Waves and Optics

Welcome to our comprehensive lesson on Waves and Optics for the CXC/CSEC Physics syllabus. This topic is essential for understanding how light and other waves behave in our physical world.

Introduction to Waves

Waves are disturbances that transfer energy from one place to another without transferring matter. They are fundamental to our understanding of many physical phenomena including light, sound, and even quantum mechanics.

Types of Waves

Mechanical Waves: Require a medium to travel through (e.g., sound waves)
Electromagnetic Waves: Can travel through vacuum (e.g., light waves)

Wave Classification by Direction

Transverse Waves: Particles move perpendicular to the direction of wave propagation
Longitudinal Waves: Particles move parallel to the direction of wave propagation

Wave Properties

Basic Wave Parameters

Wavelength (λ): Distance between two consecutive corresponding points on a wave
Frequency (f): Number of complete waves passing a point per second (measured in Hertz, Hz)
Amplitude (A): Maximum displacement from equilibrium position
Period (T): Time taken for one complete wave cycle
Wave Speed (v): Distance traveled by a wave per unit time

Wave Speed Equation: v = f λ

Where:

v = wave speed (m/s)
f = frequency (Hz)
λ = wavelength (m)

Period and Frequency Relationship: T = 1/f

Where:

T = period (s)
f = frequency (Hz)

Wave Behavior

Reflection: When waves bounce off a boundary
Refraction: When waves change direction as they pass from one medium to another
Diffraction: When waves spread out after passing through a gap or around an obstacle
Interference: When two or more waves overlap and combine

The Electromagnetic Spectrum

Electromagnetic waves are waves that can travel through a vacuum. They include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

All electromagnetic waves travel at the same speed in a vacuum: 3 × 10⁸ m/s (the speed of light).

Light and Optics

Optics is the branch of physics that studies the behavior and properties of light, including its interactions with matter and the instruments used to detect it.

Reflection of Light

When light hits a surface, it can be reflected. There are two types of reflection:

Regular/Specular Reflection: Light rays are reflected at the same angle (smooth surfaces like mirrors)
Diffuse Reflection: Light rays are scattered in many directions (rough surfaces)

Law of Reflection: Angle of incidence = Angle of reflection

Where:

Angle of incidence (i): angle between incident ray and normal
Angle of reflection (r): angle between reflected ray and normal

Refraction of Light

Refraction occurs when light passes from one medium to another, causing a change in direction due to the change in speed.

Snell's Law: n₁sin(θ₁) = n₂sin(θ₂)

Where:

n₁: refractive index of first medium
n₂: refractive index of second medium
θ₁: angle of incidence
θ₂: angle of refraction

Total Internal Reflection

Total internal reflection occurs when light traveling in a denser medium hits the boundary with a less dense medium at an angle greater than the critical angle.

Critical Angle: sin(θc) = n₂/n₁

Where:

θc: critical angle
n₁: refractive index of denser medium
n₂: refractive index of less dense medium

Example: Calculate the Critical Angle

When light travels from water (n = 1.33) to air (n = 1.00), what is the critical angle?

Solution:

Using the formula: sin(θc) = n₂/n₁

sin(θc) = 1.00/1.33 = 0.752

θc = sin⁻¹(0.752) = 48.6°

Therefore, the critical angle is 48.6°. Any light ray incident at an angle greater than this will undergo total internal reflection.

Lenses

Lenses are transparent objects that refract light to form images. There are two main types:

Convex (Converging) Lenses: Thicker in the middle than at the edges; converge light rays
Concave (Diverging) Lenses: Thinner in the middle than at the edges; diverge light rays

Lens Formula: 1/f = 1/u + 1/v

Where:

f: focal length of the lens
u: object distance from the lens
v: image distance from the lens

Magnification: m = -v/u = h'/h

Where:

m: magnification
v: image distance
u: object distance
h': height of image
h: height of object

Ray Diagrams for Lenses

Three principal rays are used to locate the image formed by a lens:

A ray parallel to the principal axis, which passes through the focal point after refraction
A ray passing through the optical center of the lens, which continues undeviated
A ray passing through the focal point before refraction, which emerges parallel to the principal axis

Optical Instruments

Various optical instruments use lenses and mirrors to observe objects:

Simple Microscope: Uses a single convex lens to magnify small objects
Compound Microscope: Uses multiple lenses to achieve higher magnification
Telescope: Uses lenses or mirrors to view distant objects
Camera: Uses a convex lens to form a real image on film or digital sensor
Human Eye: A natural optical system with a convex lens that forms a real image on the retina

Wave Phenomena

Diffraction

Diffraction is the bending of waves around obstacles or through openings. The amount of diffraction depends on the wavelength of the wave and the size of the obstacle or opening.

