Refraction of Light

Refraction

Light normally travels in a straight line when it is in a uniform medium with consistent density. Refraction of light occurs when light transitions from one medium to another with a different optical density. It results in a change in the direction of the light ray.

Key Aspects of Refraction:

1. Angle of Incidence and Angle of Refraction: When a light ray passes from one medium to another, it does so at an angle. The angle between the incident ray and the normal (perpendicular line) to the boundary at the point of incidence is called the angle of incidence (denoted as 'i'). The angle between the refracted ray and the normal is called the angle of refraction (denoted as 'r').



2. Change in Speed of Light: Refraction occurs because the speed of light is different in different media. Light travels more slowly in denser media (e.g., glass) and faster in less dense media (e.g., air).
3. Wave Theory of Light: Light consists of tiny waves, and these waves change their speed at different times when transitioning from one medium to another. This difference in speed between the left and right sides of a light beam causes the light to change direction, leading to refraction.
4. Applications: Refraction of light is essential in various optical instruments, including cameras, microscopes, and telescopes. It plays a significant role in the behaviour of light in these devices.
5. Optically Rarer and Denser Mediums: Different transparent substances, known as mediums, have different optical densities. A medium in which the speed of light is greater is called an optically rarer medium (or less dense medium). Air is an example of an optically rarer medium compared to glass and water. Conversely, a medium in which the speed of light is lower is called an optically denser medium. Glass is optically denser than air and water, but water is optically denser than air.
   When a ray of light travels from a rarer medium to a denser medium, it bends toward the normal (the perpendicular line to the surface at the point of incidence).
   When a ray of light travels from a denser medium to a rarer medium, it bends away from the normal.
6. Explanation for Bending of Light: The bending of light during refraction is due to differences in the speed of light in different media. Light travels faster in optically rarer mediums and slower in optically denser ones. When light enters a denser medium, it slows down, causing it to bend toward the normal. Conversely, when it enters a rarer medium, it speeds up, leading to a bend away from the normal.



7. Lateral Displacement: Lateral displacement refers to the perpendicular distance between the original path of the incident ray and the emergent ray. This displacement occurs when light enters and exits a parallel-sided glass slab. Lateral displacement depends on factors such as the angle of incidence, thickness of the medium, and refractive index.
8. Angle of Emergence: The angle of emergence is the angle formed by the emergent ray and the normal to the surface. In cases where the incident and emergent rays are parallel (such as light passing through a parallel-sided glass slab), the angle of emergence is equal to the angle of incidence.
9. No Refraction When Light Falls Perpendicularly: If a ray of light falls perpendicularly (normal incidence) to the surface of a medium, there is no bending of the light, and it continues in a straight line. This is because all parts of the light waves reach and exit the medium simultaneously, resulting in no change in direction.

Refraction through a Rectangular Glass Slab

Refraction through a rectangular glass slab is often used to illustrate the principles of refraction and how light changes direction as it moves from one medium to another with a different refractive index.

1. Incident Ray: Imagine a ray of light (incident ray) approaching the surface of a rectangular glass slab. The incident ray travels from a less dense medium (usually air) into a denser medium (glass).
2. Angle of Incidence: The angle between the incident ray and the normal (an imaginary line perpendicular to the surface of the glass at the point of entry) is called the "angle of incidence" (i).
3. Refraction at Entry: As the incident ray enters the glass slab, it bends toward the normal. This bending occurs because light travels more slowly in the denser medium (glass) than in the less dense medium (air). The angle between the refracted ray and the normal is called the "angle of refraction" (r).
4. Inside the Glass: Inside the glass slab, the ray continues to travel in a straight line since it is not encountering any other refracting surfaces. It may, however, change its direction slightly due to the change in the refractive index between air and glass.
5. Refraction at Exit: When the ray reaches the other surface of the glass slab (the exit surface), it once again encounters a change in medium, this time from glass to air. As it exits the glass, it bends away from the normal. The angle of refraction at this exit surface is also denoted as "r."
6. Emergent Ray: The ray of light that emerges from the glass slab is called the "emergent ray." The emergent ray is laterally displaced from the incident ray. This lateral displacement is the perpendicular distance between the paths of the incident and emergent rays as they exit the glass slab.
   It's important to note that the lateral displacement of the emergent ray depends on factors such as the angle of incidence, the thickness of the glass slab, and the refractive index of the glass. The greater the angle of incidence or the thicker the glass slab, the greater the lateral displacement will be.
7. Parallel Shift: Importantly, the emergent ray remains parallel to the incident ray, which means that the image viewed through the glass slab appears shifted laterally but not distorted or tilted.

