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Wave Optics

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Chapter Notes

Overview
This chapter introduces the wave theory of light, contrasting it with the earlier corpuscular model. It explains how Huygens' principle can be used to derive the laws of reflection and refraction, and delves into the phenomena of interference, diffraction, and polarisation. The central idea a student must take away is that light is a transverse electromagnetic wave, and its wave nature is responsible for effects like interference and diffraction that cannot be explained by ray optics alone.
Key Concepts (14)
Wavefront
A surface of constant phase; the locus of all points oscillating in phase.
e.g. When a stone is dropped in a calm pool, the circular rings on the water surface are wavefronts.
Huygens Principle
Each point on a wavefront is the source of a secondary disturbance (secondary wavelets) that spread out in all directions with the speed of the wave. The new wavefront at a later time is the forward envelope (common tangent) of these secondary wavelets.
e.g. Using Huygens principle to determine the shape of a plane wavefront after time t by drawing spheres of radius vt from each point on the initial wavefront.
Coherent sources
Two sources that have the same frequency and a constant phase difference (i.e., a stable phase relationship) with respect to time.
e.g. Two needles S1 and S2 moving periodically up and down in an identical fashion in a trough of water produce coherent water waves.
Interference
The phenomenon of superposition of two or more waves from coherent sources, resulting in a redistribution of energy in space, producing alternate bright (constructive) and dark (destructive) fringes.
e.g. Young's double-slit experiment produces interference fringes on a screen.
Constructive interference
When the path difference between two waves from coherent sources is an integral multiple of wavelength (nλ), they arrive in phase, and the resultant intensity is maximum (4I₀).
e.g. At the central point on the screen in Young's experiment, where S₁P = S₂P, constructive interference occurs.
Destructive interference
When the path difference between two waves from coherent sources is an odd multiple of half-wavelength ( (n+1/2)λ ), they arrive out of phase, and the resultant intensity is minimum (zero).
e.g. At a point on the screen where S₂P - S₁P = 2.5λ, destructive interference occurs.
Diffraction
The phenomenon of bending of light waves around the corners of an obstacle or aperture, and the spreading of light into the geometrical shadow region.
e.g. Viewing a straight filament bulb through a narrow slit formed by two razor blades shows diffraction bands.
Polarisation
The phenomenon of restricting the vibrations of the electric field vector of a light wave to a single plane (i.e., one direction) perpendicular to the direction of propagation.
e.g. When unpolarised light passes through a polaroid, the transmitted light is linearly polarised.
Malus' law
When a beam of plane polarised light is incident on an analyser (polaroid), the intensity (I) of the transmitted light varies as I = I₀ cos²θ, where θ is the angle between the pass-axis of the analyser and the plane of polarisation of the incident light.
e.g. If the pass-axes of two polaroids are at 60°, the intensity of transmitted light is I₀/4.
Critical angle
The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°. It is given by sin i_c = n₂/n₁.
e.g. For a glass-air interface (n_glass = 1.5), the critical angle is about 41.8°.
Total internal reflection
When the angle of incidence in the denser medium exceeds the critical angle, the incident wave is completely reflected back into the denser medium, with no refracted wave.
e.g. Optical fibres work on the principle of total internal reflection.
Secondary wavelets
The spherical waves emanating from each point on a wavefront, as per Huygens principle.
e.g. In Huygens construction, spheres of radius vt drawn from each point on the initial wavefront are the secondary wavelets.
Ray
The path of energy propagation in the limit of wavelength tending to zero; it is perpendicular to the wavefront.
e.g. In geometrical optics, a ray is drawn perpendicular to the plane wavefront.
Pass-axis
The direction along which a polaroid transmits the electric vector of light; the electric vector oscillating perpendicular to the aligned molecules of the polaroid is transmitted.
e.g. If the pass-axis of a polaroid is vertical, only the vertical component of the electric field is transmitted.
Formulas & Equations (13)
sin i / sin r = v₁ / v₂
Reads as: Sine of angle of incidence divided by sine of angle of refraction equals speed of light in first medium divided by speed of light in second medium.
Variables: i = angle of incidence (degrees or radians), r = angle of refraction (degrees or radians), v₁ = speed of light in medium 1 (m/s), v₂ = speed of light in medium 2 (m/s).
Use when: Deriving Snell's law from Huygens principle or relating angles to wave speeds.
⚠ Students often write v₂/v₁ instead of v₁/v₂. Remember: sin i / sin r = v₁ / v₂.
n₁ sin i = n₂ sin r
