Chapter 13: Light and Optics

Overview

Optics is the study of light and its interaction with matter, including reflection, refraction, and the behavior of lenses and mirrors. This chapter covers fundamental optical principles that explain how we see, how cameras work, and how various optical instruments function. Understanding optics is essential for understanding vision, photography, astronomy, and many modern technologies.

Learning Objectives

After completing this chapter, you will be able to:

• Apply Snell's Law to refraction problems
• Understand and calculate total internal reflection
• Use lens formulas to determine image formation
• Analyze mirror types and their applications
• Explain optical instruments and their principles

Refraction of Light

Main Concept

Refraction is the bending of light when it passes from one medium to another with different optical densities.

Refraction Process Visualization

Key Principles

• Snell's Law: The ratio of sine of angle of incidence to sine of angle of refraction is a constant called refractive index.
• Light bends towards the normal when entering a denser medium
• Light bends away from the normal when entering a less dense medium
• Frequency remains constant during refraction

Key Formulas

Refractive Index, n:

$$n = \frac{c}{v}$$

Snell's Law:

$$n_1 \sin θ_1 = n_2 \sin θ_2$$

Real Depth and Apparent Depth:

$$n = \frac{\text{Real Depth}}{\text{Apparent Depth}} = \frac{D}{d}$$

Where:

• n = Refractive index
• c = Speed of light in vacuum (3 × 10⁸ m s⁻¹)
• v = Speed of light in medium
• θ₁ = Angle of incidence
• θ₂ = Angle of refraction

Refraction in Different Media

Important Terms

• Optical Density: Property of medium that determines speed of light in it
• Refractive Index: Measure of degree of light bending when entering a medium
• Normal: Imaginary line perpendicular to interface
• Critical Angle: Angle of incidence for which angle of refraction is 90°

Refraction Examples

Air to Glass:

• n_air ≈ 1.00, n_glass ≈ 1.50
• Light bends towards normal
• Speed decreases, wavelength decreases

Glass to Air:

• Light bends away from normal
• Speed increases, wavelength increases

Worked Example

Problem: Light passes from water (n = 1.33) to glass (n = 1.50) at an angle of 30° to the normal. Calculate the angle of refraction.

Solution:

• $n_1$ = 1.33 (water), $n_2$ = 1.50 (glass)
• θ₁ = 30°

Using Snell's Law:

$$n_1 \sin θ_1 = n_2 \sin θ_2$$ $$1.33 \sin 30° = 1.50 \sin θ_2$$ $$1.33 \times 0.5 = 1.50 \sin θ_2$$ $$\sin θ_2 = \frac{0.665}{1.50} = 0.4433$$ $$θ_2 = \sin^{-1}(0.4433) = 26.3°$$

Answer: Angle of refraction = 26.3°

Total Internal Reflection

Main Concept

Total internal reflection is the phenomenon where light is completely reflected within a denser medium when it tries to pass to a less dense medium at an angle greater than the critical angle.

Total Internal Reflection Process

Key Principles

• Two Conditions:
  1. Light must travel from denser to less dense medium
  2. Angle of incidence must be greater than critical angle (i > c)
• Critical Angle (c): Angle of incidence in denser medium where angle of refraction in less dense medium is 90°

Key Formulas

Critical Angle, c:

$$n = \frac{1}{\sin c}$$

Total Internal Reflection Condition:

$$\sin c = \frac{n_2}{n_1} \quad \text{where} \quad n_1 > n_2$$

Where:

• $n_1$ = Refractive index of denser medium
• $n_2$ = Refractive index of less dense medium
• $c$ = Critical angle

Critical Angles for Different Interfaces

Important Terms

• Optical Fibre: Thin tube of glass or plastic that uses total internal reflection to transmit light
• Mirage: Optical phenomenon caused by refraction and total internal reflection of light by different temperature air layers
• Prism: Transparent optical element with polished flat surfaces

Critical Angle Examples

Applications of Total Internal Reflection

Optical Fibres:

• Core: High refractive index material
• Cladding: Lower refractive index material
• Applications: Communication, medical endoscopes, sensors

Prisms:

• Reflecting Prisms: Use total internal reflection for image erection
• Dispersing Prisms: Separate white light into component colors

Atmospheric Optics:

• Mirages: Hot air creates layers with different refractive indices
• Rainbows: Refraction, dispersion, and internal reflection in water droplets

Images

Main Concept

Images are formed when light rays from an object either converge to or appear to diverge from a point.

Key Principles

• Real Image: Formed when light rays actually converge to a point. Can be formed on a screen. Inverted.
• Virtual Image: Formed when light rays appear to diverge from a point. Cannot be formed on a screen. Upright.

Important Terms

• Object: Source of light rays
• Image: What is seen after reflection or refraction
• Real Image: Actually formed by light convergence
• Virtual Image: Apparent divergence of light rays

Image Formation

Real vs Virtual Images:

Thin Lens Formula

Main Concept

The thin lens formula relates object distance (u), image distance (v), and focal length (f) for a lens.

