Scattering of Light​ Definition, Reasons and Formula

Scattering of light is a phenomenon due to which light stops travelling in a straight path. Instead, it takes up a zig-zag trajectory. There are different types of scattering of light that occur because of the wavelength and particle size. The most commonly used formula for the scattering intensity of light is the Rayleigh Scattering, which is as mentioned below. Conceptual questions based on this formula are often asked in exams like JEE Main and IIT JAM:

Scattering intensity:  $I\propto \frac{1}{{\lambda }^4}$

Cross‑section:  $\sigma =\frac{8{\pi }^3}{3}\times \frac{{(n^2-1)}^2}{{(n^2+2)}^2}\times \frac{V^2}{{\lambda }^4}$

Where:

• $\lambda$ = wavelength of light
• $n$ = refractive index of the particle
• $V$ = volume of the scattering particle

There are many methods that help in calculating the scattering intensity of light. We will be discussing the different formulae used:

1. Rayleigh Scattering

This formula is used for particles that are much smaller as compared to the light wavelength. 

Scattering intensity:  $I\propto \frac{1}{{\lambda }^4}$

Cross‑section:  $\sigma =\frac{8{\pi }^3}{3}\times \frac{{(n^2-1)}^2}{{(n^2+2)}^2}\times \frac{V^2}{{\lambda }^4}$

Where:

• I = intensity of scattering of light
• $\lambda$ = wavelength of light
• $n$ = refractive index of the particle
• $V$ = volume of the scattering particle

2. Mie Scattering

It is used for particles that have a size comparable to the wavelength of light or larger than the wavelength of light:

Scattering cross‑section:  $\sigma =\left(\frac{2\pi }{k^2}\right)\times \underset{}{\sum }(2l+1)({\mid a{\mathrm{}}_{}l\mid }^2+{\mid b{\mathrm{}}_{}l\mid }^2)$

Where:

• $k$ = 2π/λ (wave number)
• $a{\mathrm{}}_l$, $b{\mathrm{}}_{}l$ = Mie scattering coefficients (complex functions of particle size and refractive index)
• $l$ = mode number

3. Thomson Scattering

For free electrons:

Cross‑section:  ${\sigma }_t=\left(\frac{8\pi }{3}\right)\times {\left(\frac{e^2}{4\pi {\epsilon }_{0}m{c}^2}\right)}^2=6.65\times {10}^{-29}{m}^2$

Where:

• $e$ = electron charge
• $m$ = electron mass
• $c$ = speed of light
• ${\epsilon }_0$ = permittivity of free space

Let us take a look at the step-by-step derivation of scattering of light formula:

1. An oscillating dipole forced by the incident wave

Incident field (plane wave, frequency  $\omega$):

$$E{\mathrm{inc}}_{\mathrm{}}(t)=E0 \stackrel{\to }{e}{e}^{-i\omega t}$$

Induced dipole moment

$p(t)=\alpha E{\mathrm{inc}}_{\mathrm{}}(t)$

For a homogeneous sphere (radius  $a$, index ratio $m$):

$$\alpha =4\pi {ϵ}_0{a}^3\frac{m^2-1}{m^2+2}$$2. Radiation from a Hertzian dipole

$$E{\mathrm{sc}}_{\mathrm{}}(r,\theta )=\frac{k^2}{4\pi {ϵ}_0}\frac{\left(\left|,p,\right|\right)\sin \theta }{r}{e}^{ikr},k=\frac{\mathrm{2\pi }}{\lambda }$$

3. Scattered intensity at angle $\theta$

$$I(\theta )=\frac{c}{2}{ϵ}_0\left|E{\mathrm{sc}}_{\mathrm{}}\right|{}^2=\frac{k^4\left|p\right|{}^2}{32{\pi }^2{ϵ}^{02}}\frac{1+{\cos }^{\theta }}{r^2}$$

Substituting  $p=\alpha E0$:

$I(\theta )=I0\frac{k^4}{16{\pi }^2{ϵ}^{02}}\left|\alpha \right|{}^{2}1+{\cos }^{\theta }\frac{1}{r^2}$

The factor with ${\lambda }^{-4}$ gives I ∝ 1/λ4 dependence.

4. Total scattering cross‑section

$${\sigma }_{\text{Rayleigh}}=\frac{k^4}{\left|}\frac{1}{6\pi {ϵ}^{02}}=\frac{\mathrm{8\pi }}{3}{k}^4\left|\alpha \right|{}^2$$

Substituting $\alpha$:

${\sigma }_{\text{Rayleigh}}=\frac{\mathrm{8\pi }}{3}{\frac{\mathrm{2\pi }}{\lambda }}^4{a}^6{\left(\frac{m^2-1}{m^2+2}\right)}^2$

5. Rayleigh law in one line

$$I_{\text{sc}}\propto \frac{1}{{\lambda }^4}$$

This explains why the sky is blue and the sunset may be red.

Exams like NEET and CUET may ask questions around the factors that affect the phenomenon. The following factors are responsible for impacting the scattering of light:

• Wavelength of Light: Shorter wavelengths are scattered more than longer wavelengths. This is why blue light, which has a shorter wavelength, is scattered more than red light, which has a longer wavelength.
• Particle Size: The size of particles compared to wavelength of light impacts the type and amount of scattering. If the size of the particles is very small, Rayleigh scattering occurs, whereas Mie scattering occurs with particles as compared to the wavelength.
• Concentration of Particles: Higher concentration of particles in a medium leads to an increase in scattering since there are more particles available to interact with the light.
• Angle of Incidence: Another factor that impacts scattering of light is the angle at which the light hits a particle. This affects both direction and intensity of scattered light.
• Polarization of Light: The polarization state of an incident light can influence scattering pattern of light. Even if the incident light is unpolarized, it is possible that the scattered light becomes polarized.
• Refractive Index: The difference in refractive index between particles and surrounding medium impacts the intensity of the scattering light. If the difference is large, scattering will also be large

The chapter 'Optics' holds a weightage of 7 marks in total. It comprises 2 questions, one objective type, and one short question of 2 and 5 marks simultaneously.

The Tyndall effect is a part of the scattering of light. It occurs when colloidal particles come in contact with the scattering of light. It is also used to determine the colloidal solution of the particle.

Some of the examples of the Tyndall effect are listed below:

1. In a dark room, we often see a beam of light coming through from the window, and the path of the light could be seen distinctly. The dust particles in the room also could be seen with the help of scattering of light.
2. Similarly, when a projector projects the light in the cinema hall or the wall, the pictures become visible to everyone.

This is a common question often asked in the CBSE board exam. We all must have heard the reason behind the sky’s colour in a word, i.e. Refraction. But what does it exactly mean?

The sky can be blue, but it could be seen as red in some places. How does this colour change? 

When the light hits the gas particles present in the air, the light scatters at different wavelengths. Wavelengths scatter the light in different amounts in different directions. It all depends on the wavelength factor of the light.  It can be written as:

 = 1/ λ4

The same thing happens when we see the sky as red. The wavelengths shift and create a different angle for the scattering of the light causing refraction. Low and high wavelengths change the sky’s colour from red to blue and blue to dark or light blue.