Effects of the Refraction of Light

1. Apparent Shifting of Objects: Refraction can cause objects submerged in water to appear displaced from their actual positions. For example, a stick or a pencil partially submerged in water appears to be bent at the water's surface. This effect occurs because the light rays from the submerged part of the object change direction upon entering the air, creating an optical illusion.



2. Apparent Elevation of Submerged Objects: When you look at an object placed at the bottom of a clear container filled with water (like a coin in a glass of water), the object appears to be raised higher than its actual position. This is due to the bending of light as it passes from water to air, creating a virtual image of the object that seems elevated.



3. Altered Depth Perception: When you look into a pool of water or a fish tank, the depth of the water may appear shallower than it actually is. This is because the refraction of light at the water's surface makes the bottom of the pool or tank seem closer to the surface than it truly is. This effect can be misleading and should be considered when judging the actual depth of water.



4. Reduced Thickness of Transparent Objects: When you view a thick glass slab or block from above, it appears thinner than its actual thickness. This happens because the refraction of light within the glass slab causes the light to travel a shorter distance.
5. Raised Appearance of Text or Images Under a Glass Plate: When you place a glass plate over a piece of paper with text or images, the content under the glass may appear raised and closer than it is. This is another consequence of light refraction within the glass.
6. Twinkling of Stars: Stars appear to twinkle in the night sky due to the refraction of their light as it passes through Earth's atmosphere. Variations in air density and temperature cause the starlight to bend slightly, creating the twinkling effect.
7. Mirages: A mirage is a fascinating optical illusion caused by the refraction of light in layers of air with varying temperatures. It can make distant objects appear to be reflected or displaced, often creating the illusion of water on a road or desert.
8. Colour Dispersion: The refraction of light through a prism or a raindrop can lead to the dispersion of light into its component colours, creating a rainbow or a spectrum. This effect is due to the different wavelengths of light bending by varying amounts, separating the colours.

Laws of Refraction

The laws of refraction, also known as Snell's laws, describe how light behaves when it passes from one medium to another with a different refractive index. These laws are fundamental principles in optics and govern the behaviour of light at the boundary between two substances.

1. First Law of Refraction (Law of Incidence)

The incident ray, the refracted ray, and the normal (an imaginary line perpendicular to the boundary) at the point of incidence all lie in the same plane. In simpler terms, they share a common plane.

2. Second Law of Refraction (Snell's Law)

The ratio of the sine of the angle of incidence (sin i) to the sine of the angle of refraction (sin r) is a constant value for a given pair of media. This constant is known as the refractive index. Snell's law is mathematically represented as:

Where:
n is the refractive index of the medium.
i is the angle of incidence, the angle between the incident ray and the normal.
r is the angle of refraction, the angle between the refracted ray and the normal.

Different media have different refractive indices, and this property is fundamental in understanding the behaviour of light when it passes through different substances.

Refractive Index and Speed of Light

When light travels from one medium to another, such as from air to water or from one substance to another, its behaviour changes due to differences in the speed of light in those media. This change in behaviour is described by the concept of refractive index (n), which is a fundamental property in optics.

