Refraction of light and its dispersion through a prism

1 Speed of light in a medium

In a medium (a substance like air, water, glass and others) the speed of light is smaller than "c":

v_m < c

The speed of light in a substance v_m depends on the density of the substance the light passes through: the higher the density, the smaller the speed.
For instance in air, light travels slower than in a vacuum, but faster than in glass. The speed reduction in a medium is due to the fact that the wavefronts distance between waves, the wavelength, reduces :

l_m < l_vacuum     hence     v_m = f ·l_m < c = f ·l_vacuum

Since the maximum speed of light is constant and equal to "c" , it's useful to relate the speed of light in a medium to "c" by making the ratio with it. This ratio is called index of refraction n :

n = c
v_m

The difference in the speed of light in different substances is usually described by this index. The speed of light v_m is related to the speed of light "c" in a vacuum by:

v_m = c
n
.

Hence, to find the speed of light in some transparent material, we divide the speed of light in a vacuum c=3·10⁸ m/s by the index of refraction of that material. Typical values for the index of refraction are shown in the table below:

EXERCISE: By referring to the previous table, compute the speed of light in air, in water, in glass and in diamond.
As you can observe by solving the exercise, the higher the index of refraction of a substance, the smaller the speed of light in that substance.

2  Law of refraction
The change in speed and wavelength occurring when light passes from a transparent medium to another, cause a change in the direction of light too. The change in rays direction depends on the angle of incidence of rays and on the value of the index of refractions of the mediums (a larger difference in indices of the two substances produces a larger bend in wavefronts and waves).
A very meaningful java applet clarifies the idea at: http://ww2.unime.it/dipart/i_fismed/wbt/ita/RefractionOfLight/lightrefract_ita.htm

The law of refraction [we often refer to it as the Snell law ] explains how a ray of light changes its direction of travel when passing from a substance to another.

In mathematical terms the law can be summarized by a formula:

n₁·sinq_i = n₂·sinq_r

where:
n₁ is the index of refraction of the substance from whom the rays of light approach;
n₂ is the index of refraction of the substance in which the rays of light go through.;
q_i is the angle of incidence of rays;
q_r is the angle of refraction of rays;

but the Snell law can also be described in qualitative terms by stating that:

when light passes from one substance to another:

1) the rays are bent towards the surface normal if the speed of light in the second substance is smaller than in the first (n₁ < n₂); [such a situation is shown in the previous figure]

2) the rays are bent away form the surface normal if the speed of light in the second substance is greater than in the first; (n₁ > n₂) [such a situation is shown in the figure below]

Sample exercise on the law of refraction

3  Dispersion of the light passing through a prism
The refraction law underlies the phenomenon of the dispersion of light, that is the decomposition of the white light into the component colors when passing through a prism or through a transparent object delimited by non parallel surfaces [see the picture 16.14 on page 321 in the text-book].
A beam of light containing all the visible spectrum of the light is white, because the sum of all the colors generates the white color. Normally the light we use is white. It's light containing all the colors mixed together. We can realize this fact when the beam of light passes through a glass prism: the light is decomposed in all the component colors. The phenomenon can be explained by thinking that light of different color (different wavelength) has different velocity while travelling in a medium:

v_m = f ·l_m

Hence, the change in velocity of light observed when the light passes from the air to the glass, depends on the wavelength.

By passing the interface air-glass, lower is the wavelength lower becomes the velocity of the light, so , for ex., red light rays are faster then violet light rays.
This change in velocity coupled with the direction of the light beam to the air-glass interface explains the decomposition of a white light ray in the component colours while it is passing through a prism.
This phenomenon is known as "dispersion of the light through a prism" and it's also responsible for rainbows during storms: as a matter of fact, each raindrop can be regarded as a little prism; when a light ray strikes a raindrop it is refracted and decomposed, spreading out all the visible colors ranging from red to violet.
A good java applet showing the dispersion of light through a raindrop is available at: http://users.erols.com/renau/rainbow.html

File translated from T_EX by T_TH, version 3.22.
On 23 Mar 2003, 12:24.