Lens Types and Properties

Key Principles

• Convex Lens: Converges parallel light rays. Positive focal length.
• Concave Lens: Diverges parallel light rays. Negative focal length.
• Lens formula applies to both types with appropriate sign conventions

Key Formulas

Lens Formula:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Linear Magnification, m:

$$m = \frac{v}{u} \quad \text{or} \quad m = \frac{\text{Image height}}{\text{Object height}}$$

Sign Convention:

• u is always positive
• v is positive for real images, negative for virtual images
• f is positive for convex lenses, negative for concave lenses

Important Terms

• Focal Length (f): Distance from optical center to focal point
• Optical Centre: Point in center of lens where light passes through undeviated
• Focal Point: Point where parallel rays converge (convex) or appear to diverge (concave)

Lens Types and Applications

Image Formation by Lenses

Concave Lens Characteristics

Worked Example

Problem: A convex lens has focal length 15 cm. An object is placed 25 cm from the lens. Calculate: a) Image distance b) Magnification c) Image characteristics

Solution:

• f = 15 cm, u = 25 cm

a) Image distance:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$ $$\frac{1}{15} = \frac{1}{25} + \frac{1}{v}$$ $$\frac{1}{v} = \frac{1}{15} - \frac{1}{25} = \frac{5 - 3}{75} = \frac{2}{75}$$ $$v = 37.5 \text{ cm}$$

b) Magnification:

$$m = \frac{v}{u} = \frac{37.5}{25} = 1.5$$

c) Image characteristics:

• Real image (v positive)
• Inverted (m negative, but we take magnitude)
• Magnified (m > 1)

Answer: a) 37.5 cm, b) 1.5, c) Real, inverted, magnified

Optical Instruments

Main Concept

Optical instruments use lenses and mirrors to aid vision and enhance our ability to observe the world.

Optical Instruments Overview

Key Principles

• Magnifying Glass: Uses single convex lens. Object placed within focal length for virtual, upright, magnified image.
• Compound Microscope: Uses two convex lenses: objective lens (short focal length) and eyepiece lens (longer focal length). Final image is virtual, inverted, and highly magnified.
• Telescope: Uses two convex lenses: objective lens (long focal length) and eyepiece lens (short focal length). Final image is virtual, inverted, and magnified (at infinity).

Key Formulas

Telescope Magnification (Normal Adjustment):

$$M = \frac{f_o}{f_e}$$

Distance between Lenses (Normal Adjustment):

$$d = f_o + f_e$$

Where:

• f_o = Objective focal length
• f_e = Eyepiece focal length

Important Terms

• Objective Lens: Lens closest to object
• Eyepiece Lens: Lens closest to observer's eye
• Angular Magnification: Ratio of viewing angles

Optical Instrument Types

Microscope and Telescope Comparison

Spherical Mirrors

Main Concept

Spherical mirrors (concave and convex) form images through reflection of light rays.

Spherical Mirror Types

Key Principles

• Concave Mirror: Converges light. Can form real or virtual images.
• Convex Mirror: Diverges light. Always forms virtual, upright, diminished images.
• Mirror formula similar to lens formula with appropriate sign conventions

Key Formulas

Mirror Formula:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Relationship f and r:

$$f = \frac{r}{2}$$

Sign Convention:

• f is positive for concave mirrors, negative for convex mirrors
• v is positive for real images, negative for virtual images

Important Terms

• Centre of Curvature (C): Center of sphere from which mirror is cut
• Radius of Curvature (r): Distance from pole to centre of curvature
• Pole: Center of mirror surface
• Principal Axis: Line through pole and center of curvature

Mirror Image Formation

Mirror Applications

Applications of Optics

Real-World Applications

Modern Optical Technologies

Applications of Optics

Real-World Applications

1. Vision: Eyes as optical instruments
2. Photography: Cameras and lens systems
3. Medicine: Endoscopes, microscopes, lasers
4. Astronomy: Telescopes and spectrographs
5. Communication: Fibre optic networks

Everyday Optical Devices

Corrective Lenses:

• Convex Lenses: Farsightedness correction
• Concave Lenses: Nearsightedness correction

Display Technologies:

• LCD Screens: Liquid crystal optics
• OLED Displays: Organic light-emitting diodes
• VR/AR Headsets: Virtual and augmented reality

Scientific Instruments:

• Spectrometers: Analyze light composition
• Interferometers: Measure small distance changes
• Polarimeters: Measure optical rotation

SPM Exam Tips

Common Mistakes to Avoid

1. Sign Conventions: Remember positive and negative values for different situations
2. Lens vs Mirror: Don't confuse lens and mirror formulas
3. Image Characteristics: Determine real/virtual and upright/inverted correctly
4. Units: Use consistent units (usually centimeters or meters)

Problem-Solving Strategies

1. Identify Components: Determine lens or mirror type and setup
2. Apply Sign Convention: Use consistent sign conventions
3. Use Appropriate Formula: Choose lens or mirror formula as needed
4. Verify Results: Check if image characteristics make sense

Important Formula Summary

Summary

This chapter covered essential optics concepts:

• Refraction: Light bending at interfaces
• Total Internal Reflection: Complete reflection at critical angle
• Lenses: Image formation by converging and diverging lenses
• Mirrors: Reflection and image formation
• Optical Instruments: Microscopes, telescopes, and vision

Master these concepts to understand how we see, how cameras work, and how optical technologies function in our daily lives.

Practice Questions

1. Light travels from glass (n = 1.5) to air at an angle of incidence of 40°. Calculate the angle of refraction.

2. A convex lens has focal length 20 cm. An object is placed 30 cm from the lens. Calculate: a) Image distance b) Magnification c) Image characteristics

3. Explain the difference between real and virtual images, giving one example of each.

4. Calculate the critical angle for light traveling from water (n = 1.33) to air.

5. Describe how a compound microscope works and name its main components.

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