Ray Optics And Optical Instruments Class 12 Physics Chapter 9 Notes

Ray Optics and Optical Instruments Class 12 Physics Chapter 9 Notes

Sign Convention for Spherical Mirrors

Measurement from Pole or Optical Center: All distances are measured from either the pole of the mirror or the optical center of the lens.
Direction of Measurement: Distances measured in the same direction as the incident light are considered positive, while those measured in the direction opposite to the incident light are considered negative.
Height Convention: Heights measured upwards with respect to the x-axis and normal to the principal axis (x-axis) of the mirror or lens are considered positive. Conversely, heights measured downwards are considered negative.
Unification of Formulae: By adopting this common sign convention, a single set of formulae can be used to handle various cases involving spherical mirrors and lenses.

This sign convention is crucial for correctly applying the formulae and understanding the mathematics associated with the reflection and refraction of light by spherical mirrors and lenses. It ensures consistency and simplifies calculations by providing a standardized framework for measurements.

Focal Length of Spherical Mirrors

Convergence and Divergence of Rays: When a parallel beam of light is incident on a spherical mirror (a) concave mirror or (b) convex mirror, the reflected rays behave differently. In the case of a concave mirror (a), the reflected rays converge at a point F on the principal axis. For a convex mirror (b), the reflected rays appear to diverge from a point F on the principal axis.
Principal Focus: The point F where the parallel rays either converge or appear to diverge is called the principal focus of the mirror.
Focal Plane: If the incident beam of light makes an angle with the principal axis, the reflected rays will converge (or appear to diverge) from a point in a plane through F that is normal to the principal axis. This plane is known as the focal plane of the mirror.
Focal Length (f): The focal length of the mirror (f) is defined as the distance between the focus (F) and the pole (P) of the mirror.
Focal Length Formula: The formula for the focal length of a spherical mirror is given as f = R/2, where R represents the radius of curvature of the mirror. This formula relates the focal length to the mirror’s curvature.

Mirror Equation and the formation of images by spherical mirrors

Image Formation: When dealing with spherical mirrors, images are formed by tracing the paths of light rays. An image is considered real if the rays actually converge to a point, and it’s considered virtual if the rays appear to diverge from a point when produced backward.
Ray Diagrams: To determine the position and nature of an image formed by a spherical mirror, you can use ray diagrams. Commonly used rays include:
- Ray 1: A ray parallel to the principal axis reflects through the focus (for concave mirrors) or appears to come from the focus (for convex mirrors).
- Ray 2: A ray passing through the center of curvature reflects back on itself (for both concave and convex mirrors).
- Ray 3: A ray directed toward the focus reflects parallel to the principal axis (for concave mirrors) or appears to be directed toward the focus (for convex mirrors).
- Ray 4: A ray incident at any angle at the pole follows the laws of reflection.
Mirror Equation: The mirror equation relates the object distance (u), image distance (v), and the focal length (f) of the spherical mirror. The mirror equation is given by:
- 1/v+1/u=1/f1
- This equation is valid for all cases of reflection by spherical mirrors, whether the image formed is real or virtual.
Linear Magnification: Linear magnification (m) is defined as the ratio of the height of the image (h’) to the height of the object (h). The formula for linear magnification is:
- m=−h′/h
- The negative sign indicates that the image can be inverted compared to the object’s orientation.
General Applicability: The mirror equation and magnification formula are applicable to all cases of reflection by spherical mirrors, including concave and convex mirrors and situations where the image is real or virtual. Proper use of the sign convention is essential for accurate calculations.