The degree of diffraction increases as:

The wavelength increases
The size of the opening decreases

Interference

Interference occurs when two waves meet and combine. There are two types:

Constructive Interference: Waves are in phase and amplitudes add up
Destructive Interference: Waves are out of phase and amplitudes subtract

Young's Double-Slit Experiment

This famous experiment demonstrated the wave nature of light. When light passes through two closely spaced slits, an interference pattern of bright and dark bands appears on a screen.

For constructive interference (bright fringes): d sin θ = mλ

For destructive interference (dark fringes): d sin θ = (m + 1/2)λ

Where:

d: distance between slits
θ: angle to the fringe
m: order of fringe (0, 1, 2, ...)
λ: wavelength of light

Doppler Effect

The Doppler effect is the change in frequency of a wave in relation to an observer who is moving relative to the wave source.

When the source moves toward the observer, the observed frequency increases (wavelength decreases)
When the source moves away from the observer, the observed frequency decreases (wavelength increases)

For a moving source (approaching observer): f' = f × (v/(v-vs))

For a moving source (receding from observer): f' = f × (v/(v+vs))

Where:

f': observed frequency
f: emitted frequency
v: speed of waves in the medium
vs: speed of the source

Polarization

Polarization is a property of transverse waves where the vibrations occur in a single plane. Light can be polarized by:

Reflection: Light reflected from non-metallic surfaces becomes partially polarized
Selective Absorption: Some materials (polarizing filters) only allow vibrations in one direction to pass through
Scattering: Light scattered by small particles can become partially polarized

Applications of Waves and Optics

Fiber Optics

Fiber optic cables use the principle of total internal reflection to transmit light signals over long distances with minimal loss.

Medical Applications

Endoscopy: Using fiber optics to view inside the body
Laser Surgery: Using focused light to cut or destroy tissue
X-ray Imaging: Using high-energy electromagnetic waves to see inside the body

Communications

Radio and Television: Using electromagnetic waves to transmit information
Mobile Phones: Using microwaves for wireless communication
Internet: Using fiber optics for high-speed data transmission

Glossary of Terms

Amplitude: The maximum displacement of a point on a wave from its equilibrium position.
Angle of Incidence: The angle between the incident ray and the normal to the surface.
Angle of Reflection: The angle between the reflected ray and the normal to the surface.
Angle of Refraction: The angle between the refracted ray and the normal to the surface.
Concave Lens: A lens that is thinner in the center than at the edges, causing light rays to diverge.
Constructive Interference: When two waves combine to form a wave with larger amplitude.
Convex Lens: A lens that is thicker in the center than at the edges, causing light rays to converge.
Critical Angle: The angle of incidence beyond which total internal reflection occurs.
Destructive Interference: When two waves combine to form a wave with smaller amplitude.
Diffraction: The bending of waves around obstacles or through openings.
Doppler Effect: The change in frequency of a wave in relation to an observer moving relative to the wave source.
Electromagnetic Spectrum: The range of all types of electromagnetic radiation, from radio waves to gamma rays.
Focal Length: The distance from the center of a lens or mirror to the focal point.
Focal Point: The point at which parallel rays of light converge after passing through a convex lens or reflecting from a concave mirror.
Frequency: The number of complete wave cycles passing a point per second, measured in Hertz (Hz).
Interference: The phenomenon that occurs when two or more waves meet and combine.
Longitudinal Wave: A wave in which the particles of the medium move parallel to the direction of wave propagation.
Polarization: The property of waves that describes the orientation of their oscillations.
Reflection: The change in direction of a wave at an interface between two different media so that the wave returns into the medium from which it originated.
Refraction: The change in direction of a wave passing from one medium to another caused by its change in speed.
Refractive Index: A dimensionless number that describes how fast light travels through a material.
Total Internal Reflection: The complete reflection of a light ray reaching an interface with a less dense medium when the angle of incidence exceeds the critical angle.
Transverse Wave: A wave in which the particles of the medium move perpendicular to the direction of wave propagation.
Wavelength: The distance between two successive points in phase on a wave, such as two adjacent crests or troughs.

Self-Assessment Questions

1. A light ray passes from air (n = 1.00) into water (n = 1.33). If the angle of incidence is 30°, what is the angle of refraction?

Using Snell's Law: n₁sin(θ₁) = n₂sin(θ₂)
1.00 × sin(30°) = 1.33 × sin(θ₂)
sin(θ₂) = (1.00 × sin(30°)) ÷ 1.33
sin(θ₂) = 0.5 ÷ 1.33 = 0.376
θ₂ = sin⁻¹(0.376) = 22.1°
Therefore, the angle of refraction is 22.1°.

2. Calculate the wavelength of a wave with a frequency of 5 MHz traveling at a speed of 300 m/s.

Using the wave equation: v = f λ
Rearranging to find wavelength: λ = v/f
λ = 300 m/s ÷ 5,000,000 Hz
λ = 0.00006 m = 0.06 mm
Therefore, the wavelength is 0.06 mm or 6 × 10⁻⁵ m.