Reads as: n one sine i equals n two sine r.
Variables: n₁ = refractive index of medium 1 (dimensionless), i = angle of incidence (degrees or radians), n₂ = refractive index of medium 2 (dimensionless), r = angle of refraction (degrees or radians).
Use when: Solving refraction problems using Snell's law in terms of refractive indices.
⚠ Forgetting that the angle is measured from the normal, not from the surface.
n = c / v
Reads as: Refractive index equals speed of light in vacuum divided by speed of light in the medium.
Variables: n = refractive index of the medium (dimensionless), c = speed of light in vacuum (3 × 10⁸ m/s), v = speed of light in the medium (m/s).
Use when: Calculating refractive index from speed or vice versa.
⚠ Using the speed in air instead of vacuum; the difference is negligible but conceptually wrong.
v₁ / λ₁ = v₂ / λ₂
Reads as: Speed of light in medium 1 divided by wavelength in medium 1 equals speed of light in medium 2 divided by wavelength in medium 2.
Variables: v₁ = speed of light in medium 1 (m/s), λ₁ = wavelength in medium 1 (m), v₂ = speed of light in medium 2 (m/s), λ₂ = wavelength in medium 2 (m).
Use when: Finding the change in wavelength when light passes from one medium to another.
⚠ Assuming frequency changes; frequency remains constant during refraction.
sin i_c = n₂ / n₁
Reads as: Sine of critical angle equals refractive index of rarer medium divided by refractive index of denser medium.
Variables: i_c = critical angle (degrees or radians), n₂ = refractive index of rarer medium (dimensionless), n₁ = refractive index of denser medium (dimensionless).
Use when: Calculating the critical angle for total internal reflection.
⚠ Writing n₁/n₂ instead of n₂/n₁. The denser medium has the higher refractive index.
S₁P ~ S₂P = nλ (for constructive interference)
Reads as: Path difference equals n times lambda for constructive interference.
Variables: S₁P = distance from source 1 to point P (m), S₂P = distance from source 2 to point P (m), n = integer (0, 1, 2, ...), λ = wavelength (m).
Use when: Determining positions of bright fringes in Young's double-slit experiment.
⚠ Forgetting that n can be zero, giving the central bright fringe.
S₁P ~ S₂P = (n + 1/2)λ (for destructive interference)
Reads as: Path difference equals n plus half times lambda for destructive interference.
Variables: S₁P = distance from source 1 to point P (m), S₂P = distance from source 2 to point P (m), n = integer (0, 1, 2, ...), λ = wavelength (m).
Use when: Determining positions of dark fringes in Young's double-slit experiment.
⚠ Writing (2n+1)λ/2 is also correct, but students often forget the factor of 1/2.
I = 4 I₀ cos²(φ/2)
Reads as: Intensity equals 4 I naught times cos squared of phi by 2.
Variables: I = resultant intensity at a point (W/m²), I₀ = intensity due to one source alone (W/m²), φ = phase difference between the two waves (radians).
Use when: Calculating intensity at any point in an interference pattern.
⚠ Using I₀ as the total intensity from both sources; I₀ is the intensity from a single source.
x_n = n λ D / d
Reads as: x n equals n lambda D over d.
Variables: x_n = distance of nth bright fringe from central maximum (m), n = fringe order (0, 1, 2, ...), λ = wavelength of light (m), D = distance from slits to screen (m), d = separation between slits (m).
Use when: Calculating the position of bright fringes in Young's double-slit experiment.
⚠ Using d as the slit width instead of slit separation.
x_n = (n + 1/2) λ D / d
Reads as: x n equals n plus half times lambda D over d.
Variables: x_n = distance of nth dark fringe from central maximum (m), n = fringe order (0, 1, 2, ...), λ = wavelength of light (m), D = distance from slits to screen (m), d = separation between slits (m).
Use when: Calculating the position of dark fringes in Young's double-slit experiment.
⚠ Forgetting that the first dark fringe corresponds to n=0.
I = I₀ cos²θ
Reads as: I equals I naught cos squared theta.
Variables: I = intensity of transmitted light (W/m²), I₀ = intensity of incident polarised light (W/m²), θ = angle between pass-axis of analyser and plane of polarisation of incident light (degrees or radians).
Use when: Calculating intensity after light passes through a polaroid (Malus' law).
⚠ Forgetting to square the cosine; writing I = I₀ cos θ.
y(x,t) = a sin(kx - ωt)
Reads as: y of x and t equals a sine of k x minus omega t.
Variables: y = displacement (m), a = amplitude (m), k = wave number (2π/λ) (rad/m), x = position (m), ω = angular frequency (2πν) (rad/s), t = time (s).
Use when: Describing a linearly polarised transverse wave propagating in the +x direction.
⚠ Confusing k (wave number) with propagation constant; they are the same.
k = 2π / λ
Reads as: k equals 2 pi over lambda.
Variables: k = wave number (rad/m), λ = wavelength (m).
Use when: Calculating wave number from wavelength.
⚠ Using k = λ/2π instead of the correct relation.
How It Works
Huygens Construction for Refraction of a Plane Wave
  1. Consider a plane wavefront AB incident on the interface PP' separating medium 1 and medium 2 at an angle i.
  2. Let the time taken for the wavefront to travel from B to C be τ, so BC = v₁τ.
