class 12 notes

Class 12 Physics Notes Chapter 10 (Wave Optics) – Examplar Problems (English) Book

Philoid

Philoid

— 8 min read

Examplar Problems (English)
Alright class, let's delve into Chapter 10: Wave Optics. This is a fascinating chapter that explores light's behaviour not as rays, but as waves. Understanding these concepts is crucial for many competitive exams.

Wave Optics: Detailed Notes

1. Introduction: The Wave Nature of Light

  • While Geometrical Optics (ray optics) explains phenomena like reflection and refraction adequately on a macroscopic scale, it fails to explain phenomena like interference, diffraction, and polarisation.
  • Wave Optics treats light as an electromagnetic wave, propagating through space. Key proponents were Huygens, Young, Fresnel, and Maxwell.

2. Huygens' Principle

  • Concept: Explains how waves propagate.
  • Statement:
    • Every point on a given wavefront (called the primary wavefront) acts as a source of new disturbances, called secondary wavelets.
    • These secondary wavelets travel outwards in all directions with the speed of light in that medium.
    • The new wavefront at any later time is the forward envelope (tangential surface) of these secondary wavelets.
  • Wavefront: The locus of all points in a medium vibrating in the same phase.
    • Spherical Wavefront: From a point source. Intensity (I) ∝ 1/r². Amplitude (A) ∝ 1/r.
    • Cylindrical Wavefront: From a linear source. Intensity (I) ∝ 1/r. Amplitude (A) ∝ 1/√r.
    • Plane Wavefront: From a point or linear source at a very large distance. Intensity and Amplitude remain constant (ideally).
  • Applications:
    • Proof of Laws of Reflection: Using Huygens' construction, we can show that the angle of incidence (i) equals the angle of reflection (r), and the incident ray, reflected ray, and normal lie in the same plane.
    • Proof of Laws of Refraction (Snell's Law): Similarly, Huygens' principle leads to Snell's Law: sin(i) / sin(r) = v₁ / v₂ = n₂ / n₁ (where v is speed and n is refractive index). It also explains why light bends towards the normal when entering a denser medium (v₂ < v₁) and away from the normal when entering a rarer medium (v₂ > v₁).

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 individual displacements produced by each wave at that point. (y = y₁ + y₂)
  • Interference: The modification in the distribution of light intensity in the region of superposition of two or more waves.
  • Sources:
    • Coherent Sources: Sources that emit light waves having a constant phase difference (or zero phase difference) and the same frequency/wavelength. Essential for sustained interference. Usually derived from a single parent source (e.g., using slits).
    • Incoherent Sources: Sources with fluctuating phase differences. They produce illumination but not a sustained interference pattern. (e.g., two independent bulbs).
  • Conditions for Sustained Interference:
    1. Sources must be coherent.
    2. Sources must emit waves of the same frequency/wavelength.
    3. Waves should preferably have nearly equal amplitudes for good contrast between bright and dark fringes.
    4. Sources should be narrow.
    5. Sources should be close to each other.
  • Young's Double Slit Experiment (YDSE):
    • Setup: A monochromatic source illuminates a single slit S, which then illuminates two narrow, parallel slits S₁ and S₂ (acting as coherent sources). An interference pattern of alternate bright and dark fringes is observed on a screen placed at a distance D. Let 'd' be the distance between S₁ and S₂.
    • Path Difference (Δx): For a point P on the screen at a distance 'y' from the center O, the path difference Δx = S₂P - S₁P ≈ yd/D (for D >> d and D >> y).
    • Constructive Interference (Bright Fringes/Maxima): Occurs when Δx = nλ (where n = 0, ±1, ±2, ...).
      Position: y<0xE2><0x82><0x99> = nλD/d
    • Destructive Interference (Dark Fringes/Minima): Occurs when Δx = (n + 1/2)λ or (2n+1)λ/2 (where n = 0, ±1, ±2, ...).
      Position: y<0xE2><0x82><0x99> = (n + 1/2)λD/d = (2n+1)λD/2d
    • Fringe Width (β): The separation between two consecutive bright or dark fringes.
      β = y<0xE2><0x82><0x99>₊₁ - y<0xE2><0x82><0x99> = λD/d
      Fringe width is independent of the order 'n'. It is directly proportional to λ and D, and inversely proportional to d.
    • Intensity: If I₀ is the intensity from each slit, the resultant intensity I at a point with phase difference φ is I = 4I₀ cos²(φ/2). Phase difference φ = (2π/λ) * Δx.
      • Maxima: φ = 2nπ, I<0xE2><0x82><0x99><0xE2><0x82><0x90><0xE2><0x82><0x93> = 4I₀
      • Minima: φ = (2n+1)π, I<0xE2><0x82><0x98><0xE2><0x82><0x91><0xE2><0x82><0x99> = 0
    • Effect of White Light: Central fringe is white. Fringes on either side are coloured, with violet closest to the center and red farthest (since β ∝ λ).

