Notes on Unit 1: Reflection and Refraction of Light

Unit 1: Reflection and Refraction of Light

1. Nature of Light

• Light is an electromagnetic radiation and a form of energy transfer from source to observer.
• Light shows a dual nature: acts like a particle in some experiments and like a wave in others.
• Historical perspectives:
  • Newton: particle theory of light (emission from objects into eyes).
  • Huygens: wave theory proposing light as wave motion.
  • Thomas Young (1801): demonstrated interference of light, supporting the wave model.
  • Maxwell (1865): electromagnetic wave theory; light as a high-frequency wave with fixed speed in vacuum, c.
• Speed of light in vacuum: $c \,\approx \,3 \times 10^{8} \ \mathrm{m\,s^{-1}}.$
• Einstein (1905): quantum model introducing photons; later work by Planck and Bohr refined this idea.
• Timeline activity mentioned: Physicists and contributions to light (dates: 1800, 1780, 1850, 1900, 1950, 1970).

2. Reflection and Refraction (Overview)

• Reflection and refraction occur at boundaries between transparent media.
• Reflection involves light returning into the original medium; refraction involves bending into the new medium due to a change in light speed.
• Focus areas: reflection at boundaries, refraction at boundaries, Snell’s Law, Huygens’s Principle, total internal reflection, and practical applications (e.g., fiber optics).

3. Reflection

• Definition: Reflection occurs when light meets a boundary and bounces back.
• Two major types:
  • Specular reflection: reflection from a smooth surface; reflected rays are parallel.
  • Examples: mirror, glass surface, calm water.
  • Diffuse reflection: reflection from a rough surface; reflected rays scatter in many directions.
  • Examples: bricks, wood, wall paint, skin.
• Law of Reflection:
  • The angle of incidence equals the angle of reflection when measured with respect to the normal.
  • Notation: $\theta<em>i = \theta</em>r$ where angles are measured from the normal, not the surface plane.
• Example types:
  • Simple incidence on a flat surface (e.g., 30° with the plane yields a reflected angle of 30° with the normal).
  • Two mirrors at right angles problem: determine outgoing directions after successive reflections.

4. Refraction and Snell’s Law

• Refraction: bending of light as it passes from one medium to another due to speed change.
• Why refraction occurs: difference in optical density (speed of light differs in media).
• Key relation between speed and refractive index:
  • Refractive index: $n = \frac{c}{v}$ where $c$ is the speed of light in vacuum and $v$ is the speed in the medium.
  • Light speed in medium: $v = \frac{c}{n}$
• Snell’s Law (law of refraction):
  • $n<em>1 \sin \theta</em>1 = n<em>2 \sin \theta</em>2$ where $\theta<em>1$ is the angle of incidence, $\theta</em>2$ is the angle of refraction, and $n<em>1, n</em>2$ are the refractive indices of the two media.
• Invariance of frequency across boundary:
  • Frequency remains constant when light crosses a boundary: $f<em>1 = f</em>2$.
  • Wavelength changes with medium because the wave speed changes while frequency stays fixed:
  • $\lambda<em>1 = \frac{v</em>1}{f}$ and $\lambda<em>2 = \frac{v</em>2}{f}$.
  • Therefore, $\frac{\lambda<em>1}{\lambda</em>2} = \frac{v<em>1}{v</em>2}$ and since $v = c/n$, \frac{\lambda1}{\lambda2} = \frac{n2}{n1}\n.
• Consequences of refraction:
  • If light enters a medium where speed is lower (higher n), the angle of refraction is smaller (bends toward the normal).
  • If light enters a medium where speed is higher (lower n), the angle of refraction is larger (bends away from the normal).
  • These behaviors reverse when light moves from lower to higher density (lower to higher n).

5. Huygens’s Principle

• Proposed by Huygens (1678) as a geometrical method to derive reflection and refraction laws.
• Core idea: every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is the tangent to these wavelets after time (\Delta t).
• It supports the wave-model view of light rather than the particle view.