  3. From point A in the second medium, draw a sphere of radius v₂τ (the speed in medium 2 is v₂).
  4. Draw a tangent plane CE from point C to the sphere. This tangent plane CE represents the refracted wavefront.
  5. From triangles ABC and AEC, derive sin i = BC/AC and sin r = AE/AC, leading to sin i / sin r = v₁ / v₂.
  6. Using n = c/v, obtain Snell's law: n₁ sin i = n₂ sin r.
💡 The refracted wavefront bends towards the normal if v₂ < v₁ (denser medium), which is opposite to the prediction of the corpuscular model.
Huygens Construction for Reflection of a Plane Wave
  1. Consider a plane wavefront AB incident on a reflecting surface MN at an angle i.
  2. Let the time taken for the wavefront to travel from B to C be τ, so BC = vτ.
  3. From point A, draw a sphere of radius vτ in the same medium.
  4. Draw a tangent plane CE from point C to the sphere. This tangent plane CE represents the reflected wavefront.
  5. Triangles EAC and BAC are congruent, so angle i = angle r, proving the law of reflection.
💡 The incident and reflected wavefronts are on the same side of the surface, and the speed of light remains the same.
Young's Double-Slit Experiment
  1. A single pinhole S is illuminated by a bright source. Light from S falls on two pinholes S₁ and S₂ very close to each other on an opaque screen.
  2. S₁ and S₂ act as coherent sources because they are derived from the same original source S.
  3. Spherical waves from S₁ and S₂ superpose on a screen GG' placed at a distance D from the slits.
  4. For a point P on the screen, the path difference is S₂P - S₁P ≈ xd/D, where x is the distance from the central point and d is the slit separation.
  5. Constructive interference (bright fringe) occurs when xd/D = nλ, giving x = nλD/d.
  6. Destructive interference (dark fringe) occurs when xd/D = (n+1/2)λ, giving x = (n+1/2)λD/d.
  7. The fringes are equally spaced on the screen.
💡 The two slits must be coherent; using two independent light sources (e.g., two sodium lamps) will not produce interference fringes.
Single Slit Diffraction
  1. A parallel beam of monochromatic light falls normally on a single slit of width a.
  2. The diffracted light falls on a screen. The midpoint of the slit is M, and a line through M perpendicular to the slit meets the screen at C.
  3. The slit is divided into many smaller parts, each acting as a secondary source (Huygens-Fresnel principle).
  4. The path difference between waves from the top and bottom of the slit is a sin θ.
  5. Minima (zero intensity) occur at angles θ such that a sin θ = nλ, i.e., θ ≈ nλ/a for small angles.
  6. Secondary maxima occur at angles θ ≈ (n+1/2)λ/a, with decreasing intensity as n increases.
  7. The central maximum is the brightest and broadest, with intensity falling off on either side.
💡 The condition for minima is a sin θ = nλ, not (n+1/2)λ. The central maximum is twice as wide as the other maxima.
Polarisation by a Polaroid
  1. Unpolarised light from an ordinary source (e.g., sodium lamp) is incident on a polaroid sheet P₁.
  2. The polaroid transmits only the component of the electric field parallel to its pass-axis, producing linearly polarised light.
  3. The intensity of the transmitted light is half the incident intensity.
  4. A second polaroid P₂ (analyser) is placed in the path of the polarised light.
  5. If the pass-axis of P₂ makes an angle θ with the plane of polarisation, the transmitted intensity is I = I₀ cos²θ (Malus' law).
  6. When θ = 0°, maximum intensity is transmitted; when θ = 90°, no light is transmitted (crossed polaroids).
💡 Rotating a single polaroid does not change the intensity of transmitted light; the effect is only seen with a second polaroid.
Exam Traps — Don't Make These Mistakes
⚠ Confusing slit separation (d) with slit width (a) in Young's experiment
✗ Students write the fringe width formula as β = λD/a, using the slit width instead of the slit separation.
→ In Young's double-slit experiment, d is the distance between the centres of the two slits, not the width of each slit. The formula for fringe width is β = λD/d.
✓ Fringe width β = λD/d, where d is the separation between the two slits.
⚠ Using the wrong condition for minima in single-slit diffraction
✗ Students write the condition for minima as a sin θ = (n+1/2)λ, which is actually the condition for secondary maxima.
→ In single-slit diffraction, minima occur when the path difference between the two ends of the slit is an integral multiple of λ: a sin θ = nλ. Secondary maxima occur at (n+1/2)λ.
✓ For minima: a sin θ = nλ (n = ±1, ±2, ...). For secondary maxima: a sin θ = (n+1/2)λ.
⚠ Assuming frequency changes during refraction
✗ Students write that the frequency of light changes when it enters a different medium.
→ Frequency is determined by the source and remains constant during refraction. Only speed and wavelength change.
✓ When light enters a denser medium, speed and wavelength decrease, but frequency remains unchanged.
⚠ Misapplying Malus' law for unpolarised light
✗ Students directly apply I = I₀ cos²θ to unpolarised light incident on a polaroid.
