Chapter 10: Wave Optics Long Questions | physics Class 12 CBSE

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Chapter 10: Wave Optics Long Questions | physics Class 12 CBSE | ByteNotes.in

Long Answer

Medium difficulty • Structured explanation

Medium

Question 1

Long Form

Using Huygens principle, derive the laws of reflection for a plane wave incident on a plane reflecting surface and explain why the backwave does not exist.

Draw plane wavefront AB incident at angle i on reflecting surface MN. In time τ, B travels distance BC = vτ along the surface; draw a sphere of radius vτ from point A and let CE be the tangent from C to this sphere, so AE = BC = vτ.
Triangles EAC and BAC share hypotenuse AC; AE = BC and both have a right angle, so they are congruent. Therefore angle of incidence i (angle BAC's normal) equals angle of reflection r (angle ECA's normal), proving the law of reflection.
Both the incident and reflected rays lie in the same plane (the plane containing the incident ray and the normal to the surface at the point of incidence), which is the second law of reflection.
Huygens argued that secondary wavelets have maximum amplitude in the forward direction and zero amplitude in the backward direction as an ad hoc assumption; the backwave D1D2 (which would appear behind the original wavefront) has zero amplitude and thus does not physically exist.
The absence of the backwave is fully justified by the more rigorous electromagnetic wave theory, confirming that energy flows only in the forward direction in the absence of a boundary.

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Question 2

Long Form

Compare the corpuscular model and the wave model of light with respect to the speed of light in a denser medium, and explain which model was validated by experiment. Also describe how Maxwell further advanced the wave model.

The corpuscular model (Descartes, Newton) predicted that when light bends toward the normal on entering a denser medium, its speed must increase in that medium (v_denser > v_rarer), because the greater speed was assumed to cause the bending.
The wave model (Huygens, 1678) predicted the opposite: refraction toward the normal corresponds to a decrease in speed in the denser medium (v_denser < v_rarer), derived from the geometrical wavefront construction.
Foucault's 1850 experiment directly measured the speed of light in water and air, finding light is slower in water — confirming the wave model's prediction and disproving the corpuscular model.
Maxwell's electromagnetic theory (~1855) resolved the further difficulty of light propagating through vacuum by showing that mutually regenerating changing electric and magnetic fields can sustain electromagnetic waves in free space without any material medium.
Maxwell calculated the theoretical speed of electromagnetic waves and found it matched the measured speed of light, compelling the conclusion that light is an electromagnetic wave — thus unifying optics with electromagnetism.

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Long Answer

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Medium

Question 3

Long Form

Derive the expression for the positions of bright and dark fringes in Young's double-slit experiment and hence obtain the fringe width. State how fringe width changes with wavelength, slit separation, and screen distance.

Two coherent slits S1 and S2 separated by d are at distance D from screen. For a point P at height x, the path difference is S2P − S1P ≈ xd/D (for D >> d and D >> x).
Constructive interference (bright fringe) occurs when xd/D = nλ, giving xn = nλD/d for n = 0, ±1, ±2, …; destructive interference (dark fringe) occurs when xd/D = (n + 1/2)λ, giving xn = (n + 1/2)λD/d.
The fringe width β = distance between consecutive bright (or dark) fringes = λD/d, showing that all fringes are equally spaced.
Fringe width β is directly proportional to wavelength (larger λ → wider fringes), directly proportional to D (farther screen → wider fringes), and inversely proportional to d (closer slits → wider fringes).
Since fringes are equally spaced and alternately bright and dark, the pattern is a series of parallel equidistant bands; the central bright fringe is at x = 0 (path difference = 0).

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Long Answer

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Hard

Question 4

Long Form

Analyse the phenomenon of coherent addition of two waves with an arbitrary phase difference φ. Show how this leads to the general intensity formula, and deduce the special cases of constructive and destructive interference.

Let two coherent sources produce displacements y1 = a cos ωt and y2 = a cos(ωt + φ) at point P. By superposition, y = y1 + y2 = a[cos ωt + cos(ωt + φ)].
Using the identity cos A + cos B = 2cos((A+B)/2)cos((A−B)/2), we get y = 2a cos(φ/2) cos(ωt + φ/2). The resultant amplitude is 2a cos(φ/2).
Since intensity ∝ (amplitude)², the resultant intensity is I = 4I0 cos²(φ/2), where I0 is the intensity due to each individual source alone.
For constructive interference: φ = 0, ±2π, ±4π, … (path difference = nλ), cos²(φ/2) = 1, so I = 4I0 — four times the individual intensity.
For destructive interference: φ = ±π, ±3π, … (path difference = (n+1/2)λ), cos²(φ/2) = 0, so I = 0. For incoherent sources, the time-averaged intensity is I = 2I0 at all points.

