Refractive index of red and blue light

Question

if $\frac{\sin(I)}{n}=\sin(r)$ where $n$ is the refractive index, $I$ is angle of incidence and $r$ is angle of refraction (these are relative to the normal).

How come blue light refracts at a higher angle than red light if red travels faster in glass than blue, surely $n$ for blue is larger than $n$ for red making $\sin(r)$ for red larger than that for blue, and the larger $\sin(r)$, the greater the angle (to a point)?

Answer 1

You have it backward, faster speed of light in a material corresponds to less refraction, not more. In the limit that the speed of light in a material was the same as the speed of light in vacuum, there would be no refraction at all.

It can be shown that in a material the index of refraction is the speed of light in vacuum, $c$ divided by the speed of light in the material $c_m$.

$ n ={ c \over c_m} $

So, slower speed in a material corresponds to a larger index of refraction ,and higher speed to a lower index of refraction. The index of refraction is always greater than or equal to 1, because $c$, the speed of light in vacuum, is always greater than the speed in a material.

So, as you have stated, red light has a lower index of refraction than blue light since it also has a shorter wavelength, so lower index of refraction corresponds to higher speed in a material.

Now Snell's Law is stated

$${n_1 \over n_2} = {\sin \theta_2\over \sin \theta_1}$$

where the geometry is as shown:

So if $n_1=1.0$ and $\theta_1 = 20$ degrees we have

$n_2 =1.51$ for red light and $\theta_2 = 13.09$ degrees,

$n_2 = 1.53$ for blue light, and $\theta_2 = 12.91$ degrees.

As you can see from the diagram, a smaller value of $\theta_2$ corresponds to a larger amount of refraction because it is a larger change in angle and thus a larger change in direction of the incident light.

Answer 2

If the question is about why red and blue light refract at different angles, this is because the refractive index, $n$ for many materials depend upon the frequency of light and they are referred to as dispersive. More often, the variation is given in terms of the wavelength ($\lambda$) and in the optics industry people quote the refractive index for the Helium d line ($\lambda = 587.6 nm$), and for the Mercury e line ($\lambda = 546.1 nm$), and a few other properties at other wavelengths, see Kaye and Laby.

The reason why the refractive index depends on frequency is related to the atomic structure of matter.