Dispersion and Scattering

Dispersion Of White Light By A Glass Prism

White light, such as sunlight, is a mixture of light of different colours (wavelengths). When a beam of white light passes through a transparent medium, it splits into its constituent colours. This phenomenon is called dispersion of light.

Dispersion occurs because the refractive index of a transparent medium is slightly different for different wavelengths of light. This dependence of refractive index on wavelength is called dispersion. When light enters a medium, its speed changes. Since $n = c/v$ and $v = \nu\lambda$, with $\nu$ constant, $n \propto 1/\lambda$. Generally, for most transparent media, the refractive index is higher for shorter wavelengths (like violet) and lower for longer wavelengths (like red). This means violet light slows down more than red light when entering the medium.

Dispersion by a Prism

When a beam of white light is incident on a prism, each colour in the white light is refracted by a different amount because each colour has a different refractive index in the prism material. According to Snell's Law ($n_1 \sin\theta_1 = n_2 \sin\theta_2$), the bending of light depends on the refractive index $n_2$ (of the prism). The angle of deviation ($\delta$) for a prism also depends on the refractive index ($n$) and the angle of the prism ($A$). For small angles and minimum deviation, $n \approx ( (A + \delta)/2 ) / \sin(A/2) $, suggesting $\delta$ depends on $n$.

Since the refractive index is higher for violet light ($n_V > n_R$), violet light is deviated more than red light ($\delta_V > \delta_R$). The other colours in the spectrum (Indigo, Blue, Green, Yellow, Orange - forming the sequence VIBGYOR) are deviated by angles between those of violet and red.

When a beam of white light enters one face of a prism, it separates into its constituent colours, and these colours are spread out into a band called a spectrum as they emerge from the other face. The order of colours in the spectrum is typically VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange, Red), corresponding to increasing wavelength and decreasing deviation.

Diagram illustrating dispersion of white light by a prism, showing the spectrum.

(Image Placeholder: A triangular prism. A beam of white light is shown incident on one face. Inside the prism, the beam is shown splitting and spreading out into different colours. The emergent rays are shown diverging, separated into the visible spectrum (VIBGYOR order), with violet bent most and red bent least.)

Cause of Dispersion

Dispersion is a property of the medium, not the light itself. Different media exhibit different degrees of dispersion. Dispersion occurs because the interaction of light waves with the atoms and molecules of the medium depends on the frequency (or wavelength) of the light. The speed of light in the medium is influenced by these interactions, and this interaction strength varies with frequency.

In a vacuum, the speed of light is the same for all wavelengths, so there is no dispersion in a vacuum.

Recombination of Spectrum

Isaac Newton demonstrated that the spectrum produced by a prism can be recombined to form white light again by passing it through an identical second prism placed in an inverted position relative to the first. This shows that dispersion is a reversible process.

Atmospheric Refraction

The Earth's atmosphere is a gaseous envelope whose density and refractive index vary with altitude. The atmosphere is densest near the Earth's surface and becomes progressively less dense at higher altitudes. The refractive index of air is slightly greater than 1 and also decreases with altitude (as density decreases).

When light from celestial objects (like the Sun, Moon, or stars) enters the Earth's atmosphere, it passes from a region of very low density (near vacuum) to regions of increasing density. Since the refractive index increases as density increases, light bends towards the normal. Because the Earth's atmosphere is arranged in layers of decreasing density upwards, the path of light from a celestial object bends continuously towards the Earth's surface.

$Diagram illustrating atmospheric refraction causing apparent position shift of a star.$

(Image Placeholder: Earth's surface and atmosphere layered with increasing density downwards. A ray of light from a distant star enters the atmosphere at an angle, bending downwards continuously as it passes through denser layers. An observer on Earth's surface receives this curved ray. A dotted line extends straight back from the direction the ray enters the observer's eye. The apparent position of the star (along the dotted line) is higher than the actual position.)