→ Malus' law applies only to plane polarised light incident on an analyser. For unpolarised light incident on a polaroid, the transmitted intensity is always half the incident intensity, regardless of the angle of rotation.
✓ For unpolarised light through a polaroid: I_transmitted = I_incident / 2. For polarised light through an analyser: I = I₀ cos²θ.
⚠ Confusing coherent and incoherent sources in interference
✗ Students state that two separate sodium lamps can produce interference fringes.
→ Two independent light sources are incoherent because they undergo random phase changes (~10⁻¹⁰ s), so no stable interference pattern is formed. Only sources derived from the same original source (like in Young's experiment) are coherent.
✓ Two independent sources of light are incoherent and cannot produce a stable interference pattern. Coherent sources must have the same frequency and a constant phase difference.
⚠ Forgetting the backwave in Huygens principle
✗ Students draw the secondary wavelets and include the backwave (D₁D₂) as part of the new wavefront.
→ Huygens made an ad hoc assumption that the amplitude of secondary wavelets is maximum in the forward direction and zero in the backward direction, so the backwave does not exist. Including it is incorrect.
✓ The new wavefront is the forward envelope of the secondary wavelets. The backwave is not observed and is not part of the wavefront.
Compare & Contrast
Number of sources
Interference
Diffraction
Interference involves superposition from a few (typically two) coherent sources, while diffraction involves superposition from a large number of secondary sources (continuous distribution) from a single wavefront.
Prediction of speed of light in denser medium
Corpuscular model of light
Wave model of light
The corpuscular model predicts that light travels faster in a denser medium (v₂ > v₁ when ray bends towards normal), while the wave model predicts that light travels slower in a denser medium (v₂ < v₁ when ray bends towards normal). Experiments (Foucault, 1850) confirmed the wave model.
Path difference condition
Constructive interference
Destructive interference
Constructive interference occurs when path difference is an integral multiple of λ (nλ), resulting in maximum intensity (4I₀). Destructive interference occurs when path difference is an odd multiple of λ/2 ((n+1/2)λ), resulting in zero intensity.
Direction of electric field vector
Plane polarised wave
Unpolarised wave
In a plane polarised wave, the electric field vector oscillates in a single plane (one direction) perpendicular to the direction of propagation. In an unpolarised wave, the electric field vector takes all possible directions in the transverse plane, changing rapidly and randomly.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on Huygens principle or interference, 1 LA (5 marks) on Young's double-slit experiment or diffraction. Polarisation typically appears as a 1-2 mark question.
Most asked: Young's double-slit experiment: derivation of fringe width, position of fringes, and effect of changing parameters. · Huygens principle and its application to derive laws of reflection and refraction. · Conditions for constructive and destructive interference. · Single slit diffraction: condition for minima and maxima, width of central maximum. · Malus' law and polarisation by polaroids.
Don't skip: Derivation of Snell's law using Huygens principle (5 marks).; Derivation of fringe width in Young's double-slit experiment and numerical problems (5 marks).; Single slit diffraction: condition for minima and expression for angular width of central maximum (3-5 marks).
One-Line Revision (read aloud before the exam)
  1. A wavefront is a surface of constant phase; rays are perpendicular to wavefronts.
  2. Huygens principle: each point on a wavefront is a source of secondary wavelets; the new wavefront is the forward envelope.
  3. Snell's law derived from Huygens principle: n₁ sin i = n₂ sin r.
  4. The wave model predicts light slows down in a denser medium; this was confirmed by Foucault's experiment.
  5. Coherent sources have the same frequency and a constant phase difference.
  6. In Young's double-slit experiment, constructive interference occurs when path difference = nλ.
  7. In Young's double-slit experiment, destructive interference occurs when path difference = (n+1/2)λ.
  8. Fringe width in Young's experiment: β = λD/d.
  9. In single slit diffraction, minima occur at a sin θ = nλ.
  10. The central maximum in single slit diffraction is twice as wide as other maxima.
  11. Diffraction is the bending of light around obstacles; it sets the limit of resolution of optical instruments.
  12. Polarisation proves light is a transverse wave.
  13. Malus' law: I = I₀ cos²θ for polarised light through an analyser.
  14. Unpolarised light through a polaroid gives half the incident intensity.
  15. Total internal reflection occurs when angle of incidence exceeds the critical angle.
  16. Frequency of light remains constant during refraction; speed and wavelength change.
  17. Two independent light sources are incoherent and cannot produce stable interference.
  18. The backwave in Huygens construction does not exist; amplitude of secondary wavelets is zero in the backward direction.

