NCERT Solutions Class 12 Physics Chapter 10 Wave Optics

Why does frequency stay constant? At the boundary, the electromagnetic wave on both sides must oscillate in lock-step - otherwise the wave fields wouldn't match across the interface at every instant. The "ticking rate" of the wave (frequency) is therefore continuous across boundaries. Wavelength and speed can both change; only their product (frequency) is fixed.

Why colour doesn't change underwater. Although the wavelength of light shortens in water (589 nm → 443 nm), our eyes register colour by the frequency of the photon, which is unchanged. So a yellow sodium lamp viewed from inside a swimming pool still looks yellow - even though its in-water wavelength corresponds to violet on the air-wavelength scale.

Huygens' picture of refraction. Picture a planar wavefront hitting the water at an angle. The bottom edge of the wavefront enters the water first, where it slows down. While the top edge is still racing through air, the bottom edge crawls - pivoting the wavefront and bending the propagation direction toward the normal. Snell's law n₁sin₁ = n₂sin₂ is just the geometry of this pivoting.

Common mistake. Students often write _water = n_air (multiply by n instead of dividing). Sanity check: light slows down in water, so its wavelength must shorten. If you got a number larger than 589 nm, your formula is upside-down.

Huygens' principle in action. Each point on a wavefront acts as a secondary source that emits its own little spherical wavelet. The envelope of all these wavelets, a moment later, is the new wavefront. For a spherical wavefront, the envelope of secondary wavelets stays spherical (just bigger). For a plane wavefront, the envelope is another parallel plane. This is why wavefronts maintain their shape as they propagate in a uniform medium.

Why a convex lens "collimates" a point source. The lens delays the central rays (which pass through more glass) more than the edge rays. For a source at the focus, this differential delay is exactly the right amount to convert outgoing spherical wavefronts into flat ones. Reverse-running: a plane wavefront incident on a lens converges to a point at the focus.

Real-world - parabolic mirrors and antennas. The same principle underlies satellite dishes, car headlamp reflectors, and astronomical telescopes. A parabolic mirror takes the spherical wavefronts from a source at its focus and reflects them as a parallel beam (plane wavefront). For receiving, the reverse: parallel incoming wavefronts (from a distant satellite or star) converge to the focus, where the detector sits.

Common mistake. Students sometimes say "cylindrical" for case (b). Cylindrical wavefronts come from line sources (like a slit), not from a point source after passing through a lens. A line source plus a cylindrical lens would give plane wavefronts; a point source plus a spherical (ordinary) lens gives plane wavefronts.

Why dispersion happens. Inside glass, light interacts with the electrons in the atoms, which resonate at certain natural frequencies (in the ultraviolet for most glasses). Light frequencies closer to a resonance interact more strongly and are slowed down more. Visible violet is closer to the UV resonance than red - so violet "feels" the glass more, and travels slower.

Why a prism separates colours. When white light enters a prism, each colour refracts by a slightly different angle (because each has a different n). Violet bends most (largest deviation), red least. After passing through both refracting surfaces, the colours fan out into a spectrum. This is exactly what creates a rainbow when sunlight hits raindrops.

Real-world - chromatic aberration. Simple convex lenses focus different colours at slightly different points (because f depends on n which depends on colour). This is why old camera lenses produced colour fringes around bright objects. Modern lenses use combinations of glass types (crown + flint) whose dispersions cancel - called "achromatic doublets."

Common mistake. Students sometimes say "red is faster in vacuum than violet." No - in vacuum, ALL colours travel at exactly c = 310⁸ m/s. Dispersion is a medium effect; it disappears in vacuum.

Quick numerical check. The numerical difference in speed between red and violet in glass is tiny (~1%), but enough over the path length of a prism to create a clearly visible separation of colours.

Where the formula comes from (Huygens picture). Light from the two narrow slits S₁ and S₂ acts as two coherent point sources. At a point P on the screen at perpendicular distance y from the central axis, the path difference is approximately Δ = S₂ P - S₁ P ≈ y dD (using the small-angle approximation). For constructive interference (a bright fringe), Δ = nλ, giving y_n = nλ D/d.

Alternative method - use fringe width directly. β = y₄4 = 1.24 cm = 0.3 cm = 310^-3 m. Then λ = β d/D = 310^-32.810^-4/1.4 = 610^-7 m - same answer.

Common mistake - confusing path difference with phase difference. Be sure to use Δ = nλ for bright fringes (path difference is an integer number of wavelengths) and Δ = (n+1/2)λ for dark fringes. The corresponding phase difference is φ = 2Δ/λ.

Common mistake - fringe width vs distance to n-th fringe. Some students write β = 1.2 cm instead of β = y₄/4. Remember: y_n is the distance from the centre to the n-th bright fringe, NOT the spacing between adjacent fringes.

Real-world. The same physics underlies thin-film interference (soap films, oil slicks, anti-reflection coatings on camera lenses), Michelson interferometers used to measure tiny distances, and the Laser Interferometer Gravitational-wave Observatory (LIGO), which first detected gravitational waves in 2015.