3. A convex lens has a focal length of 15 cm. If an object is placed 40 cm from the lens, where will the image be formed?

Using the lens formula: 1/f = 1/u + 1/v
1/15 = 1/40 + 1/v
1/v = 1/15 - 1/40
1/v = (40 - 15)/(15 × 40)
1/v = 25/600
v = 600/25 = 24 cm
Therefore, the image will be formed 24 cm from the lens.

4. Explain why a blue sky appears during the day while sunsets often appear red.

The blue sky during the day is caused by Rayleigh scattering. Sunlight contains all colors of the visible spectrum, but blue light has a shorter wavelength and is scattered more efficiently by air molecules in the atmosphere. This scattered blue light reaches our eyes from all directions in the sky.

During sunset, sunlight must travel through more of the atmosphere to reach our eyes. The shorter wavelengths (blue, violet) get scattered away from our line of sight, leaving primarily the longer wavelengths (red, orange) to reach our eyes directly, causing the reddish appearance of the sun and surrounding sky during sunset.

5. What is the critical angle for light traveling from diamond (n = 2.42) to air (n = 1.00)?

Using the formula for critical angle: sin(θc) = n₂/n₁
sin(θc) = 1.00/2.42
sin(θc) = 0.413
θc = sin⁻¹(0.413) = 24.4°
Therefore, the critical angle is 24.4°. This explains why diamonds sparkle; most light hitting the internal surfaces at greater than 24.4° is totally internally reflected.

6. Two coherent light sources produce an interference pattern. The distance between adjacent bright fringes on a screen is 1.5 mm. If the wavelength of light is 600 nm and the screen is 3 m from the sources, what is the distance between the two sources?

For small angles, the fringe spacing y is given by: y = λL/d
Where λ is the wavelength, L is the distance to the screen, and d is the separation between sources.
Rearranging to find d: d = λL/y
d = (600 × 10⁻⁹ m × 3 m) ÷ (1.5 × 10⁻³ m)
d = 1.8 × 10⁻⁶ m ÷ 1.5 × 10⁻³ m
d = 1.2 × 10⁻³ m = 1.2 mm
Therefore, the distance between the two sources is 1.2 mm.

7. A police car's siren emits a frequency of 500 Hz. If the car is moving toward a stationary observer at 20 m/s, what frequency will the observer hear? (Speed of sound in air = 340 m/s)

Using the Doppler effect formula for a source approaching an observer:
f' = f × (v/(v-vs))
f' = 500 Hz × (340 m/s/(340 m/s - 20 m/s))
f' = 500 Hz × (340/320)
f' = 500 Hz × 1.0625
f' = 531.25 Hz
Therefore, the observer will hear a frequency of 531.25 Hz, which is higher than the emitted frequency.

8. Explain why a spoon appears to bend when placed in a glass of water.

A spoon appears to bend at the water's surface due to refraction. Light rays from the part of the spoon underwater travel from water (higher refractive index) to air (lower refractive index). When these light rays cross the boundary, they bend away from the normal according to Snell's Law.

When our eyes trace these refracted rays back, they do so in straight lines, creating the illusion that the spoon is bent or displaced from its actual position. This is the same principle that makes objects underwater appear closer to the surface than they actually are.

9. If white light passes through a prism, it separates into different colors. Explain why this happens and state which color deviates the most.

White light separates into different colors when passing through a prism due to dispersion. Dispersion occurs because different wavelengths of light travel at different speeds through the prism material, causing them to refract at different angles.

Each color in white light has a different wavelength, with violet having the shortest wavelength and red having the longest. The refractive index of the prism material varies with wavelength (called dispersion). Shorter wavelengths (violet and blue) have a higher refractive index and thus bend more than longer wavelengths (orange and red).

Therefore, violet light deviates the most when passing through a prism, while red light deviates the least.

10. A polarizing filter is placed in front of an unpolarized light source, and a second polarizing filter is placed after it. If the transmission axes of the two filters are at 45° to each other, what percentage of the original light intensity will be transmitted through both filters?

According to Malus' Law, when polarized light passes through a polarizing filter, the intensity of transmitted light is given by:
I = I₀cos²θ, where θ is the angle between the transmission axes of the polarizers.

When unpolarized light passes through the first polarizer, its intensity is reduced to 50% of the original intensity.
I₁ = 0.5 I₀

For the second polarizer at 45° to the first:
I₂ = I₁cos²(45°)
I₂ = 0.5 I₀ × cos²(45°)
I₂ = 0.5 I₀ × 0.5
I₂ = 0.25 I₀

Therefore, 25% of the original light intensity will be transmitted through both filters.