This phenomenon is called atmospheric refraction. It causes the apparent position of celestial objects to be slightly different from their actual position. Since light bends towards the normal, and the normal points towards the centre of the Earth, light from a celestial object is bent downwards towards the Earth. This makes the object appear higher in the sky than its true position.

Effects of Atmospheric Refraction

Apparent Position of Stars: Stars appear slightly higher in the sky than their actual position, especially when they are near the horizon (where the light travels through a longer path in the atmosphere).
Apparent Flattening of the Sun and Moon at Sunrise/Sunset: Near the horizon, the Sun and Moon appear somewhat flattened. This is because the bottom edge of the disc is slightly more refracted (appears higher) than the top edge due to the larger angle of incidence and the vertical density gradient of the atmosphere.
Advance Sunrise and Delayed Sunset: Atmospheric refraction causes the Sun to be visible about 2 minutes before the actual sunrise and remain visible for about 2 minutes after the actual sunset. Even when the Sun is slightly below the horizon, its rays are refracted downwards, making it appear above the horizon to an observer.
Twinkling of Stars: The twinkling (scintillation) of stars is primarily due to fluctuations in the apparent position and brightness of the star caused by variations in the refractive index of the turbulent atmosphere along the path of light. Planets usually do not twinkle because they are closer and appear as discs, so the average effect of the fluctuations over the disc area cancels out.

Atmospheric refraction is also responsible for phenomena like mirages and looming, as discussed in the context of Total Internal Reflection, although these specific phenomena occur under unusual atmospheric temperature gradients that create layers of significantly different densities.

Scattering Of Light

When light passes through a medium containing tiny particles (like molecules of air, dust particles, or water droplets), the light is redirected in various directions by these particles. This phenomenon is called scattering of light.

Scattering occurs because the oscillating electric field of the light wave interacts with the electrons and protons in the particles, causing them to oscillate. These oscillating charges then re-emit electromagnetic radiation in different directions. The amount and direction of scattering depend on the size of the scattering particles relative to the wavelength of light, and the wavelength itself.

Tyndall Effect

The Tyndall effect is the scattering of light as a light beam passes through a colloidal dispersion or a suspension. The individual suspension particles scatter and reflect light, making the beam visible. The amount of scattering depends on the frequency of the light and the density and size of the particles.

While scattering occurs at all wavelengths, the Tyndall effect is more pronounced for shorter wavelengths (blue light) when the scattering particles are relatively small compared to the wavelength of light (Rayleigh scattering, discussed below). If the particles are large (like in fog or clouds), all wavelengths are scattered equally, making the scattered light appear white.

Examples of Tyndall effect: The beam of a projector seen in a dusty room, the path of sunlight visible through a dusty or smoky window, the blue appearance of smoke from a cigarette.

Rayleigh Scattering

Rayleigh scattering occurs when light is scattered by particles much smaller than the wavelength of the light. The intensity of Rayleigh scattering is inversely proportional to the fourth power of the wavelength ($I \propto 1/\lambda^4$). This means shorter wavelengths are scattered much more strongly than longer wavelengths.

For example, blue light (shorter wavelength) is scattered much more than red light (longer wavelength). If the wavelength of blue light is about half that of red light, blue light is scattered $2^4 = 16$ times more intensely than red light.

Why Is The Colour Of The Clear Sky Blue?

The clear sky appears blue due to the scattering of sunlight by the molecules of gases (primarily nitrogen and oxygen) in the Earth's atmosphere. These molecules are much smaller than the wavelength of visible light, so Rayleigh scattering is dominant.

When sunlight (which is white light, a mixture of all colours) enters the atmosphere, the shorter wavelengths (blue and violet) are scattered much more effectively than the longer wavelengths (red and orange). The scattered blue light reaches our eyes from all directions in the sky, making the sky appear blue. The violet light is scattered even more, but our eyes are less sensitive to violet, and the Sun emits less violet light than blue light, so the blue colour is more prominent.

The sunlight that reaches our eyes directly from the Sun (without being scattered) is depleted in blue light, which is why the Sun appears slightly yellowish during the day.