Refractive Index (n): The refractive index of a medium is a measure of how much the speed of light changes when it enters that medium from another medium, typically air or vacuum. It quantifies how much light is bent or refracted as it crosses the boundary between two substances. Mathematically, the refractive index (n) is defined as the ratio of the speed of light in vacuum (or air) to the speed of light in the medium:

The refractive index is always greater than or equal to 1 because the speed of light in any medium is less than or equal to the speed of light in a vacuum. A higher refractive index indicates that light travels more slowly in that medium and will be bent more when it enters that medium.

Relative and Absolute Refractive Index

1. Relative Refractive Index: When light travels from one material to another (excluding vacuum or air), the refractive index is called the relative refractive index. For example, when light goes from water to glass, it is the relative refractive index of glass with respect to water.
2. Absolute Refractive Index: When light travels from vacuum to a material, the refractive index is called the absolute refractive index. It has only one subscript indicating the material's name. For example, n_glass represents the absolute refractive index of glass with respect to vacuum.
3. Relation between Refractive Indices: The refractive index for light travelling from medium 1 to medium 2 (n₁₂) is equal to the reciprocal (inverse) of the refractive index for light travelling from medium 2 to medium 1 (n₂₁):

   

   This relationship means that if you know the refractive index of a material for light entering it from another medium, you can easily calculate the refractive index for light exiting from that material back into the original medium.
   The refractive index (n) of medium 2 concerning medium 1 can be described as the ratio of the speed of light in medium 1 to the speed of light in medium 2. This is represented as n₂₁ and is calculated using the following formula:
   

   

   In simpler terms, it's a measure of how fast light travels through one medium compared to another. If medium 1 happens to be either vacuum or air, then the refractive index of medium 2 is determined in reference to vacuum. This is termed the absolute refractive index of the medium and can be calculated as:
   

   

   This ratio helps us understand how much light slows down or speeds up when moving between different substances.

Refraction of Light by Spherical Lenses

Spherical lenses refract light as it passes through them, causing it to converge or diverge based on the type of lens.

1. Convex Lens

A convex lens is thicker at its centre and thinner at its edges, causing it to bulge outward. It resembles a piece of spherical glass that bows outward.
Convex lenses are also known as converging lenses because they converge (bring together) parallel rays of light that pass through them.

Key characteristics of convex lenses:

1. Principal Focus: When parallel rays of light enter a convex lens, they converge to a single point on the opposite side of the lens. This point is called the principal focus (denoted as 'F'). Convex lenses have two principal foci (F and F’, one on each side of the lens), which are equidistant from the lens's optical centre.
2. Focal Length: The focal length ('f') of a convex lens is the distance between its optical centre ('C') and its principal focus ('F'). It determines the lens's ability to converge light.



3. Real Image: Convex lenses can form real images, which are projected onto a screen or surface. These images are inverted (upside down) if the object is beyond the focal point and upright if the object is between the lens and its focal point.
4. Applications: Convex lenses are used in various optical devices, such as cameras, binoculars, telescopes, and magnifying glasses.

2. Concave Lens

In contrast, a concave lens is thinner at its centre and thicker at its edges, giving it a hollow or "cave-like" appearance.
Concave lenses are also known as diverging lenses because they cause parallel rays of light to diverge (spread apart) after passing through them.

Key characteristics of concave lenses:

1. Principal Focus: When parallel rays of light enter a concave lens, they appear to diverge from a point on the same side of the lens as the incoming light. This point is also called the principal focus (denoted as 'F'). Like convex lenses, concave lenses have two principal foci (F and F’).
2. Focal Length: The focal length ('f') of a concave lens is the distance between its optical centre ('C') and its principal focus ('F'). It determines the lens's ability to diverge light.



3. Virtual Image: Concave lenses can form virtual images, which cannot be projected onto a screen. These images are always upright and appear on the same side of the lens as the object.
4. Applications: Concave lenses are used in correcting vision problems like nearsightedness (myopia). They are also employed in optical instruments like microscopes and cameras for specific purposes.