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Long Answer

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Medium

Question 5

Long Form

Describe Young's double-slit experiment, explain why two separate sodium lamps cannot serve as coherent sources, and discuss the role of a single source in achieving coherence.

In Young's experiment, a bright source illuminates a single pinhole S; light from S falls on two closely spaced pinholes S1 and S2 on an opaque screen. The spherical waves from S1 and S2 overlap on screen GG′ to produce interference fringes.
Light from an ordinary source (sodium lamp) undergoes random phase changes on the order of 10⁻¹⁰ s, so two independent lamps have no fixed phase relationship — they are incoherent. The intensities from incoherent sources simply add (I = 2I0), producing no fringes.
Since S1 and S2 are illuminated by the same source S, any phase change in S appears simultaneously and identically in both S1 and S2; the phase difference between them remains constant, making them coherent.
Coherent sources maintain a stable phase difference at every point on the screen, so the positions of maxima and minima are fixed in space, producing the observable fringe pattern.
The fringe pattern on screen GG′ is a superposition of the double-slit interference pattern with the single-slit diffraction envelope from each individual slit.

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Hard

Question 6

Long Form

Explain the single-slit diffraction pattern in detail: describe the central maximum, positions of minima and secondary maxima, and compare the intensity distribution with the double-slit interference pattern.

A single slit of width a is illuminated by monochromatic light. The slit is divided into many secondary sources; contributions from all parts of the wavefront at the slit are summed with their appropriate phase differences at each point on the screen.
The central maximum occurs at θ = 0 where all secondary wavelets arrive in phase; it is the brightest and widest feature of the pattern, spanning from −λ/a to +λ/a.
Minima occur at θ ≈ nλ/a (n = ±1, ±2, …) where the slit can be divided into pairs of zones that cancel. Secondary maxima occur approximately at θ ≈ (n + 1/2)λ/a and become progressively weaker with increasing n.
In double-slit interference, bright fringes are equally spaced and have nearly equal intensity (governed by 4I0 cos²(φ/2)); in single-slit diffraction, the central maximum is twice as wide as secondary maxima and intensities decrease away from centre.
The actual observed double-slit pattern is a product (superposition) of the double-slit fringe pattern and the single-slit diffraction envelope, so some double-slit maxima that coincide with single-slit minima are absent (missing orders).

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Long Answer

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Hard

Question 7

Long Form

Describe the phenomenon of polarisation of light by polaroids. State Malus' law and derive the expression for transmitted intensity when a polaroid sheet is placed between two crossed polaroids at angle θ to the first.

A polaroid contains long-chain molecules aligned along one direction; it absorbs the electric field component along the molecules and transmits only the perpendicular component (along its pass-axis), converting unpolarised light into linearly polarised light with intensity I0 = Iincident/2.
Malus' law: when linearly polarised light of intensity I0 passes through a polaroid whose pass-axis makes angle θ with the direction of polarisation, the transmitted intensity is I = I0 cos²θ.
For a polaroid P2 placed between crossed polaroids P1 and P3 (P1 ⊥ P3), let P2 make angle θ with P1. By Malus' law, intensity after P2 is I1 = I0 cos²θ.
The angle between P2 and P3 is (90° − θ); by Malus' law again, intensity after P3 is I = I1 cos²(90° − θ) = I0 cos²θ sin²θ = (I0/4) sin²2θ.
Transmitted intensity is maximum when sin²2θ = 1, i.e., when 2θ = 90°, so θ = 45°. When P2 is at 45° to both P1 and P3, the transmitted intensity through the three-polaroid system is maximum at I0/4.

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Hard

Question 8

Long Form

Explain how Huygens principle is used to understand the behaviour of wavefronts through a prism, a convex lens, and a concave mirror, and show that the total travel time from object to image is the same along every ray.

Prism: a plane wavefront passing through a prism travels more slowly in glass; the lower portion (passing through greater thickness) is delayed more, tilting the emerging wavefront and thus deviating the ray — this is the origin of refraction by a prism.
Convex lens: the central part of a plane wavefront traverses the thickest portion of the lens and is delayed the most; the emerging wavefront becomes spherical with a central depression, converging to the focal point F — the lens focuses parallel rays.
Concave mirror: a plane wave reflected by a concave mirror produces a spherical reflected wavefront that converges to the focal point F = R/2; the curvature of the mirror imposes converging curvature on the reflected wavefront.
For a convex lens, although the central ray traverses the shorter geometric path through air, it passes through the thickest glass (lower speed); the extra delay in glass exactly offsets the shorter air path, making total travel time equal to that of edge rays.
This equal-time principle is a direct consequence of the requirement that the image is a proper wavefront (surface of constant phase); since all points on a wavefront have the same phase, they must have taken equal time from the source point.

