Comprehensive Study Notes on the Refraction of Light

Introduction to the Phenomena of Light

Many common experiences are related to the various phenomena of light. One such observation is that of a straw placed in a glass of lemonade, which appears to be bent at the surface of the liquid. We perceive objects when light falling on them is reflected and enters our eyes. In the case of the straw, light rays reflected from the submerged part of the straw must travel through different mediums (water and then air) before reaching the observer's eyes. Another common observation is the appearance of water logging on roads during hot summer days, even when there has been no rain. These phenomena are explained by the way light behaves when moving between different substances.

The Path of Light and Medium Transitions

When light travels through a single, uniform medium, its path is a straight line. However, the direction of light can change when it encounters the boundary between two different mediums. To investigate this, an experiment can be conducted using a trough filled three-fourths with water and a few drops of milk (to make the path visible), with the remaining space filled with smoke and covered by a transparent sheet.

When a laser beam is flashed obliquely (at an angle) from air into the water, a deviation in the direction of the ray occurs at the surface of separation between the air and the water. Conversely, if the light from the laser torch falls normally (perpendicularly) to the surface of the water, there is no deviation in its path.

As observed in these experiments, light rays undergo a deviation at the surface of separation when they enter obliquely from one medium to another. No such deviation occurs if the ray is incident normally. The specific point of direction change is always at the surface of separation between the two distinct mediums.

Speed of Light and Optical Density

The deviation in the path of a light ray when moving between mediums is caused by a change in the speed of light. To understand this, one can compare it to a toy car moving from a smooth surface to a rough surface. The car undergoes a change in direction at the boundary because its speed changes upon hitting the rougher surface. Similarly, light travels at different speeds in different materials.

The approximate speeds of light in various mediums are as follows:

Air: $3 \times 10^8\,m/s$
Water: $2.25 \times 10^8\,m/s$
Glass: $2 \times 10^8\,m/s$
Diamond: $1.25 \times 10^8\,m/s$

The ability of a medium to influence the speed of light through it is known as its optical density. In a medium with higher optical density (an optically denser medium), the speed of light is lower. Conversely, in a medium with lower optical density (an optically rarer medium), the speed of light is higher. It is important to note that optical density is a specific property related to light and has no direct relationship with the material density (mass per unit volume) of the substance. Based on the speeds provided, the mediums can be arranged in increasing order of optical density: Air < Water < Glass < Diamond.

The Phenomenon of Refraction

When a ray of light enters obliquely from one medium to another of different optical densities, it undergoes a deviation at the surface of separation. This phenomenon is defined as refraction.

This phenomenon explains why a straw appears bent in water. In an experiment where a straw is kept obliquely in a glass that is then filled with water, light rays from the immersed part of the straw (e.g., point B) travel from water into air. As they cross the surface of separation, they undergo refraction and deviate away from their original path. To an observer, these refracted rays appear to originate from a higher position (point C) rather than their actual source (point B). This causes the submerged part of the straw to appear elevated, resulting in the "bent" appearance.

Refractive Index

The ability of light to undergo refraction in a medium depends on its optical density, which is quantified as the refractive index. The refractive index ( $n$ ) of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that specific medium.

The formula for refractive index is: $n = \frac{c}{v}$ Where: $c$ = the speed of light in vacuum ( $3 \times 10^8\,m/s$ ) $v$ = the speed of light in the medium

Calculated refractive indices for various mediums (using $c = 3 \times 10^8\,m/s$ ):

Air: $n = \frac{3 \times 10^8\,m/s}{3 \times 10^8\,m/s} = 1$
Water: $n = \frac{3 \times 10^8\,m/s}{2.25 \times 10^8\,m/s} \approx 1.33$
Glass: $n = \frac{3 \times 10^8\,m/s}{2 \times 10^8\,m/s} = 1.5$
Kerosene: $n \approx 1.44$
Turpentine oil: $n \approx 1.47$
Crown glass: $n \approx 1.52$
Diamond: $n \approx 2.42$

There is an inverse relationship between the speed of light and the refractive index: the speed of light is lower in a medium with a higher refractive index. Optical density is directly proportional to the refractive index.

Geometric Parameters of Refraction

In the study of refraction, specific terms are used to describe the geometry of the light path. The ray falling on the surface of separation of the two mediums is the incident ray. The ray that undergoes a change in direction is the refracted ray. The angle between the incident ray and the normal (an imaginary line perpendicular to the surface at the point of incidence, denoted as $NN^\prime$ ) is the angle of incidence ( $i$ ). The angle between the refracted ray and the normal is the angle of refraction ( $r$ ).

The behavior of the refracted ray depends on the relative optical densities of the two mediums:

From Rarer to Denser: When light enters from an optically rarer medium (e.g., air) to an optically denser medium (e.g., water), the refracted ray deviates towards the normal. In this case, the angle of refraction ( $r$ ) is less than the angle of incidence ( $i$ ).
From Denser to Rarer: When light enters from an optically denser medium (e.g., water) to an optically rarer medium (e.g., air), the refracted ray deviates away from the normal. In this case, the angle of refraction ( $r$ ) is greater than the angle of incidence ( $i$ ).
General Rules:

The incident ray, the refracted ray, and the normal at the point of incidence all lie on the same plane.
A ray incident normally (perpendicularly) at the surface of separation does not undergo refraction.