Rules for Obtaining Images Formed by Convex Lenses

By using certain rules, you can draw ray diagrams to locate the position and characteristics of the image formed by a convex lens. These rules are used to find the position and nature of images formed by convex lenses. By applying these rules, you can determine whether the image is real or virtual, where it is located relative to the lens, and whether it's upright or inverted. Different combinations of these rules help analyse various object positions in front of a convex lens and predict the characteristics of the resulting images.

Rule 1: Parallel Ray Rule

When a ray of light is parallel to the principal axis of a convex lens, it will pass through the lens and refract in such a way that it appears to come from the focal point (F) on the opposite side of the lens.

Rule 2: Optical Centre Rule

When a ray of light passes through the optical centre (O) of a convex lens, it continues in a straight line without any deviation. The optical centre is the point on the lens where light passes through without bending. This rule is particularly useful for determining the path of the ray but doesn't directly reveal information about the image's location.

Rule 3: Focal Point Rule

If a ray of light initially travels towards the focal point (F’) of a convex lens, it will refract through the lens and emerge parallel to the principal axis. This rule is the opposite of Rule 1, where a parallel ray converges to the focal point.

Formation of Images by Convex Lenses

The formation of images by a convex lens involves the refraction of light rays as they pass through the lens. Convex lenses are thicker at the centre than at the edges and are capable of creating both real and virtual images, depending on the object's position relative to the lens.

Primary cases of image formation by a convex lens based on the object's position are:

Case 1: Object between the Optical Center (C) and Focus (F’)

1. Position of Image: When an object is placed between the optical centre (C) and the focus (F’) of a convex lens, the image is formed on the same side as the object, which is behind the lens (on the left side).
2. Size of Image: The image is enlarged (magnified) compared to the actual object.
3. Nature of Image: The image is virtual and erect (upright). This means it appears on the same side as the object and looks larger than the actual object. It is a virtual image because the light rays do not actually converge to form the image but only appear to converge when extended backwards.
4. Application: This configuration is commonly used when using a magnifying glass or simple microscope. By placing an object within the focus of the lens, a larger and upright virtual image is produced, making it easier to examine small details.

Case 2: Object at the Focus (F’)

1. Position of Image: When the object is positioned exactly at the focus (F’) of the convex lens, the image is formed at infinity.
2. Size of Image: The image is highly enlarged.
3. Nature of Image: The image is real and inverted. It is highly magnified but located at an infinite distance from the lens. This type of image is often used in astronomical telescopes.

Case 3: Object between F’ and 2F’ (or between f and 2f)

1. Position of Image: Placing an object between the focus (F’) and twice the focus (2F’), or between twice the focal length (2f) and the lens, results in an image formed beyond twice the focal length (2F) on the opposite side of the lens.
2. Size of Image: The image is enlarged compared to the object.
3. Nature of Image: The image is real and inverted. It is larger than the object and is located on the opposite side of the lens.
4. Application: This configuration is used in cameras and projectors. In cameras, an object is positioned between the focus and twice the focus to form a real and enlarged image on the film or sensor. In projectors, a slide or film is placed between the focus and twice the focus, and an enlarged image is projected onto a screen.

Case 4: Object at 2F’ (or at 2f)

1. Position of Image: When the object is situated at twice the focus (2F’), or at twice the focal length (2f), the image is formed at twice the focal length (2F) on the other side of the lens.
2. Size of Image: The image is the same size as the object.
3. Nature of Image: The image is real and inverted. It has the same size as the object and is located at a distance of twice the focal length from the lens.
4. Application: This type of image formation is used in slide projectors, where a slide is placed exactly at twice the focal length to project an image of the same size as the slide.

Case 5: Object Beyond 2F’ (or Beyond 2f)

1. Position of Image: When the object is positioned beyond twice the focus (2F’), or beyond twice the focal length (2f), the image is formed between the focus (F) and twice the focus (2F) on the opposite side of the lens.
2. Size of Image: The image is diminished (smaller) compared to the object.
3. Nature of Image: The image is real and inverted. It is smaller than the object and is located on the opposite side of the lens.
4. Application: This is a common configuration for taking photographs with cameras. The distant object (e.g., a landscape) forms a real, smaller, and inverted image on the camera's film or image sensor.