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Long Answer

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Hard

Question 9

Long Form

Analyse the role of coherence in determining whether an interference pattern is observed. Contrast coherent and incoherent addition of waves, and explain the time-averaged intensity for incoherent sources.

Two sources are coherent if the phase difference between them at any observation point remains constant with time; they are incoherent if the phase difference changes randomly and rapidly (within ~10⁻¹⁰ s for ordinary light sources).
For coherent sources with fixed phase difference φ, the resultant intensity I = 4I0 cos²(φ/2) varies from point to point on the screen, creating stable bright and dark fringes.
For incoherent sources, the phase difference φ at each point varies randomly in time; the time average of cos²(φ/2) over many random values of φ equals 1/2.
Therefore, the time-averaged intensity for incoherent sources is I = 4I0 × (1/2) = 2I0 at every point — equal to the simple sum of the individual intensities with no fringe pattern.
This explains why two independent sodium lamps produce uniform illumination (I = 2I0) on a screen rather than fringes; only when sources share a common origin (like Young's two pinholes from one source) can they be phase-locked and produce stable interference.

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Long Answer

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Hard

Question 10

Long Form

Using Huygens principle, derive Snell's law of refraction. Also show that when light passes from a denser to a rarer medium at an angle greater than the critical angle, total internal reflection occurs.

Plane wavefront AB in medium 1 (speed v1) hits surface PP′ at angle i. In time τ, B travels BC = v1τ; a secondary sphere of radius v2τ is drawn from A in medium 2 (speed v2). Tangent CE from C to this sphere gives refracted wavefront.
From triangles ABC and AEC: sin i = BC/AC = v1τ/AC and sin r = AE/AC = v2τ/AC. Dividing: sin i / sin r = v1/v2. Writing n = c/v gives n1 sin i = n2 sin r — Snell's law.
For refraction from denser to rarer medium (v2 > v1, n2 < n1), the refracted angle r > i. As i increases, r approaches 90°; when r = 90°, sin ic = n2/n1, defining the critical angle ic.
For i > ic, sin r = (n1/n2) sin i > 1, which has no real solution for r — no refracted wave can exist. All the incident energy is reflected back into the denser medium: total internal reflection.
Total internal reflection requires (a) light travelling from a denser to a rarer medium, and (b) angle of incidence exceeding ic; it is complete with no energy loss, unlike partial reflection at normal incidence.

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Chapter 10: Wave Optics | Long Questions

Using Huygens principle, derive the laws of reflection for a plane wave incident on a plane reflecting surface and explain why the backwave does not exist.

Compare the corpuscular model and the wave model of light with respect to the speed of light in a denser medium, and explain which model was validated by experiment. Also describe how Maxwell further advanced the wave model.

Derive the expression for the positions of bright and dark fringes in Young's double-slit experiment and hence obtain the fringe width. State how fringe width changes with wavelength, slit separation, and screen distance.

Analyse the phenomenon of coherent addition of two waves with an arbitrary phase difference φ. Show how this leads to the general intensity formula, and deduce the special cases of constructive and destructive interference.

Describe Young's double-slit experiment, explain why two separate sodium lamps cannot serve as coherent sources, and discuss the role of a single source in achieving coherence.

Explain the single-slit diffraction pattern in detail: describe the central maximum, positions of minima and secondary maxima, and compare the intensity distribution with the double-slit interference pattern.

Describe the phenomenon of polarisation of light by polaroids. State Malus' law and derive the expression for transmitted intensity when a polaroid sheet is placed between two crossed polaroids at angle θ to the first.

Explain how Huygens principle is used to understand the behaviour of wavefronts through a prism, a convex lens, and a concave mirror, and show that the total travel time from object to image is the same along every ray.

Analyse the role of coherence in determining whether an interference pattern is observed. Contrast coherent and incoherent addition of waves, and explain the time-averaged intensity for incoherent sources.

Using Huygens principle, derive Snell's law of refraction. Also show that when light passes from a denser to a rarer medium at an angle greater than the critical angle, total internal reflection occurs.

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