Practical Applications and Observations of Refraction

Refraction is responsible for several everyday observations and practical challenges:

Visibility of Objects: A coin placed in a vessel that is hidden from view by the vessel's rim becomes visible once water is poured into the vessel. This is because light from the coin refracts at the water-air boundary and reaches the eye.
Apparent Elevation: Letters in a textbook appear raised when a glass slab is placed over them. Similarly, the bottom of a pond appears shallower than it actually is, especially when viewed from a distance.
Bow Fishing: People engaged in bow fishing must aim at a point slightly below the perceived position of the fish because the refraction of light at the water's surface causes the fish to appear higher in the water than its actual depth.

Atmospheric Refraction

The Earth's atmosphere is composed of layers of air with varying physical conditions, such as pressure and temperature, which continuously change. These variations lead to changes in the optical density of the atmosphere at different altitudes.

Twinkling of Stars: Stars are so far away that they appear as point sources of light. As their light passes through the atmosphere, it undergoes multiple, irregular refractions due to constantly changing air layers. Consequently, the star's apparent position shifts slightly and its intensity fluctuates, creating the twinkling effect. Planets, being closer, do not appear as point sources and thus do not twinkle in the same way.
Position of the Sun: Atmospheric refraction allows us to see the Sun a few seconds before it actually reaches the eastern horizon in the morning (advanced sunrise) and for some time after it has crossed below the western horizon in the evening (delayed sunset). The light from the Sun refracts as it enters the denser layers of the Earth's atmosphere, curving over the horizon toward the observer.

Total Internal Reflection

Refraction does not occur in every instance where light moves from one medium to another. Specifically, when light travels from an optically denser medium to an optically rarer medium, a different phenomenon can occur called Total Internal Reflection (TIR).

As the angle of incidence ( $i$ ) in the denser medium increases, the angle of refraction ( $r$ ) in the rarer medium also increases, moving further away from the normal. Eventually, a specific angle of incidence is reached where the angle of refraction becomes exactly $90^\circ$ . This specific angle of incidence is called the critical angle. For a glass-air pair, the critical angle is approximately $42^\circ$ .

Total Internal Reflection occurs when a ray of light enters from an optically denser medium to an optically rarer medium at an angle of incidence greater than the critical angle. In this scenario, the ray is not refracted but is instead reflected back completely into the same denser medium.

Conditions required for Total Internal Reflection:

Light must travel from an optically denser medium to an optically rarer medium.
The angle of incidence must be greater than the critical angle for the given pair of mediums.

Applications of Total Internal Reflection

Total Internal Reflection is utilized in various technologies and explains several natural phenomena:

Aquarium Observations: The bottom of an aquarium can appear reflected above the surface of the water when viewed from certain angles because light undergoes TIR at the water-air interface.
Mirage: In summer, air near the hot road is less dense than the cooler air above. Light from objects undergoes refraction as it moves down through layers of decreasing density, eventually hitting a layer at an angle greater than the critical angle. This causes TIR, reflecting the light upward. The observer sees a reflected image of the sky or surrounding objects on the road, which looks like water logging.
Reflectors: Vehicle tail lamps and cycle reflectors contain numerous small prisms. Light entering these prisms normally through one face strikes the internal surface at an angle ( $45^\circ$ ) greater than the critical angle of glass ( $42^\circ$ ), undergoing TIR twice to reflect back toward the source.
Periscopes: High-quality periscopes use prisms instead of mirrors. Prisms utilize TIR to provide images with greater visual clarity and less loss of light intensity.
Optical Fibers: These are thin fibers used primarily in telecommunications and medicine. Light enters the fiber and strikes the inner walls at an angle greater than the critical angle, causing successive total internal reflections. This allows signals to travel long distances at the speed of light with minimal loss of intensity. In medicine, this technology is used in endoscopy to visualize internal organs. Charles K. Kao was awarded the Nobel Prize in Physics in 2009 for his contributions to the development of optical fiber transmission.

Comparison: TIR vs. Plane Mirror Reflection

There are distinct differences between Total Internal Reflection and reflection from a plane mirror:

Total Internal Reflection: The ray of light is completely reflected (virtually 100% of light energy is retained). It occurs only when light moves from a denser to a rarer medium at an angle greater than the critical angle.
Reflection from a Plane Mirror: The ray of light is not completely reflected (some light is absorbed or transmitted by the mirror surface). Reflection occurs at the surface for any angle of incidence.

Exercises and Assessment Data

Key problems and data from the assessment include:

If light travels from medium X to Y and the angle of refraction is greater than the angle of incidence ( $r > i$ ), medium Y is optically rarer than medium X. Therefore, the speed of light is higher in medium Y, and medium X has a higher refractive index.
Comparing mediums by refractive index: crown glass ( $1.52$ ), glycerine ( $1.47$ ), sunflower oil ( $1.47$ ), water ( $1.33$ ), and flint glass ( $1.62$ ). Light travels fastest in water (lowest $n$ ). Light will not deviate when moving from glycerine to sunflower oil at an angle because their refractive indices are identical ( $1.47$ ), meaning their optical densities are the same.
Calculating speed in a medium: If the speed of light in air is $3 \times 10^8\,m/s$ and the refractive index of kerosene is $1.44$ , the speed in kerosene is $v = \frac{3 \times 10^8\,m/s}{1.44} \approx 2.08 \times 10^8\,m/s$ .