Refraction

Laws of Refraction: Refraction of light is governed by two fundamental laws, Snell’s Law and Plane of Incidence.
Snell’s Law: The ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is constant and depends on the refractive indices of the two media:
- sini/sinr=n21
- where n21 is the refractive index of the second medium with respect to the first medium.
Plane of Incidence: The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.
Optical Density: When n21>1, the refracted ray bends toward the normal, indicating that the second medium is optically denser than the first. Conversely, when n21<1, the refracted ray bends away from the normal, and the first medium is optically denser.
Refractive Indices: The refractive index (n) of a medium is a measure of how much light slows down when passing through that medium compared to vacuum. If n21 is the refractive index of medium 2 with respect to medium 1, then n12=1/n21 is the refractive index of medium 1 with respect to medium 2.
Rectangular Slab: When light passes through a rectangular slab with two interfaces (e.g., air-glass and glass-air), there is no deviation in the direction of the emerging ray. However, there is a lateral displacement or shift compared to the incident ray.
Apparent Depth: When viewing objects in a denser medium, such as water, from air, the objects appear closer to the surface. The apparent depth (h1) is related to the real depth (h2) by the refractive index of the denser medium (water, in this case).

Total Internal Reflection

Total internal reflection is a fascinating phenomenon that occurs when light travels from a denser medium to a rarer medium at the interface between the two media. Here are the key points related to total internal reflection:

Internal Reflection: When light passes from a denser medium (higher refractive index) to a rarer medium (lower refractive index), it can undergo partial reflection and partial refraction at the interface. As the angle of incidence increases, the angle of refraction also increases.
Critical Angle: The critical angle (ic) is the angle of incidence at which the refracted ray would make an angle of 90 degrees with the normal. This angle corresponds to the maximum angle at which total internal reflection can occur.
Total Internal Reflection: When the angle of incidence exceeds the critical angle (i>ic), refraction is not possible, and the incident ray is totally reflected back into the denser medium. This phenomenon is known as total internal reflection.
Demonstration: Total internal reflection can be easily demonstrated with a laser pointer and a container of turbid water. By adjusting the angle of incidence, you can observe total internal reflection occurring at the water-air interface.
Applications:
- Prisms: Certain prisms are designed to make use of total internal reflection to bend light by 90 or 180 degrees. They are also used to invert images without changing their size.
- Optical Fibers: Optical fibers are extensively used for transmitting signals, including audio and video, over long distances. They utilize total internal reflection to guide light along their length with minimal loss in intensity. Optical fibers find applications in telecommunications, medical instruments, and decorative lighting.
Optical Fiber Construction: Optical fibers consist of a core and cladding. The core has a higher refractive index than the cladding, ensuring that light undergoes total internal reflection within the core, allowing for efficient signal transmission.

Refraction at Spherical Surfaces and by Lenses

Refraction at spherical surfaces and by lenses is an essential topic in optics. It involves understanding how light behaves when it passes through or interacts with spherical interfaces and transparent optical media, such as lenses. Here’s an overview of the key concepts related to this topic:

Spherical Interfaces: When light passes from one transparent medium into another with a curved or spherical interface, refraction occurs. At each point on the spherical surface, you can apply the same laws of refraction as you would at a flat interface. The normal at the point of incidence is perpendicular to the tangent plane at that point and passes through the center of curvature.
Spherical Surface Refraction: To understand refraction at a single spherical surface, you can apply the formula for image formation at that surface. The behavior of light as it passes through the curved surface can be analyzed to determine the properties of the resulting image.
Thin Lenses: A thin lens is an optical medium with two surfaces, at least one of which is spherical. Lenses are often used to bend and focus light, and they come in two main types: converging (convex) and diverging (concave) lenses.
Lens Maker’s Formula: The lens maker’s formula relates the focal length of a lens to the refractive indices of the lens material and the surrounding medium, as well as the radii of curvature of the two lens surfaces. This formula is crucial for designing lenses with specific optical properties.
Lens Formula: The lens formula describes the relationship between the object distance (u), the image distance (v), and the focal length (f) of a lens. It is based on the principles of refraction and can be used to predict the location and characteristics of the image formed by a lens.

Refraction at a spherical surface

Refraction at a spherical surface is a fundamental concept in optics, especially when dealing with lenses and curved optical elements. The key equation for this phenomenon is given by Equation below:

n2(1/v−1/u)=(n2−n1)1/R

Here’s what each of these variables represents:

n1 and n2 are the refractive indices of the two media separated by the spherical surface. n1 is the refractive index of the initial medium (from which the light is coming), and n2 is the refractive index of the medium into which the light is entering.
u is the object distance, which is the distance between the object and the spherical surface. The object is where the light rays originate.
v is the image distance, which is the distance between the image formed by the spherical surface and the surface itself.
R is the radius of curvature of the spherical surface. It’s the radius of the sphere from which the surface is a part.