Where the cosine-squared formula comes from. Two coherent waves at the screen are y₁ = acosω t, y₂ = acos(ω t + φ). Their sum is y₁ + y₂ = 2acos(φ/2) cos(ω t + φ/2). The amplitude of the resultant is A = 2acosφ/2, and intensity I ∝ A². With I₀ = (2a)² ∝ maximum intensity, we get I = I₀cos²φ/2.

Alternative - phasor method. Represent each wave as a phasor (vector) of length a. Two phasors at angle φ apart sum to a resultant of length R = √a² + a² + 2a²cosφ = 2a |cos(φ/2)|. Same result.

Common mistake - confusing path difference and phase difference. The factor of 2π/λ is essential. Path difference is measured in metres; phase difference in radians. Many students forget the conversion and use φ = λ/3, which is dimensionally wrong (radians, not metres!).

Pattern of intensities. Worth memorising for quick MCQs:

• Δ = 0 or λ: I = K (maximum).
• Δ = λ/6: φ = π/3, I = Kcos²π/6 = (3/4)K.
• Δ = λ/4: φ = π/2, I = Kcos²π/4 = K/2.
• Δ = λ/3: φ = 2π/3, I = Kcos²π/3 = K/4.
• Δ = λ/2: φ = π, I = 0 (dark fringe).

Real-world - anti-reflection coatings. A camera lens coated with a thin transparent film of thickness λ/4 where λ is the dominant visible wavelength inside the coating produces a path difference of λ/2 for the wave reflected from the front and back surfaces - destructive interference. This reduces reflection from the lens, transmitting more light to the sensor.

The arithmetic of the ratio. Whenever you have two wavelengths in a YDSE problem, the bright fringes coincide when n₁₁ = n₂₂. The smallest solution is found by reducing the wavelength ratio to lowest terms - here 650:520 = 5:4, so n₁ = 4, n₂ = 5. In general, if ₁/₂ = p/q in lowest terms, then n₁ = q and n₂ = p are the smallest orders that coincide.

What happens between coincidences. Between the central maximum and the first coincidence, you'd see bright fringes of ₁ (red) and ₂ (green) at different positions, creating a complicated overlapping pattern. The fringes appear to "beat" - bright regions where they overlap, washed-out regions where they're staggered.

Common mistake - coinciding dark fringes vs bright fringes. For dark fringes, the condition is n₁ + 1/2₁ = n₂ + 1/2₂, which has different solutions. Don't apply the bright-fringe condition to dark fringes.

Real-world - white-light fringes. If the source is white light (a mixture of all visible wavelengths), only the central maximum where Δ = 0 for all wavelengths simultaneously is white. Away from the centre, fringes of different colours fall at different positions, producing rainbow-coloured fringes near the centre that quickly wash out. The "central white fringe" is therefore a signature of zero-path-difference geometry - useful for calibrating interferometers.

Pitfall. Make sure to convert all units consistently. Wavelengths in nm, distances in m - mixing them up by a factor of 10⁹ is a common slip.

Why angular width is convenient. Linear fringe width β = λ D/d depends on the screen distance D. Angular fringe width θ = β/D = λ/d does not - it's a property only of the slits and the wavelength. So if a problem doesn't mention D, you should immediately suspect angular width.

Alternative - small-angle radians. Always remember the conversion: 1 degree = π/180 rad ≈ 0.01745 rad. Always convert before plugging into formulae that assume radians.

Order of magnitude. Slit separations in YDSE are typically 0.1-1 mm. Our answer (0.34 mm) is well in this range - a sanity check that nothing has gone wrong.

Real-world. Diffraction gratings (used in spectrometers) work by the same principle. The angular position of a bright spectral line is set by sinθ = nλ/d, where d is now the grating period. Knowing d (engraved precisely) and measuring θ, one can determine λ - this is how astronomers identify spectral lines of distant stars.

Why Brewster's law works. Light is an electromagnetic wave with electric field vector E. Inside the glass, the field drives electrons to oscillate; oscillating electrons re-radiate, producing the reflected beam. But oscillating dipoles don't radiate along their own oscillation axis. When the angle of incidence is such that the reflected ray would be along the dipole oscillation direction for the parallel-polarised (p) component, that component is suppressed - leaving only the perpendicular-polarised (s) component in the reflection. Geometrically, this occurs when the reflected and refracted rays are 90° apart, leading to tan i_B = n₂/n₁.

Alternative derivation. Use the condition that reflected and refracted rays are perpendicular: i_B + r = 90^∘, so r = 90^∘ - i_B. Apply Snell's law: n₁sin i_B = n₂sin r = n₂sin(90^∘ - i_B) = n₂cos i_B. Divide both sides by cos i_B: n₁tan i_B = n₂, giving tan i_B = n₂/n₁.

Real-world - polarising sunglasses. When sunlight reflects off a horizontal surface (road, water, snow) at an angle near 56^∘, the reflected glare is strongly horizontally polarised. Polaroid sunglasses with vertically oriented transmission axes block this horizontal glare, dramatically reducing the brightness while preserving most other light. Fishermen and pilots wear them for exactly this reason.