6. Refractive Index (Table) and Notable Values (n)

• Common refractive indices (approximate, 20°C, 1 atm unless noted):
  • Air: $n \approx 1.000$
  • Water: $n \approx 1.333$
  • Diamond: $n \approx 2.419$
  • Glass (crown): $n \approx 1.52$
  • Glass (flint): $n \approx 1.66$
  • Fluorite: $n \approx 1.434$
  • Fused quartz (SiO₂): $n \approx 1.458$
  • Ethyl alcohol: $n \approx 1.361$
  • Glycerine: $n \approx 1.473$
  • Ice (H₂O) at 0°C: $n \approx 1.309$
  • Benzene: $n \approx 1.501$
  • Carbon disulfide: $n \approx 1.628$
  • Carbon tetrachloride: $n \approx 1.461$
  • Sodium chloride (NaCl): $n \approx 1.544$
  • Zircon: $n \approx 1.923$
  • Ammonia: not listed in the table but often around 1.000–1.35 depending on phase and wavelength
• Note on air index for gases: shown as 1.000 at 0°C, 1 atm in the table.

7. Frequency vs Wavelength Across Boundaries

• Frequency remains constant across media:
  • $f<em>1 = f</em>2.$
• Wavelength changes because speed changes:
  • $\lambda = \frac{v}{f}.$
• Because $v = \frac{c}{n}$, the wavelength ratio relates to refractive indices:
  • $\frac{\lambda<em>1}{\lambda</em>2} = \frac{v<em>1}{v</em>2} = \frac{n<em>2}{n</em>1}.$
• In short, crossing a boundary changes wavelength but not frequency.

8. Total Internal Reflection (TIR)

• Occurs when light attempts to move from a medium with a higher refractive index (denser) to a medium with a lower index (less dense) and hits the boundary at a sufficiently large incidence angle.
• Critical angle $\theta_c$ is defined by the condition that the refracted ray travels along the boundary (angle of refraction = 90°):
  • From Snell’s Law: $n<em>1 \sin \theta</em>c = n<em>2 \sin 90° = n</em>2.$
  • Therefore, $\theta<em>c = \arcsin\left(\frac{n</em>2}{n<em>1}\right)$, valid for n1 > n_2.
• For incidence angles greater than the critical angle, the light is totally internally reflected.
• Applications:
  • Fiber optics: light is guided through cores with total internal reflection, minimal loss.
  • Medical uses and telecommunications.
• Example problems (typical outcomes):
  • Example: Flint glass (n = 1.66) in air (n ≈ 1): $\theta_c = \arcsin \left(\frac{1}{1.66}\right) \approx 37^ ext{o}.$
  • Sound analog: Air (v ≈ 342 m/s) to concrete (v ≈ 1840 m/s) boundary yields:
  • $\sin \theta<em>c = \frac{v</em>{ ext{air}}}{v<em>{ ext{concrete}}} = \frac{342}{1840} \approx 0.186 \Rightarrow \theta</em>c \approx 10.7^ ext{o}.$
• Note: For TIR, the incident light must originate in the denser (slower) medium.