MCQs — 10 Questions with Answers

Q1. Which scientist experimentally confirmed that the speed of light in water is less than in air, validating the wave model?

  • A. Foucault in 1850 ✓
  • B. Newton in 1687
  • C. Young in 1801
  • D. Maxwell in 1855

Answer: A — Foucault's 1850 experiment directly measured light speed in different media and confirmed wave model's prediction that light is slower in denser media.

Q2. A wavefront is best defined as:

  • A. The path along which light energy travels
  • B. The locus of points oscillating in the same phase at any instant ✓
  • C. A surface perpendicular to the direction of light propagation
  • D. The boundary between two different media

Answer: B — By definition, a wavefront is a surface of constant phase where all points vibrate in phase, not just a geometric boundary.

Q3. According to Huygens Principle, the new wavefront is found by drawing a common tangent to secondary wavelets of radius:

  • A. λ (wavelength)
  • B. vt (speed × time) ✓
  • C. c/n × t (speed in medium × time)
  • D. λf (wavelength × frequency)

Answer: B — Huygens Principle uses vt as the radius of secondary wavelets, where v is wave speed in the medium and t is time elapsed.

Q4. A plane wave approximation is valid for spherical waves when the observer is:

  • A. Very close to the point source
  • B. At a distance comparable to one wavelength
  • C. At a large distance from the source ✓
  • D. Exactly one focal length away from the source

Answer: C — At large distances, the curvature of a spherical wavefront becomes negligible and a small portion appears flat like a plane wave.