Case 6: Object at Infinity

1. Position of Image: When the object is at a considerable distance (infinity) from the convex lens, the image is formed at the focus (F).
2. Size of Image: The image is highly diminished (very small).
3. Nature of Image: The image is real and inverted. It is tiny and located at the focal point of the lens.
4. Application: This type of image formation is used in telescopes to observe distant celestial objects. The objective lens of the telescope forms a real, highly diminished, and inverted image at the focal point for further magnification by the eyepiece.

Rules for Obtaining Images Formed by Concave Lenses

These rules are crucial for constructing ray diagrams to determine the position and characteristics of images formed by concave lenses. Concave lenses, unlike convex lenses, produce virtual, erect (upright), and diminished images in various scenarios depending on the position of the object relative to the lens.

Rule 1: Parallel Ray Rule

When a ray of light parallel to the principal axis of a concave lens strikes the lens, it refracts in such a way that it appears to diverge from a specific point on the same side of the lens. This point is known as the focus (F) of the concave lens.
The ray diagram for this rule involves drawing the incoming parallel ray of light, refracting it at the lens, and extending it backwards to show that it appears to come from the focus.
This rule demonstrates how concave lenses disperse parallel rays of light.

Rule 2: Optical Centre Rule

When a ray of light passes through the optical centre (C) of a concave lens, it continues in a straight line without any deviation.
The optical centre is the midpoint of the lens, and light passing through it does not refract.
Even if the ray of light initially had a direction away from the optical centre, it will proceed in a straight line through the centre without bending.
This rule is useful for understanding the path of light when it goes through the optical centre of a concave lens.

Rule 3: Focus-to-Parallel Ray Rule

When a ray of light is directed towards the focus (F’) of a concave lens, it refracts and emerges from the lens parallel to the principal axis.
The ray diagram for this rule involves drawing a ray aimed at the focus of the concave lens, showing its refraction at the lens, and indicating that it exits parallel to the principal axis.
Rule 3 is essentially the reverse of Rule 1 for convex lenses, where Rule 1 converges parallel rays to the focus, while Rule 3 for concave lenses diverges rays directed toward the focus.

Formation of Images by Concave Lenses

When a concave lens is used, images are formed based on the position of the object.
Primary cases of image formation by a concave lens based on the object's position are:

Case 1: Object between Optical Centre (C) and Infinity

1. Position of Image: If the object is anywhere between the optical centre (C) and infinity in front of a concave lens the image is formed between the optical centre (C) and the focus (F) of the lens.
2. Size of the Image: Diminished, the size of the image is smaller than that of the object.
3. Nature of the Image: The image is virtual but appears to be on the same side as the object. The image is upright, having the same orientation as the object.

As the object is moved further away from the optical centre towards infinity, the image becomes smaller and moves closer to the focus (F).

Case 2: Object at Infinity

1. Position of Image: When the object is placed at infinity from a concave lens, the image is formed at the focus (F) of the concave lens.
2. Size of the Image: Highly Diminished, the size of the image is much smaller than that of the object.
3. Nature of the Image: Virtual and erect.

Sign Convention for Spherical Lenses

The New Cartesian Sign Convention for spherical lenses is a standardised method for measuring various distances and characteristics in ray diagrams for both convex and concave lenses.

Measurement from Optical Centre (C): All distances are measured from the optical centre (C) of the lens. The optical centre is the point in the lens where light rays passing through it do not undergo any deviation.

Positive and Negative Distances:

1. Distances measured in the same direction as that of the incident light (right side of the lens) are taken as positive.
2. Distances measured against the direction of the incident light (left side of the lens) are taken as negative.
3. Distances measured upward and perpendicular to the principal axis (the central axis of the lens) are taken as positive.
4. Distances measured downward and perpendicular to the principal axis are taken as negative.