This equation relates the object distance (u), the image distance (v), the refractive indices of the media (n1 and n2), and the radius of curvature (R) of the spherical surface. It describes how light rays change direction and converge or diverge when they pass through or interact with the spherical surface. This equation is fundamental in the design and analysis of lenses and other optical systems.

Refraction by a lens

The refraction by a lens is described by the lens maker’s formula and the thin lens formula, which are fundamental in understanding how lenses form images. Here’s a summary of the key equations and concepts involved:

Lens Maker’s Formula (Lens Equation):
- 1/f=(n−1)(1/R1−1/R2)
  - f is the focal length of the lens. n is the refractive index of the material of the lens. R1 is the radius of curvature of the first lens surface (positive if the center of curvature is on the same side as the incoming light). R2 is the radius of curvature of the second lens surface (positive if the center of curvature is on the same side as the incoming light).
  This formula relates the focal length of the lens to its refractive index and the radii of curvature of its two surfaces.
Thin Lens Formula: 1/f=1/v−1/u
- f is the focal length of the lens.v is the image distance (positive if the image is on the same side as the outgoing light, negative if on the opposite side).u is the object distance (positive if the object is on the same side as the incoming light, negative if on the opposite side).
This formula gives the relationship between the focal length, object distance, and image distance for a thin lens. It’s valid for both convex and concave lenses and for real and virtual images.
Focal Points of a Lens:
- A lens has two focal points, one on each side.
- The focal point on the side from which light enters the lens is called the first focal point (F).
- The focal point on the opposite side, where parallel rays converge after passing through the lens, is called the second focal point (F’).
- For convex lenses, both focal points are real and are on the same side as the incoming light.
- For concave lenses, the first focal point (F) is virtual, and the second focal point (F’) is real.
Magnification (m): m=h′/h =v/u
- m is the magnification.
- h′ is the height of the image.
- h is the height of the object.
- v is the image distance.
- u is the object distance.
Magnification tells you how much larger or smaller the image is compared to the object. It can be positive (for upright images) or negative (for inverted images).

To analyze the formation of images by lenses, you can use these equations along with the sign conventions for distances and focal lengths. The choice of signs depends on whether the distances and focal lengths are measured in the direction of incoming or outgoing light rays and whether the image is real or virtual, erect or inverted. Ray diagrams can also help in visualizing the image formation process.

Remember that these formulas apply to thin lenses (lenses with negligible thickness compared to their focal length). For thick lenses, you would need to consider the lens as a combination of two or more thin lenses and apply the formulas accordingly.

Power of a lens

The power of a lens, denoted by P, is a measure of how much the lens converges or diverges a beam of light. It is defined as the reciprocal of the focal length of the lens and is expressed in dioptres (D), where 1 dioptre is equal to 1 meter^(-1).

The formula for calculating the power of a lens is:

P=1/f

Where:

P is the power of the lens in dioptres (D).
f is the focal length of the lens in meters (m).

Key points about lens power:

Positive Power (Converging Lens): A lens with a positive power is a converging lens. It converges incoming parallel rays of light to a focal point. The focal length (f) is positive for such lenses.
Negative Power (Diverging Lens): A lens with a negative power is a diverging lens. It causes parallel rays of light to diverge as if they came from a virtual focal point. The focal length (f) is negative for such lenses.
Unit of Power: The SI unit for lens power is the dioptre (D), which is equivalent to 1 meter^(-1).
Focal Length and Power Relationship: The relationship between focal length and power is inversely proportional. Lenses with shorter focal lengths have higher powers, while lenses with longer focal lengths have lower powers.
Prescription Lenses: Opticians use lens power to prescribe corrective lenses for vision problems. A positive power lens corrects for farsightedness (hyperopia), while a negative power lens corrects for nearsightedness (myopia).