9. Worked Examples and Applications (Snell’s Law and TIR)

• Example: Water entry from air at 24.5° with refractive water index around 1.36.
  • Given: incident angle $\theta<em>1 = 24.5^ ext{o}$, $n</em>1 \approx 1.00$ (air), $n_2 \approx 1.36$ (water).
  • Compute $\theta<em>2$ using Snell’s Law: $\sin \theta</em>2 = \frac{n<em>1}{n</em>2} \sin \theta_1 = \frac{1}{1.36} \sin 24.5^ ext{o}.$
  • With $\sin 24.5^ ext{o} \approx 0.416$, we get $\sin \theta<em>2 \approx 0.306$, so $\theta</em>2 \approx 17.8^ ext{o}$.
• Example: A beam in air incident at 37° on a surface to a material with refracted angle 25°.
  • Using Snell’s Law to find the material’s refractive index: $n<em>2 = \frac{n</em>1 \sin \theta<em>1}{\sin \theta</em>2} = \frac{1.0 \times \sin 37^ ext{o}}{\sin 25^ ext{o}}.$
  • Since $\sin 37^ ext{o} \approx 0.6018$ and $\sin 25^ ext{o} \approx 0.4226$, $n_2 \approx \frac{0.6018}{0.4226} \approx 1.425.$
  • Speed in the material: $v<em>2 = \frac{c}{n</em>2} \approx \frac{3.00 \times 10^8}{1.425} \approx 2.11 \times 10^8\ \mathrm{m\,s^{-1}}.$
• Example: Coconut oil (n ≈ 1.48) and incidence 25.5° with a given diagram.
  • Method: apply Snell’s Law to determine corresponding angles at interfaces using the given refractive index.
• Example: An acrylic cube under water (n = 1.49) and total internal reflection at the top face: determine the incident angle for TIR.
  • Method: use the critical angle formula with n1 = 1.49 (acrylic) and n2 ≈ 1.33 (water) or the relative boundary as given to find the threshold for TIR.

10. Summary of Key Formulas

• Refractive index: $n = \frac{c}{v}$
• Speed in medium: $v = \frac{c}{n}$
• Snell’s Law: $n<em>1 \sin \theta</em>1 = n<em>2 \sin \theta</em>2$
• Frequency continuity: $f<em>1 = f</em>2$
• Wavelength relation across boundary (with constant frequency): $\frac{\lambda<em>1}{\lambda</em>2} = \frac{v<em>1}{v</em>2} = \frac{n<em>2}{n</em>1}$
• Wavelength: $\lambda = \frac{v}{f}$
• Total Internal Reflection: \thetac = \arcsin\left(\frac{n2}{n1}\right)\quad (n1 > n_2)
• Condition for TIR: $\theta<em>1 > \theta</em>c$

11. Applications of Internal Reflection

• Fiber optics: light transmitted with minimal loss along fibers; loss mainly at ends or due to absorption.
• Medical applications: endoscopy and diagnostic fiber-optic tools.
• Telecommunications: use of fiber cables for data transmission.

12. Quick Practice Problems (Representative Solutions)

• Problem: An optical fiber core (n1 = 1.55) and cladding (n2 = 1.42).
  • (a) Critical angle: $\theta<em>c = \arcsin\left(\frac{n</em>2}{n_1}\right) = \arcsin\left(\frac{1.42}{1.55}\right) \approx 66.5^ ext{o}.$
  • (b) If light tries to exit at angles greater than this, TIR occurs.
• Problem: Air-to-water interface: incident angle 24.5°, water speed v ≈ 2.22 × 10^8 m/s.
  • Water index: $n_2 = \frac{c}{v} = \frac{3.0 \times 10^8}{2.22 \times 10^8} \approx 1.36.$
  • Snell: $\sin \theta<em>2 = \frac{n</em>1}{n<em>2} \sin \theta</em>1 = \frac{1}{1.36} \sin 24.5^ ext{o} \approx 0.306 \Rightarrow \theta_2 \approx 17.8^ ext{o}.$
• Problem: Incident angle 37° from air into a block with refracted angle 25°.
  • Material index: $n<em>2 = \dfrac{n</em>1 \sin \theta<em>1}{\sin \theta</em>2} = \dfrac{1.0 \times \sin 37^ ext{o}}{\sin 25^ ext{o}} \approx \dfrac{0.6018}{0.4226} \approx 1.425.$
  • Speed in material: $v<em>2 = \dfrac{c}{n</em>2} \approx \dfrac{3.0 \times 10^8}{1.425} \approx 2.11 \times 10^8\ \mathrm{m\,s^{-1}}.$