Q5. The corpuscular model predicts that when light refracts toward the normal at an air-water boundary, light speed should:

  • A. Increase in water compared to air ✓
  • B. Decrease in water compared to air
  • C. Remain constant in both media
  • D. Increase in air compared to water

Answer: A — Corpuscular model incorrectly predicted that bending toward normal implies acceleration in the second medium, opposite to the correct wave model prediction.

Q6. Which of the following is NOT a valid reason why the wave model of light was initially rejected?

  • A. Newton's authority and popularity
  • B. Light could travel through vacuum without a medium
  • C. The extremely small wavelength made waves undetectable ✓
  • D. Waves were thought to always require a physical medium to propagate

Answer: C — The small wavelength of visible light (~0.6 μm) actually SUPPORTED the wave model by allowing ray approximation; it was the other three factors that caused initial rejection.

Q7. In Huygens Principle, why does the backwave not appear even though the mathematical construction predicts one?

  • A. The backwave travels at a different speed
  • B. The amplitude of secondary wavelets is maximum forward and zero backward ✓
  • C. The backwave is absorbed by the source
  • D. Destructive interference cancels the backwave only in forward direction

Answer: B — Huygens postulated that secondary wavelets have directional amplitude (maximum forward, zero backward) to resolve the backwave paradox.

Q8. Statement I: A wavefront is always perpendicular to the direction of energy propagation (ray direction). Statement II: Secondary wavelets in Huygens Principle are actual physical disturbances from the medium particles. Which is correct?

  • A. Both I and II are true
  • B. I is true, II is false ✓
  • C. I is false, II is true
  • D. Both I and II are false

Answer: B — Statement I is a fundamental definition (ray ⊥ wavefront). Statement II is false because secondary wavelets are mathematical constructs for finding the next wavefront, not physical sources.

Q9. If the wavelength of light is λ and it propagates with speed v, then using Huygens Principle, the time required for a spherical wavefront from a point source to expand by one wavelength is:

  • A. λ/v ✓
  • B. v/λ
  • C. λv
  • D. 1/(λv)

Answer: A — Distance = one wavelength λ; speed = v; time = distance/speed = λ/v using the relationship between wave propagation and Huygens construction.

Q10. Maxwell's electromagnetic theory resolved the apparent contradiction in light propagation through vacuum by proposing that:

  • A. Light does not actually propagate through vacuum
  • B. A hypothetical medium called aether fills vacuum
  • C. Changing electric and magnetic fields sustain each other without needing a medium ✓
  • D. Light is purely corpuscular and requires no medium whatsoever

Answer: C — Maxwell showed that time-varying electric fields produce time-varying magnetic fields and vice versa, allowing wave propagation in vacuum without any physical medium.

Flashcards

Define wavefront in one sentence.

A wavefront is the locus of all points oscillating in the same phase at any given instant.

State Huygens Principle in simple terms.

Each point on a wavefront acts as a source of secondary wavelets, and the envelope of these wavelets forms the new wavefront position.

Why does corpuscular model predict light travels faster in water than air?

Corpuscular model assumes if light bends toward normal on refraction, it must speed up in the denser medium.

What does wave model predict about light speed in water versus air?

Wave model predicts light travels slower in water than in air, which matches experimental observation.

Who experimentally proved light speed is less in water than air and when?

Foucault experimentally confirmed in 1850 that light speed in water is less than in air, supporting the wave model.

What is the wavelength of yellow light approximately?

The wavelength of yellow light is approximately 0.6 micrometers or 600 nanometers.

What is the relationship between a ray and a wavefront?

A ray is the direction of energy propagation and is always perpendicular to the wavefront.

What is a plane wave and when does it occur?

A plane wave occurs at a large distance from the source when a small portion of a spherical wavefront can be approximated as a flat surface.

Why does Huygens Principle have a backwave problem and how was it resolved?

The backwave (wavelet envelope in backward direction) exists mathematically but Huygens assumed secondary wavelets have maximum amplitude forward and zero backward.

State the definition of geometrical optics.

Geometrical optics is the branch of optics that completely neglects the finiteness of wavelength and assumes light travels in straight lines.

Important Board Questions

State Huygens Principle and explain how it can be used to determine the position of a wavefront at a later time. [2 marks]

Define secondary wavelets and explain the envelope construction. Use the concept that radius of each secondary wavelet = vt.