For example:

A lens with a focal length of 1 meter has a power of +1.0 D (converging lens).
A lens with a focal length of -0.25 meters (or -25 cm) has a power of -4.0 D (diverging lens).

Understanding the power of a lens is essential in the design of eyeglasses, contact lenses, and other optical devices used to correct vision problems.

Combination of thin lenses in contact

When you have two or more thin lenses placed in contact with each other, you can calculate the effective focal length of the combination using the lensmaker’s formula and the concept of power. This is useful in designing optical systems for various devices like cameras, microscopes, and telescopes.

Here’s how you can calculate the effective focal length and power of a combination of thin lenses in contact:

For each individual lens:
- Determine the focal length (f1, f2, f3, …) of each lens.
- Calculate the power (P1, P2, P3, …) of each lens using the formula P = 1/f, where P is the power in dioptres (D) and f is the focal length in meters (m). Note that for converging lenses, the power is positive, and for diverging lenses, the power is negative.
To find the effective focal length of the combination (f), use the following formula for two lenses in contact:
- 1/f=1/f1+1/f2
- This formula allows you to calculate the effective focal length of two lenses in contact. For more than two lenses, you can extend the formula by adding the reciprocals of the focal lengths for each lens.
Calculate the effective power (P) of the combination using the formula:
- P=1/f
- The effective power represents how much the entire lens combination converges or diverges light.
To find the total magnification (m) of the combination, multiply the magnifications (m1, m2, m3, …) of individual lenses:
- m=m1⋅m2⋅m3⋅…
- Each lens in the combination contributes to the overall magnification, and this product accounts for the combined effect of all the lenses.

This approach allows you to analyze and design optical systems with multiple lenses by considering their individual characteristics (focal length and power) and their combined effects on image formation and magnification. It is commonly used in optics and lens design for various applications.

Refraction through a prism

When light passes through a prism, it undergoes refraction at the two faces of the prism, resulting in a deviation in the direction of the light. This deviation is commonly referred to as the angle of deviation (d). The angle of deviation depends on several factors, including the angles of incidence and refraction at the prism’s surfaces and the refractive index of the material of the prism.

Here are the key points related to refraction through a prism:

Incident and Refracted Angles: When light enters the prism at one face (let’s call it AB), it undergoes refraction. The angles at this face are as follows:
- Angle of incidence (i): The angle between the incident ray (incoming light) and the normal (perpendicular) to face AB.
- Angle of refraction (r1): The angle between the refracted ray inside the prism and the normal to face AB.
Second Refraction: After refraction at face AB, the light travels inside the prism and eventually exits through the second face (AC). At this second refraction, the angles are:
- Angle of incidence (r2): The angle between the incident ray (inside the prism) and the normal to face AC.
- Angle of emergence (e): The angle between the emerging ray (outside the prism) and the normal to face AC.
Angle of Deviation: The angle of deviation (d) is defined as the angle between the incident ray (PQ) and the emerging ray (RS). It represents the total deviation of light as it passes through the prism.
Relation Between Angles: The angles of incidence and refraction at the first face AB satisfy the relationship:
- r1 + r2 = A, where A is the apex angle of the prism (angle between faces AB and AC).
Angle of Deviation and Incidence: The angle of deviation (d) is given by:
- d = i + e – A
Minimum Deviation: At the minimum deviation (Dm), the refracted ray inside the prism becomes parallel to the base of the prism. In this case:
- d = Dm
- i = e
- r1 = r2
Refractive Index of Prism: The refractive index (n) of the prism material can be determined using the angle of minimum deviation (Dm) and the apex angle (A) of the prism:
- n = (sin[(A + Dm)/2]) / (sin(A/2))
Thin Prism: For small-angle prisms or thin prisms, Dm is small, and you can approximate the formula as:
- Dm ≈ (n21 – 1)A

Microscope

In microscopy, a simple magnifier or microscope consists of a converging lens with a small focal length. Its purpose is to produce an enlarged, virtual, and erect image of an object. Here are the main points related to simple microscopes:

Design of Simple Microscope: A simple microscope typically consists of a converging lens. To use it effectively, the lens is held close to the object being observed, within one focal length or less, and the eye is positioned near the lens on the opposite side.
Image Formation: When the object is placed at a distance slightly less than the focal length of the lens, it forms a virtual image that is closer than infinity but still comfortable for viewing.
Linear Magnification (m): The linear magnification of a simple microscope can be calculated using the formula:
- m = D/f, where D is the near point of distinct vision (about 25 cm) and f is the focal length of the lens.
Angular Magnification: Angular magnification (M) can be defined as the ratio of the angle subtended by the image when observed through the microscope to the angle subtended by the object when viewed without the microscope.
Angular Magnification (M) When the Image is at Infinity: When the image is formed at infinity, the angular magnification (M) can be expressed as:
- M = θ/θo, where θ is the angle subtended by the image, and θo is the maximum angle subtended by the object at the near point.
Comparison Between Linear and Angular Magnification: The angular magnification (M) when the image is at infinity is usually one less than the linear magnification (m) when the image is formed at the near point.
Compound Microscope: For achieving larger magnifications, a compound microscope is used. It consists of two lenses: an objective lens that produces a real, inverted, and magnified image of the object and an eyepiece that further magnifies this image to make it visible to the observer.
Tube Length: The distance between the second focal point of the objective lens and the first focal point of the eyepiece lens is called the tube length (L).
Total Magnification of a Compound Microscope: The total magnification of a compound microscope is the product of the magnification due to the objective (m_o) and the magnification due to the eyepiece (m_e):
- Total Magnification (M_total) = m_o * m_e = (L/fo) * (D/fe), where fo is the focal length of the objective and fe is the focal length of the eyepiece.
Factors Influencing Magnification: To achieve a large magnification, both the objective and eyepiece lenses should have small focal lengths. However, this is limited by practical constraints in lens manufacturing.
Improving Image Quality: Modern microscopes use multi-component lenses for both the objective and eyepiece to enhance image quality and reduce optical aberrations.

Telescope

A telescope is an optical instrument used to observe distant objects and provide angular magnification. Telescopes are commonly used for astronomical observations, terrestrial observations, and various other applications. Here are the key points related to telescopes:

Components of a Telescope: A telescope typically consists of two main components:
- Objective Lens or Mirror: The objective is a large optical element with a long focal length and a large aperture. It is responsible for collecting and forming an initial real image of the distant object.
- Eyepiece: The eyepiece is a smaller optical element that further magnifies the real image formed by the objective, producing a final image that is visible to the observer.
Real Image Formation: When light from a distant object enters the objective of the telescope, it forms a real and inverted image near the second focal point of the objective.
Magnifying Power (m): The magnifying power of a telescope is defined as the ratio of the angle (b) subtended at the eye by the final image to the angle (a) subtended by the object at the lens or the eye. Mathematically, it can be expressed as:
- m = (fo / fe), where fo is the focal length of the objective and fe is the focal length of the eyepiece.
- The length of the telescope tube is usually the sum of the focal lengths of the objective and eyepiece (fo + fe).
Terrestrial Telescopes: Telescopes designed for terrestrial observations often include inverting lenses to make the final image erect.
Angular Magnification: Angular magnification is defined as the ratio of the angle subtended by the final image when observed through the telescope to the angle subtended by the object when viewed without the telescope.
Light Gathering Power: The light-gathering power of a telescope depends on the area of the objective. Larger-diameter objectives can collect more light and are capable of observing fainter objects.
Resolution (Resolving Power): The resolving power of a telescope is its ability to distinguish two closely spaced objects. It also depends on the diameter of the objective. Telescopes with larger objectives have better resolving power and can separate closely spaced objects.
Reflecting Telescopes: Many modern telescopes use concave mirrors for the objective rather than lenses. These telescopes are known as reflecting telescopes. They offer advantages such as reduced chromatic aberration and lighter weight.
Cassegrain Telescopes: Cassegrain telescopes are a type of reflecting telescope that uses a convex secondary mirror to focus the incident light. This design allows for a large focal length in a shorter telescope.
Large Telescopes: Large telescopes with large-diameter objectives are used in observatories worldwide for advanced astronomical research. Examples include the 2