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Class 12 Physics Notes Chapter 2 (Wave Optics) – Physics Part-II Book

Philoid

Philoid

— 8 min read

Physics Part-II
Alright class, let's dive into Chapter 10: Wave Optics. We've seen light behave like a ray in the previous chapter, but many phenomena can only be explained by considering light as a wave. This chapter is crucial, especially the concepts of interference, diffraction, and polarisation, which frequently appear in competitive exams. Pay close attention to the principles and formulas.

Wave Optics: Detailed Notes for Government Exam Preparation

1. Introduction to Wave Nature of Light

  • Light exhibits dual nature: wave and particle. Wave optics deals with phenomena explained by considering light as a wave (e.g., interference, diffraction, polarisation).
  • Wavefront: The locus of all points in a medium vibrating in the same phase.
    • Spherical Wavefront: Source is a point source.
    • Cylindrical Wavefront: Source is a linear source (like a slit).
    • Plane Wavefront: Source is at a very large distance (effectively infinity). Rays are perpendicular to the wavefront.

2. Huygens' Principle (Foundation of Wave Optics)

  • Statement:
    1. Every point on a given wavefront (called the primary wavefront) acts as a fresh source of new disturbances, called secondary wavelets, which travel in all directions with the speed of light in the medium.
    2. The surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new position of the wavefront at that instant (the secondary wavefront).
  • Applications:
    • Proof of Laws of Reflection: Using Huygens' construction on a reflecting surface, we can show that the angle of incidence (i) equals the angle of reflection (r). (∠i = ∠r).
    • Proof of Laws of Refraction (Snell's Law): Using Huygens' construction on an interface between two media, we can derive Snell's law:
      • sin i / sin r = v₁ / v₂ = n₂ / n₁ = ¹n₂
      • Where v₁, v₂ are speeds of light in medium 1 and 2, and n₁, n₂ are their refractive indices. This also shows that light bends towards the normal when entering a denser medium (v₂ < v₁, so r < i) and vice-versa. Frequency (ν) remains unchanged during refraction (λ = v/ν).

3. Interference of Light

  • Principle of Superposition: When two or more waves overlap in a medium, the resultant displacement at any point is the vector sum of the displacements due to individual waves. y = y₁ + y₂ + ...
  • Coherent Sources: Sources that emit light waves of the same frequency (or wavelength), same amplitude (preferably), and have a constant phase difference between them. Interference patterns are stable only if sources are coherent.
    • Laser light is highly coherent.
    • Two independent sources (like bulbs) are incoherent.
    • Coherent sources are usually derived from a single parent source (e.g., using a double slit).
  • Interference: The modification in the distribution of light intensity in the region of superposition of two or more coherent waves.
  • Young's Double Slit Experiment (YDSE):
    • Setup: A monochromatic light source illuminates two narrow, parallel slits (S₁ and S₂) close to each other. An interference pattern of alternate bright and dark bands (fringes) is observed on a screen placed some distance away.
    • Path Difference (Δx): For a point P on the screen at a distance 'y' from the center O, the path difference between waves from S₁ and S₂ is Δx = S₂P - S₁P.
      • For D >> d and D >> y: Δx ≈ y d / D
      • Where d = distance between slits, D = distance from slits to screen.
    • Phase Difference (φ): φ = (2π/λ) * Δx = (2π/λ) * (y d / D)
    • Constructive Interference (Bright Fringes / Maxima):
      • Condition: Path difference Δx = nλ (where n = 0, ±1, ±2, ...)
      • Condition: Phase difference φ = 2nπ
      • Position on screen: y_n = n λ D / d
    • Destructive Interference (Dark Fringes / Minima):
      • Condition: Path difference Δx = (n + 1/2)λ or (2n - 1)λ/2 (where n = 0, ±1, ±2,... or n=1, 2, 3... respectively)
      • Condition: Phase difference φ = (2n + 1)π
      • Position on screen: y_n = (n + 1/2) λ D / d or (2n - 1)λ D / 2d
    • Fringe Width (β): The separation between two consecutive bright or dark fringes.
      • β = y_(n+1) - y_n = λ D / d
      • Fringe width is directly proportional to λ and D, and inversely proportional to d.
      • All fringes are of equal width.
    • Intensity Distribution: If I₁ and I₂ are intensities of light from the two slits, the resultant intensity I at a point with phase difference φ is:
      • I = I₁ + I₂ + 2√(I₁I₂) cos φ
      • If I₁ = I₂ = I₀, then I = 2I₀ (1 + cos φ) = 4I₀ cos²(φ/2)
      • Maximum Intensity (I_max): (√I₁ + √I₂)². If I₁=I₂=I₀, I_max = 4I₀ (at cos φ = +1).
      • Minimum Intensity (I_min): (√I₁ - √I₂)². If I₁=I₂=I₀, I_min = 0 (at cos φ = -1).
      • Ratio: I_max / I_min = [(√I₁ + √I₂)/(√I₁ - √I₂)]² = [(A₁+A₂)/(A₁-A₂)]² (where A is amplitude).
    • Effect of White Light: Central fringe is white. Fringes on either side are coloured (violet closest to the center, red farthest). Fringe pattern disappears after a few fringes due to overlapping.
    • Effect of Inserting a Thin Sheet: If a transparent sheet of thickness 't' and refractive index 'μ' is placed in the path of one beam, the optical path increases by (μ-1)t. The entire fringe pattern shifts by Δy = (D/d)(μ-1)t towards the side where the sheet is placed. Fringe width β remains unchanged.

