://refractiveindex.info/?shelf=glass&book=BK7&page=SCHOTT

The value of 4.7 is most likely in the microwave region.

• thanks for clearing that up! Ok so it's frequency dependant. Why's that? Is it something to do with polarization?
– Dory
Sep 27, 2015 at 8:56
• Polarization is involved but is not required for frequency dependance. Classically speaking, the frequency dependance comes from the fact that light is driving around internal charges which in turn create their own electromagnetic (EM) waves. What comes out is the interplay between the two. This leads to the medium having a finite response time and attenuation. The finite response time means that an EM wave switching polarity at a sluggish 6 trillion times per second (microwave frequency) can interact with the medium much differently than one which is doing so 600 trillion times (green light)
– Orko
Sep 27, 2015 at 18:20
• The dependance of the index of refraction on polarization of light is another effect, known as birefringence. This is usually the case in periodic structures such as crystals, but is not really present in glass which is an amorphous solid (i.e. it looks approximately the same in each direction). For birefringent materials, the index of refraction is a function of both polarization AND frequency.
– Orko
Sep 27, 2015 at 18:21