What is the dispersion of white light?

There are 7 component colors of white light in the visible spectrum. In dispersion, when a ray of white light is passed through a medium (for example prism) having some index of refraction, the white light is refracted in such a way that it splits into seven colors including violet, red, indigo, green, blue, orange & yellow. In this type of dispersion, most of the things are wavelength-dependent. The red light has maximum wavelength because of which it bends the least & violet light has minimum wavelength because of which it bends the most. The difference in wavelength is also due to the different speeds of the component colors. Dispersion of white light through a prism is a very common example which is also used in our day-to-day life.

How dispersion of white light happens?

Dispersion of white light is a phenomenon that includes processes that are dependent on the wavelength ($\lambda$) of light & it can occur for any wave irrespective of its type. Dispersion occurs in presence of a medium, specifically, when beams of light refract through that particular medium. In dispersion, the angle of refraction varies according to the refractive index ($\mu$), which is in turn dependent on the medium itself. But in the case of a specific medium, $\mu$ is also dependent on $\lambda$ .

Examples of dispersion of white light

Dispersion of white light in presence of a glass prism

Dispersion in the case of white light happens in such a way that when white light in form of incident ray falls on the prism made up of glass or glass prism, it splits into its 7 constituent colors as it refracts and refraction occurs. The refractive index of the prism is 1.5. As the $\lambda$ of those 7 colors is variable but the frequency $\nu$ is constant, they get deviated by the angles that are different which is mainly due to differences in their particular velocities in glass medium. Red color (possesses maximum wavelength) deviates the least & violet (possesses minimum wavelength) deviates the most out of all the colors. However, even if the refractive index of the glass slab & triangular prism (made up of glass) is the same, the glass slab cannot disperse white light because of its geometric design.

Formation of a rainbow

Rainbows are formed when light from the sun enters a water droplet and gets scattered by the process of refraction, resulting in the formation of a spectrum. Dispersion of light on a rainy day happens in such a way that when white light passes through a water droplet which is a medium denser than air, it changes direction. The refracted ray is reflected off the back and then refracted again before it exits and travels to our eyes.

Derivation

Prism formula and angle of minimum deviation

Snell’s law is a formula that gives a relation between the angle of incidence and angle of refraction and is given as:

$\mu =\frac{\sin i}{\sin r}$

Where ‘i’ is the angle of incidence and ‘r’ is the angle of refraction.

Here,

$$\begin{gathered} \delta =i_1-r_1+i_2-r_2 ......\left(1\right) \\ \delta =i_1+i_2-\left(r_1+r_2\right) \\ \angle ALO+\angle AMO=2rt\angle s \left(In quadrilateral, \angle ALO=\angle AMO=90°\right) \\ \angle LAM+\angle LOM=2rt\angle s \left(Sum ofangles of a quadrilateral is 4rt\angle s\right) ..\left(2\right) \\ \angle r_1+\angle r_2+\angle LOM=2rt\angle s ......\left(3\right) \\ LAM=\angle r_1+\angle r_2 \left(On comparing \left(2\right) and \left(3\right)\right) \\ A=\angle r_1+\angle r_2 \\ \delta =i_1+i_2-A \left(On substituting A in \left(1\right)\right) \\ {i}_1+i_2=A+\delta \\ \angle i_1=\angle i_2 \\ \angle r_1=\angle r_2=\angle r \\ \angle ALM=\angle LMA=90°-\angle r \end{gathered}$$

Therefore,

$\begin{array}{rcl}AL& =& LM and assuming that LM\parallel BC\\ \angle A& =& \angle r_1+\angle r_2\\ A& =& 2r\\ r& =& \frac{A}{2}\end{array}$

Now,

$\begin{array}{rcl}{i}_1+i_2& =& A+\delta \\ i_1+i_1& =& A+{\delta }_m\\ 2i_1& =& A+{\delta }_m\\ i_1& =& \frac{A+{\delta }_m}{2}\\ \therefore \mu & =& \frac{\sin \left({\displaystyle \frac{A+{\delta }_m}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}\end{array}$

Where

$$\begin{gathered} \mu =Refractive index of the material of the prism \\ A= Prism angle \\ \delta =Angle of deviation \\ {\delta }_m=Angle of minimum deviation \end{gathered}$$

Formulas

In the case of dispersion of white light, there are the following relevant formulas:

• The angle of minimum deviation :

$i=\frac{A+{\delta }_m}{2}$

• Prism formula:

$\mu =\frac{\sin \left({\displaystyle \frac{A+{\delta }_m}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}$

Where

$$\begin{gathered} \mu =Refractive index of the material of the prism \\ A= Prism angle \\ \delta =Angle of deviation \\ {\delta }_m=Angle of minimum deviation \end{gathered}$$

Context and Applications

The concept of dispersion of white light is very important when it comes to studying the nature of light. It is applicable in the following parts of Physics:

1. Optical Physics
2. Reflection & refraction of light
3. Particle & wave motion

Practice Problems

1. Which of the materials mentioned below will have maximum dispersive power?
   1. Flint glass
   2. Crown glass
   3. Water
   4. Air

Correct Answer: a. Flint glass

Explanation: When light passes and gets dispersed, dispersive power varies directly according to the index of refraction of the material. Hence, flint glass has maximum dispersive power.

2. Which of the following components of white light has maximum wavelength?

1. violet
2. red
3. indigo
4. blue

Correct answer: b. red

Explanation: Red light has maximum wavelength due to least frequency and hence, it deviates the least. Whereas, violet light has a minimum wavelength.

3. Which of the following components of white light has maximum frequency?

1. violet
2. red
3. indigo
4. yellow

Correct answer: a. violet

Explanation: Violet light has minimum wavelength due to maximum frequency and hence, it deviates the most.

4. If a light is passed through a hollow glass prism then which of the following conditions will be true in the case of a hollow glass prism?

1. No refraction
2. No deviation
3. Only reflection
4. Both a. and b.

Correct Answer: d. Both a. and b

Explanation: When light rays pass through a hollow glass prism they won’t get deviated because the index of refraction outside the prism and inside the prism will be the same. Hence, there will be no change.

5. Which of the following prisms is used to observe infrared radiations?

1. Nicol
2. Flint
3. Crown
4. Rock salt

Correct Answer : d. Rock salt

Explanation: The prism that is used to observe infrared radiations is the Rocksalt prism due to its ability to not absorb infrared rays.

Common Mistakes

In the dispersion of light waves, there is no actual explanation of why refraction is dependent on the wavelength of the components of light. Another common pitfall regarding dispersion is that it is always shown as the refraction and dispersion occurs only at the 2^nd surface, though it occurs at both. In CBSE class 12, refraction in lenses is very common and they do have some focal length. But sometimes, determining the focal length is not that easy.

Related Concepts

• Optics
• Electromagnetic spectrum