Common mistake - direction of light. The formula tan i_B = n₂/n₁ assumes light goes FROM medium 1 INTO medium 2. If light comes from glass into air n₁ = 1.5, n₂ = 1, then tan i_B = 1/1.5 ≈ 0.667, giving i_B ≈ 33.7^∘. The two Brewster angles (air-to-glass and glass-to-air) are complementary: 56.3^∘ + 33.7^∘ = 90^∘.

Geometry check. Draw it: incident ray hits the mirror at 45^∘ to the normal; reflected ray leaves at 45^∘ on the other side. The two rays make a 90^∘ angle at the point of incidence - exactly the condition for perpendicularity. This is the same geometry as a right-angle prism (or any corner reflector): incident and reflected rays at 90^∘ means the mirror is at 45^∘ to both.

Why ν is conserved on reflection. The electrons in the mirror surface oscillate at the frequency of the incident wave (they're driven by it). These oscillating electrons re-radiate at the SAME frequency (because they're tied to the driving frequency). So the reflected wave's frequency equals the incident wave's frequency exactly.

Real-world - corner-cube reflectors. A corner cube (three mutually perpendicular mirrors) sends any incoming ray back exactly along the direction it came from. The Apollo missions left corner-cube reflectors on the Moon's surface; laser pulses fired from Earth bounce back along the same path, allowing the Earth-Moon distance to be measured to millimetre precision. Same idea: 45^∘-style geometry, applied in three dimensions.

Common mistake - measuring angles from the surface vs the normal. The "angle of incidence" i in optics is measured from the NORMAL (perpendicular to the surface), not from the surface itself. A student who measures from the surface would get i + r = 90^∘ inside the geometry, leading to confusion. Stick with the normal as your reference.

Where the Fresnel distance comes from. Diffraction by a slit of width a produces an angular spread (half-width to first minimum) θ ≈ λa. At distance z from the slit, the linear spread is zθ ≈ zλ/a. Setting this spread equal to the original aperture width a: zλ/a = a z = a²λ. This is the distance at which the diffracted spread "catches up" with the aperture size - the rough boundary between ray-optics and wave-optics regimes.

Big aperture, small λ → big z_F. Note how strongly z_F depends on the aperture (squared). Doubling the aperture quadruples z_F. This is why we can use ray optics confidently for everyday objects (windows, lenses, eyes) - apertures of cm-scale and visible light give z_F of kilometres.

Real-world - telescope resolution. The Fresnel distance idea connects to the resolving power of telescopes. The diffraction-limited angular resolution of a telescope of aperture D is θ ∼ λ/D. The Hubble Space Telescope D ∼ 2.4 m, λ ∼ 550 nm achieves θ ∼ 0.05 arc-seconds - about ten times better than any ground telescope's view through Earth's turbulent atmosphere.

Common mistake - applying ray optics where it fails. For very small apertures e.g., the pupil of the human eye at low light, ∼ 6 mm, with light of ∼ 550 nm, z_F ≈ 65 m - still fine for everyday vision but on the edge for astronomical detail. For wavelengths of microwaves ∼ 1 cm and apertures of cm-scale, z_F is only a few cm and diffraction shows up immediately. Always check the regime.

Red shift vs blue shift. A receding source produces a longer observed wavelength (red shift, λ > 0). An approaching source produces a shorter observed wavelength (blue shift, λ < 0). Same formula, just sign of v.

Why v ≪ c lets us use the non-relativistic formula. The full relativistic Doppler formula for a receding source is _obs0 = √1 + v/c1 - v/c. For v/c ≪ 1, expanding to first order gives λ/₀ ≈ v/c - exactly what we used. Our answer v ≈ 700 km/s is only 0.23% of c, so the non-relativistic formula is accurate to better than one part in 10,000.

Hubble's law and cosmology. In the 1920s, Edwin Hubble noticed that the red shift of distant galaxies is proportional to their distance from us: v = H₀ d, where H₀ is the Hubble constant. This was the first evidence that the universe is expanding. Every distant galaxy's red shift, combined with its distance, places it on a graph that traces back to a single point - the Big Bang, some 13.8 billion years ago.

Real-world - exoplanet detection. Tiny periodic Doppler shifts of a star's spectral lines reveal the wobble caused by an orbiting planet's gravitational pull. The technique can detect planets of Earth's mass around nearby stars. The shifts involved are minute - wavelength changes of order 1 part in 10⁹ - but modern spectrographs can measure them.

Common mistake - confusing wavelength shift with frequency shift. For a receding source, λ increases AND ν decreases since λ = c. The fractional shift is the same in magnitude: Δν/₀ = -λ/₀ = -v/c (to first order). Use whichever quantity the problem provides.

Sanity check. 687 km/s sounds enormous, but is typical of stars in distant galaxies. Stars within our own Milky Way usually have line-of-sight velocities of tens of km/s - far smaller. The high value here suggests a distant extragalactic source.