4. Diffraction of Light

  • Phenomenon: The bending of light waves around the corners of obstacles or apertures and their consequent spreading into the regions of the geometrical shadow. Diffraction is significant when the size of the obstacle/aperture is comparable to the wavelength of light.
  • Difference from Interference: Interference is the superposition of waves from two (or more) distinct coherent sources. Diffraction is the superposition of secondary wavelets originating from different points of the same wavefront.
  • Diffraction at a Single Slit:
    • Setup: Monochromatic light passes through a narrow slit of width 'a'. A diffraction pattern of a central bright maximum, flanked by secondary minima and weaker secondary maxima, is observed on a screen at distance D.
    • Explanation: Each point on the wavefront within the slit acts as a source of secondary wavelets. These interfere to produce the pattern.
    • Condition for Minima: Path difference between wavelets from the edges of the slit is nλ.
      a sin θ = nλ (where n = ±1, ±2, ±3, ...)
      Position: y<0xE2><0x82><0x99> = nλD/a (using sin θ ≈ tan θ ≈ y/D for small θ)
    • Condition for Secondary Maxima (Approximate): Path difference is approximately (n + 1/2)λ.
      a sin θ ≈ (n + 1/2)λ (where n = ±1, ±2, ±3, ...)
      Position: y<0xE2><0x82><0x99> ≈ (n + 1/2)λD/a
    • Width of Central Maximum: The distance between the first minima on either side of the center.
      Angular width = 2θ₁ = 2λ/a (since sin θ₁ ≈ λ/a)
      Linear width = 2y₁ = 2λD/a
      The central maximum is twice as wide as the secondary maxima. Most of the light intensity is concentrated in the central maximum.
  • Resolving Power: The ability of an optical instrument to distinguish between two closely spaced objects.
    • Rayleigh's Criterion: Two images are just resolved when the central maximum of the diffraction pattern of one falls on the first minimum of the diffraction pattern of the other.
    • Telescope: Resolving Power = D / (1.22 λ), where D is the diameter of the objective lens/mirror. Limit of resolution (dθ) = 1.22 λ / D. Larger D gives better resolution.
    • Microscope: Resolving Power = (2μ sin θ) / (1.22 λ), where μ is the refractive index of the medium between object and objective, and θ is the half-angle of the cone of light from the object collected by the objective. (μ sin θ is the Numerical Aperture, NA). Limit of resolution (d) = 1.22 λ / (2μ sin θ). Using oil immersion (high μ) and shorter λ increases resolution.

5. Polarisation of Light

  • Concept: Demonstrates the transverse nature of light waves. Light waves are electromagnetic waves with electric (E) and magnetic (B) field vectors oscillating perpendicular to the direction of propagation and perpendicular to each other.
  • Unpolarised Light: Light in which the electric field vector oscillates randomly in all possible directions perpendicular to the direction of propagation (e.g., light from the sun, a bulb).
  • Polarised Light (Linearly Polarised): Light in which the electric field vector oscillates along a single direction perpendicular to the direction of propagation.
  • 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 (Brewster's angle, iₚ or θ<0xE1><0xB5><0xA_>) is such that the reflected and refracted rays are perpendicular to each other.
      Brewster's Law: μ = tan iₚ (where μ is the refractive index of the second medium relative to the first).
    • Polarisation by Scattering: When sunlight strikes air molecules, it gets scattered. The scattered light perpendicular to the direction of incidence is plane polarised. This explains why the sky appears blue (Rayleigh scattering: Intensity ∝ 1/λ⁴, blue light scatters more).
    • Polarisation by Dichroism (Polaroids): Certain crystals (like tourmaline) and synthetic materials (Polaroids) absorb light oscillating in one direction and transmit light oscillating perpendicular to that direction. A Polaroid has a 'pass axis'.
  • Malus' Law: When completely plane polarised light passes through a polariser (analyser), the intensity (I) of the transmitted light varies as the square of the cosine of the angle (θ) between the pass axis of the polariser and the plane of polarisation of the incident light.
    I = I₀ cos²θ (where I₀ is the maximum intensity of transmitted light when θ = 0°).
    • If unpolarised light of intensity I<0xE1><0xB5><0xA_> passes through a polariser, the transmitted intensity is I<0xE1><0xB5><0xA_>/2.
  • Plane of Vibration: The plane containing the direction of vibration (E-vector) and the direction of propagation.
  • Plane of Polarisation: The plane perpendicular to the plane of vibration.