Explain why the corpuscular model and wave model of light make opposite predictions about the speed of light in water compared to air. Which prediction is correct and why? [5 marks]

For corpuscular model: bending toward normal → speed increases in water. For wave model: bending toward normal → speed decreases in water. Foucault's 1850 experiment confirmed wave model is correct because v_water < v_air; use Snell's law: n = c/v to justify.

Using Huygens Principle, derive the law of reflection for a plane wavefront striking a plane mirror. Show that angle of incidence equals angle of reflection. [6 marks]

Draw a plane mirror with incident wavefront AB at angle θ to mirror. Construct secondary wavelets of radius vt from all points. The tangent to these wavelets (reflected wavefront) makes equal angle on opposite side. Use geometry of right triangles to prove θ_i = θ_r. Show that the reflected wavefront position follows the law of reflection.

Assertion-Reason Questions for Wave Optics

Q1. Assertion (A):

A wavefront is always perpendicular to the direction of energy propagation in a wave.

Reason (R):

The energy of a wave travels in a direction normal to the wavefront because that is the direction of maximum phase gradient.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are correct. The study material states: 'The energy of the wave travels in a direction perpendicular to the wavefront.' The reason correctly explains why this is true—energy propagates normal to surfaces of constant phase. This is consistent with the definition of a wavefront as a surface of constant phase.

Q2. Assertion (A):

According to Huygens principle, a backwave theoretically exists but has zero amplitude in the forward-moving direction.

Reason (R):

The principle assumes the amplitude of secondary wavelets is maximum backward and zero in the forward direction.

Answer: C — A is true but R is false — The assertion is true—the study material confirms a backwave exists mathematically. However, the reason is false—it reverses the assumption. Huygens actually assumed amplitude is maximum in the forward direction and zero backward, exactly opposite to the reason stated. This tests whether students carefully read Huygens' actual assumption.

Q3. Assertion (A):

The speed of light in a medium determines whether light bends toward or away from the normal during refraction.

Reason (R):

When light enters an optically denser medium (lower speed), it bends toward the normal because the slower wave speed bends the wavefront, consistent with Huygens principle.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true, but the reason does NOT correctly explain the assertion. The reason describes the mechanism from Huygens wave construction, but the assertion only states a correlation, not a causal mechanism. The reason provides mechanistic detail beyond what the assertion claims. This tests distinction between true statements and true explanations.

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Case-Based Questions (CBQ) for Wave Optics

Case Study:

An optical research laboratory in India is studying the propagation of laser light through different transparent materials. They use a He-Ne laser (wavelength in vacuum: 633 nm) and measure its wavelength in water (refractive index n = 4/3) and in glass (refractive index n = 1.5). A student researches how Huygens principle applies to spherical wavefronts entering these media. The student creates wavefronts at the interface and applies Huygens construction to predict the refraction behavior at various incident angles.

Q1. What is the wavelength of the He-Ne laser light in water?

  • A. 474.75 nm ✓
  • B. 633 nm
  • C. 844.75 nm
  • D. 316.5 nm

Answer: A

Q2. Which statement correctly applies Huygens principle to the refraction at the vacuum-water interface?

  • A. Secondary wavelets in water travel faster than in vacuum, causing the wavefront to bend away from the normal
  • B. Secondary wavelets in water travel slower than in vacuum, causing the wavefront to bend toward the normal ✓
  • C. The wavefront remains spherical unchanged as it crosses the interface
  • D. Secondary wavelets travel at the same speed in both media, so no bending occurs

Answer: B

Q3. If the student observes the wavefront entering glass (n=1.5) at normal incidence, what can they predict about the wavefront shape immediately after entering glass?

  • A. It becomes planar because refractive index is greater than water
  • B. It remains spherical but with a smaller radius of curvature than in vacuum ✓
  • C. It becomes ellipsoidal due to the difference in indices
  • D. It splits into two separate wavefronts

Answer: B

Q4. According to the study material's discussion of Huygens principle, which optical phenomenon would this refraction experiment help visualize?