Colour Of The Sun At Sunrise And Sunset

At sunrise and sunset, the Sun appears reddish or orange. This is also explained by Rayleigh scattering.

When the Sun is near the horizon, sunlight travels through a much longer path in the atmosphere compared to when the Sun is overhead. During this longer journey, most of the shorter wavelength colours (blue, violet) are scattered away from the line of sight by the atmospheric molecules.

The light that reaches the observer's eyes is therefore enriched in the longer wavelength colours (red, orange, yellow), as these colours are scattered much less effectively. This gives the Sun and the sky around it a reddish or orange appearance.

If the atmosphere contains larger particles like dust or smoke, scattering can be more complex and less dependent on wavelength, leading to variations in the colours observed during sunsets and sunrises.

Some Natural Phenomena Due To Sunlight

Sunlight interacting with the Earth's atmosphere and water droplets produces several beautiful natural optical phenomena. Two prominent examples are the rainbow and the blue sky/red sunset, which are due to dispersion, refraction, and scattering.

The Rainbow

A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It appears as a multicoloured arc.

Formation of a Rainbow

Rainbows are formed when sunlight (white light) encounters raindrops in the atmosphere. The process involves a combination of:

Refraction: As sunlight enters a spherical raindrop, it refracts and disperses into its constituent colours. The amount of refraction and deviation depends on the wavelength (colour), similar to what happens in a prism.
Total Internal Reflection (TIR): After entering the raindrop and refracting, the light ray travels to the opposite inner surface of the raindrop. Here, if the angle of incidence is greater than the critical angle for water-air interface, the light undergoes total internal reflection.
Refraction (again): The reflected light then travels back to the front surface of the raindrop and refracts again as it exits the raindrop into the air.

$Formation of a rainbow by refraction, TIR, and dispersion in a raindrop.$

(Image Placeholder: A spherical raindrop. A ray of white sunlight enters the drop, refracts and splits into colours. The light travels across, reflects off the back inner surface, and refracts out the front surface, diverging and showing the spectrum (e.g., red ray and violet ray emerging at different angles).)

Different colours emerge from the raindrop at slightly different angles relative to the incoming sunlight due to dispersion. Red light is deviated less than violet light. For a primary rainbow, the light that reaches the observer's eyes is the light that emerges from the raindrops at a specific angle (about 40-42 degrees) relative to the direction opposite to the sunlight. This angle varies slightly with colour (red at about 42°, violet at about 40°).

The observer sees a rainbow as an arc because the conditions for seeing a particular colour at a particular angle are met by raindrops located on a circular arc in the sky. The centre of the rainbow arc is in the direction opposite to the Sun, on the line joining the Sun and the observer's eye (the anti-solar point).

A secondary rainbow is sometimes seen above the primary rainbow, with colours reversed. It is formed by light that undergoes two internal reflections within the raindrop.

Scattering Of Light

As discussed in the previous subheading, scattering of sunlight by atmospheric particles is responsible for phenomena like the blue colour of the sky and the reddish appearance of the Sun at sunrise and sunset. This section likely aims to reiterate or apply the concept of scattering to these specific natural phenomena.

Blue Sky (Recap)

The blue colour of a clear sky is due to the preferential scattering of shorter wavelengths (blue and violet) of sunlight by the tiny molecules of air ($N_2, O_2$) following Rayleigh's Law ($I \propto 1/\lambda^4$). The scattered blue light reaches our eyes from all parts of the sky.

Red Sun at Sunrise/Sunset (Recap)

At sunrise and sunset, sunlight travels through a much longer path in the atmosphere. This results in most of the blue light being scattered away, leaving the longer wavelengths (red, orange) to reach the observer's eyes, making the Sun appear reddish.

While the rainbow involves refraction and reflection within water droplets (a form of interaction with particles), the blue sky and red sunsets involve scattering of light by much smaller air molecules or other atmospheric particles.