6. Doppler Effect in Light

  • The apparent change in the frequency (and wavelength) of light due to the relative motion between the source of light and the observer.
  • Red Shift: Source moving away from the observer; apparent frequency decreases, apparent wavelength increases (shifts towards the red end of the spectrum). Δλ/λ ≈ v/c (for v << c).
  • Blue Shift: Source moving towards the observer; apparent frequency increases, apparent wavelength decreases (shifts towards the blue end of the spectrum). Δλ/λ ≈ -v/c (for v << c).
  • Used in astronomy to determine the speed of stars and galaxies.

Key Formulas Summary:

  • Huygens' Refraction: n₁ sin i = n₂ sin r
  • YDSE Path Difference: Δx ≈ yd/D
  • YDSE Bright Fringes: yd/D = nλ
  • YDSE Dark Fringes: yd/D = (n + 1/2)λ
  • YDSE Fringe Width: β = λD/d
  • Single Slit Minima: a sin θ = nλ
  • Single Slit Central Max Width: Linear = 2λD/a; Angular = 2λ/a
  • Telescope Resolution Limit: dθ = 1.22 λ / D
  • Microscope Resolution Limit: d = 1.22 λ / (2μ sin θ)
  • Brewster's Law: μ = tan iₚ
  • Malus' Law: I = I₀ cos²θ

Multiple Choice Questions (MCQs)

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

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

  3. The phenomenon of diffraction occurs when the size of the obstacle or aperture is:
    (a) Much larger than the wavelength of light
    (b) Much smaller than the wavelength of light
    (c) Comparable to the wavelength of light
    (d) Independent of the wavelength of light

  4. Two sources of light are said to be coherent if they emit light waves of:
    (a) Same amplitude and constant phase difference
    (b) Same frequency and constant phase difference
    (c) Same amplitude and same frequency
    (d) Different wavelengths and constant phase difference

  5. Brewster's angle (iₚ) for a transparent medium is related to its refractive index (μ) by the relation:
    (a) μ = sin iₚ
    (b) μ = cos iₚ
    (c) μ = tan iₚ
    (d) μ = cot iₚ

  6. In the diffraction pattern due to a single slit, the width of the central maximum is:
    (a) Equal to the width of secondary maxima
    (b) Half the width of secondary maxima
    (c) Twice the width of secondary maxima
    (d) Independent of the slit width

  7. When unpolarised light of intensity I₀ is passed through a polaroid, the intensity of the transmitted light is:
    (a) I₀
    (b) I₀/2
    (c) I₀/4
    (d) Zero

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

  9. Which phenomenon clearly demonstrates the transverse nature of light waves?
    (a) Reflection
    (b) Refraction
    (c) Interference
    (d) Polarisation

  10. In YDSE, if white light is used instead of monochromatic light, the central fringe will be:
    (a) White
    (b) Black
    (c) Coloured
    (d) Invisible

Answer Key for MCQs:

  1. (b)
  2. (b)
  3. (c)
  4. (b)
  5. (c)
  6. (c)
  7. (b)
  8. (a) (Resolving Power = D / (1.22 λ))
  9. (d)
  10. (a)

Make sure you understand the reasoning behind each answer. Go through these notes thoroughly, focusing on the conditions, formulas, and conceptual differences between interference, diffraction, and polarisation. Good luck with your preparation!

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