  • A. Polarization of light in anisotropic media
  • B. The diffraction pattern from a single slit
  • C. The creation of interference patterns from two coherent sources
  • D. The bending of spherical wavefronts at interfaces as predicted by Huygens construction ✓

Answer: D

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Previous Year Questions (PYQ) for Wave Optics

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 55/1/1 · 1 mark

The shape of the interference fringes in Young's double-slit experiment, when the distance between the slit and the screen is very large as compared to the slit-separation, is nearly

  • straight
  • parabolic
  • circular
  • hyperbolic

CBSE 2026 · Set 55/1/1 · 1 mark

Four independent waves are expressed as (i) y₁ = A₁ sin ωt (ii) y₂ = A₂ sin 2ωt (iii) y₃ = A₃ cos ωt (iv) y₄ = A₄ sin (ωt + π/3). The interference between two of these waves is possible in

  • (i) and (iii) only
  • (iii) and (iv) only
  • (i), (iii) and (iv) only
  • All of them

CBSE 2026 · Set 55/1/1 · 2 marks

A beam of light consisting of two wavelengths 400 nm and 600 nm is used to illuminate a single slit of width 1 mm. Find the least distance of the point from the central maximum where the dark fringes due to both wavelengths coincide on the screen placed 1.5 m from the slit.

CBSE 2026 · Set 55/1/1 · 3 marks

State the two conditions under which total internal reflection occurs. A transparent container contains layers of three immiscible transparent liquids A, B and C of refractive indices n, 3n/4 and 2n/3, respectively. A laser beam is incident at the interface between A and B at an angle θ as shown in figure. Prove that the beam does not enter region C at all for sin θ > 2/3.

CBSE 2026 · Set 55/1/2 · 1 mark

Four independent waves are expressed as (i) y₁ = A₁ sin ωt (ii) y₂ = A₂ sin 2 ωt (iii) y₃ = A₃ cos ωt (iv) y₄ = A₄ sin (ωt + π/3). The interference between two of these waves is possible in

  • (i) and (iii) only
  • (iii) and (iv) only
  • (i), (iii) and (iv) only
  • All of them

CBSE 2026 · Set 55/1/2 · 1 mark

The angular width of interference fringes in Young's double-slit experiment depends on

  • distance between the slits and the screen only
  • wavelength of light used only
  • both wavelength of light used and the slits separation
  • slits separation only

CBSE 2026 · Set 55/1/2 · 3 marks

State the two conditions under which total internal reflection occurs.

CBSE 2026 · Set 55/1/3 · 1 mark

The phenomenon of interference is shown by

  • longitudinal mechanical wave only
  • transverse mechanical wave only
  • electromagnetic waves only
  • all these waves

CBSE 2026 · Set 55/2/1 · 1 mark

Assertion (A): Light added to light can produce darkness. Reason (R): When two coherent light waves interfere, there is darkness at position of destructive interference.

  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
  • Assertion (A) is true, but Reason (R) is false.
  • Assertion (A) is false and Reason (R) is also false.

CBSE 2026 · Set 55/2/1 · 2 marks

In a Young's double-slit experiment, a beam of light consisting of two wavelengths 500 nm and 600 nm is used. The interference fringes are observed at a screen placed 1.8 m away from the plane of slits (slit separation 0.3 mm). Calculate the least distance from the central maximum where the bright fringes due to both the wavelengths coincide.

CBSE 2026 · Set 55/2/1 · 5 marks

(a) What are coherent sources? Why they are necessary for observing stable interference pattern? Draw a graph showing the variation of intensity of light with the position on the screen in Young's double-slit experiment. (b) Find the intensity of light at a point on the screen when two interfering waves of the same intensity (I₀) have a path difference of (i) λ/4 and (ii) λ/3.

CBSE 2026 · Set 55/2/2 · 1 mark

Assertion (A): Light added to light can produce darkness. Reason (R): When two coherent light waves interfere, there is darkness at position of destructive interference.

  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
  • Assertion (A) is true, but Reason (R) is false.
  • Assertion (A) is false and Reason (R) is also false.

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Class 12 Physics Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 12 Physics Previous Year Questions, Chapter-wise

All 962 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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Class 12 Physics Derivations

All 68 derivations, chapter-wise, step by step with diagrams and marked with the CBSE board years (2022–2026) each was asked in. Free PDF download.

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