4. Diffraction of Light

  • Definition: The phenomenon of bending of light waves around the corners of obstacles or apertures and spreading into the regions of the geometrical shadow.

  • Condition: Size of the obstacle/aperture should be comparable to the wavelength of light (a ≈ λ).

  • Diffraction at a Single Slit:

    • When monochromatic light passes through a narrow single slit, a diffraction pattern is observed on a screen: a broad central bright maximum, flanked by weaker secondary maxima and minima on both sides.
    • Explanation: Due to superposition of secondary wavelets originating from different points within the single slit.
    • Condition for Minima: Path difference between wavelets from the edges of the slit is a sin θ = nλ (where n = ±1, ±2, ±3, ...). θ is the angle of diffraction.
    • Condition for Secondary Maxima (Approximate): a sin θ = (n + 1/2)λ (where n = ±1, ±2, ±3, ...).
    • Central Maximum: Lies between the first minima on either side (n=1 and n=-1).
      • Angular width: Δθ = 2θ₁ = 2λ / a (since for small θ, sin θ ≈ θ).
      • Linear width: W = 2 y₁ = 2 (λD / a).
      • Its width is twice that of any secondary maximum. Its intensity is maximum.
    • Intensity Distribution: Most of the light intensity is concentrated in the central maximum. Intensities of secondary maxima decrease rapidly (I₁ ≈ I₀/22, I₂ ≈ I₀/61, ...).

  • Comparison between Interference and Diffraction:

    Feature Interference (YDSE) Diffraction (Single Slit)
    Cause Superposition of waves from 2 coherent sources Superposition of wavelets from different parts of the same wavefront
    Fringe Width All fringes are of equal width (β) Central max is twice as wide as secondary maxima
    Intensity All bright fringes have same intensity (ideally) Intensity falls rapidly for higher order maxima
    Minima Intensity is usually zero (perfectly dark) Intensity is minimum (not perfectly zero)
    Number of Fringes Large number usually visible Fewer maxima are usually visible

  • Resolving Power: The ability of an optical instrument to distinguish between two closely spaced objects or spectral lines.

    • Rayleigh Criterion: Two images are said to be just resolved if the central maximum of the diffraction pattern of one coincides with the first minimum of the diffraction pattern of the other.
    • Limit of Resolution (Δθ): The minimum angular separation between two objects so that they are just resolved. Smaller the limit of resolution, higher the resolving power.
    • Resolving Power of a Microscope: R.P. = 1 / d_min = 2μ sin θ / λ
      • d_min = minimum resolvable distance between objects.
      • μ = refractive index of medium between object and objective.
      • θ = semi-vertical angle of the cone of light entering the objective.
      • μ sin θ = Numerical Aperture (NA). R.P. increases with NA and decreases with λ. Use oil immersion objective and shorter wavelength light (e.g., UV) to increase resolution.
    • Resolving Power of a Telescope: R.P. = 1 / Δθ = D / (1.22 λ)
      • Δθ = limit of resolution (minimum angular separation).
      • D = diameter (aperture) of the objective lens/mirror.
      • R.P. increases with aperture D and decreases with λ. Larger objective lenses give better resolution.