Focal Length Sign:

1. The focal length (f) of a convex lens is considered positive and is written with a plus sign (+f).
2. The focal length (f) of a concave lens is considered negative and is written with a minus sign (-f).

Lens Formula

The lens formula is a fundamental equation in optics that relates the object distance (u), image distance (v), and the focal length (f) of a lens. It's used to calculate these parameters for both convex and concave lenses. The lens formula is as follows:

Where:
v is the image distance (positive for real images and negative for virtual images).
u is the object distance (positive when the object is on the same side as the incident light source and negative when it's on the opposite side).
f is the focal length of the lens (positive for convex lenses and negative for concave lenses).

This formula essentially states that the reciprocals of the image distance and the object distance are related to the reciprocal of the focal length.

Magnification Produced by a Lens

The magnification produced by a lens is a measure of how the size of an image compares to the size of the object. The linear magnification (m) is defined as the ratio of the height of the image (h₂) to the height of the object (h₁):

This formula tells you how many times larger or smaller the image is compared to the object.

Alternatively, you can express the magnification in terms of the image distance (v) and the object distance (u):

Where:
m is the magnification.
v is the image distance (positive for real images and negative for virtual images).
u is the object distance (positive when the object is on the same side as the incident light source and negative when it's on the opposite side).

Here's how to interpret the magnification:

If m is positive, the image is virtual and upright (erect).
If m is negative, the image is real and inverted.
The magnitude of m indicates whether the image is larger (m > 1), the same size (m = 1), or smaller (m < 1) than the object.

For convex lenses, the magnification can be positive or negative, and the image can be smaller, the same size, or larger than the object, depending on the object's position relative to the lens.

For concave lenses, the magnification is always positive, indicating that the image is always virtual, upright, and smaller than the object.

These magnification formulas and rules help describe how lenses create images and how those images relate to the object being viewed or photographed.

Power of a Lens

The power of a lens is a measure of its ability to converge or diverge light rays. It depends on the lens's focal length and is expressed in diopters (D), where 1 diopter is the power of a lens with a focal length of 1 metre.

The formula for calculating the power of a lens (P) is:

Where:
P is the power of the lens in diopters (D).
f is the focal length of the lens in metres (m).

Key points to remember:

1. For a convex lens (converging lens), the focal length is positive, and therefore, the power is also positive.
2. For a concave lens (diverging lens), the focal length is negative, leading to a negative power.
3. The greater the power of a lens, the stronger its ability to bend light rays. Lenses with shorter focal lengths have higher powers.
4. Diopters (D) are the units used to measure the power of lenses. One diopter is equal to the power of a lens with a focal length of 1 metre.
5. When an optometrist prescribes eyeglasses, the prescription typically includes the power of the lenses needed to correct a person's vision. These prescriptions are written in diopters, such as +2.0 D for a converging lens or -1.5 D for a diverging lens.

Power of a Combination of Lenses

The power of a combination of lenses is found by adding the powers of the individual lenses, taking into account their signs. The formula for finding the resultant power (P) of a combination of lenses with powers p₁, p₂, p₃ and so on, is:

P = p₁ + p₂ + p₃ + …

Where:
P is the resultant power of the combination.
p₁, p₂, p₃, … are the powers of the individual lenses.

Key points to remember:

1. The powers of individual lenses should be considered with their proper signs. Convex lenses have positive powers, and concave lenses have negative powers.
2. When a combination of lenses is placed in close contact with each other, the formula is used to calculate the overall power of the combination.
3. If the resultant power is positive, the combination behaves like a converging lens (convex lens), and if it's negative, the combination behaves like a diverging lens (concave lens).
4. This concept is used in designing optical instruments like cameras, microscopes, and telescopes, where combinations of lenses are used to improve image quality and correct optical aberrations.