5. Polarisation of Light

  • Transverse Nature: Polarisation demonstrates that light waves are transverse (vibrations are perpendicular to the direction of propagation). Sound waves (longitudinal) cannot be polarised.
  • Unpolarised Light: Light in which vibrations occur randomly in all possible directions perpendicular to the direction of propagation (e.g., light from sun, bulb). Represented by arrows and dots.
  • Plane Polarised Light (or Linearly Polarised Light): Light in which vibrations are confined to a single plane containing the direction of propagation.
    • Plane of Vibration: The plane containing the electric field vector and the direction of propagation.
    • Plane of Polarisation: The plane perpendicular to the plane of vibration.
  • Methods of Polarisation:
    • Polarisation by Reflection (Brewster's Law): When unpolarised light is incident on the boundary between two transparent media, the reflected light is completely plane polarised if the angle of incidence (i_p, polarising angle or Brewster's angle) is such that:
      • tan i_p = μ = n₂ / n₁ (Brewster's Law)
      • At this angle, the reflected and refracted rays are perpendicular to each other (i_p + r = 90°). The vibrations in the reflected light are perpendicular to the plane of incidence.
    • Polarisation by Scattering: When sunlight strikes air molecules, it gets scattered. Light scattered in a direction perpendicular to the direction of incident light is plane polarised. This explains why the sky appears blue (Rayleigh scattering: intensity ∝ 1/λ⁴, blue light scatters more).
    • Polarisation by Selective Absorption (Dichroism): Certain crystals (like tourmaline) absorb light vibrations along one direction more strongly than along the perpendicular direction.
    • Polaroids: Artificial polarising films based on dichroism (using long-chain molecules aligned parallel). Used in sunglasses, camera filters, LCD screens, 3D movie glasses.
      • Polariser: Produces plane polarised light from unpolarised light.
      • Analyser: Used to detect/analyse plane polarised light.
  • Malus' Law: When plane polarised light is incident on an analyser, the intensity I of the transmitted light varies with the angle θ between the transmission axes of the polariser and the analyser as:
    • I = I₀ cos²θ
    • Where I₀ is the maximum intensity transmitted (when θ = 0° or 180°).
    • Intensity is zero when θ = 90° (crossed polaroids).
    • If unpolarised light of intensity I_un falls on a polariser, the intensity of transmitted polarised light is I₀ = I_un / 2.

6. Doppler Effect in Light

  • Apparent change in frequency (or wavelength) of light due to relative motion between the source and the observer.
  • Red Shift: Wavelength increases (frequency decreases) when the source moves away from the observer. Used to infer that distant galaxies are moving away from us (expanding universe). Δλ/λ ≈ +v/c.
  • Blue Shift: Wavelength decreases (frequency increases) when the source moves towards the observer. Δλ/λ ≈ -v/c.
    • v = relative speed, c = speed of light. (Formula valid for v << c).

Multiple Choice Questions (MCQs)

  1. According to Huygens' principle, light is a form of:
    a) Particle
    b) Ray
    c) Wave
    d) Radiation

  2. In Young's double-slit experiment, the fringe width is given by β = λD/d. If the distance between the slits (d) is halved and the distance between the slits and screen (D) is doubled, the new fringe width will be:
    a) β/2
    b) β
    c) 2β
    d) 4β

  3. For constructive interference to occur between two monochromatic light waves of wavelength λ, the path difference should be:
    a) (n + 1/2)λ
    b) nλ
    c) (2n + 1)λ
    d) nλ/2
    (where n = 0, 1, 2, ...)

  4. The phenomenon of bending of light around the corners of small obstacles is called:
    a) Reflection
    b) Refraction
    c) Interference
    d) Diffraction

  5. In the diffraction pattern due to a single slit of width 'a' with light of wavelength λ, the first minimum is observed at an angle θ such that:
    a) a sin θ = λ/2
    b) a sin θ = λ
    c) a sin θ = 3λ/2
    d) a sin θ = 2λ

  6. The resolving power of a telescope increases when:
    a) The wavelength of light decreases
    b) The wavelength of light increases
    c) The diameter of the objective lens decreases
    d) The focal length of the objective lens increases

  7. Polarisation of light proves the:
    a) Corpuscular nature of light
    b) Longitudinal nature of light
    c) Transverse nature of light
    d) Quantum nature of light

  8. According to Brewster's law, when light is incident at the polarising angle i_p, the angle between the reflected and refracted rays is:
    a) 0°
    b) 90°
    c) 180°
    d) Depends on the medium

  9. Unpolarised light of intensity I₀ is passed through a polariser. The intensity of the emerging polarised light will be:
    a) I₀
    b) I₀/2
    c) I₀/4
    d) 0

  10. If plane polarised light of intensity I₁ passes through an analyser such that the angle between the transmission axes of the polariser and analyser is 60°, the intensity of the emerging light I₂ will be:
    a) I₁
    b) I₁/2
    c) I₁/4
    d) I₁/8

Answers to MCQs:

  1. c) Wave
  2. d) 4β (New β' = λ(2D)/(d/2) = 4 λD/d = 4β)
  3. b) nλ
  4. d) Diffraction
  5. b) a sin θ = λ (for first minimum, n=1)
  6. a) The wavelength of light decreases (R.P. = D / (1.22 λ))
  7. c) Transverse nature of light
  8. b) 90°
  9. b) I₀/2
  10. c) I₁/4 (Using Malus' Law: I₂ = I₁ cos²(60°) = I₁ (1/2)² = I₁/4)

Remember to focus on the conditions for maxima and minima in both interference and diffraction, the formula for fringe width, factors affecting resolving power, Brewster's law, and Malus' law. These are high-yield areas for exams. Good luck with your preparation!

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