NCERT Plus Two (Class 12) Physics • Chapter 9

Ray Optics and Optical Instruments

Q: A concave mirror of focal length 20 cm in air is completely immersed in water (n = 4/3). What will be its new focal length?

20 cm (unchanged)! Reflection obeys i = r regardless of surrounding medium, so a mirror's focal length f = R/2 does NOT change when dipped in water (unlike a lens!).

Q: A small ink dot on a paper is viewed through a glass slab of thickness 6 cm and refractive index 1.5. How much does the dot appear raised?

Normal shift Δt = t (1 − 1/n) = 6 × (1 − 1/1.5) = 6 × (1 − 2/3) = 2 cm upward.

Q: State the two necessary conditions for Total Internal Reflection (TIR). In an optical fibre, which has higher refractive index: core or cladding?

Two Conditions: (1) Light must travel from an optically denser medium to a rarer medium. (2) Angle of incidence in the denser medium must exceed the critical angle ( i > i c ). In an optical fibre, the core must have a higher refractive index than the cladding ( n core > n cladding ).

Q: An object 3 cm tall is placed 24 cm in front of a concave mirror of focal length 16 cm. Find the position, nature, and height of the image.

Given u = −24 cm , f = −16 cm , h = +3 cm . Using 1/v + 1/u = 1/f : 1/v = 1/(−16) − 1/(−24) = −1/16 + 1/24 = −1/48 ⇒ v = −48 cm (48 cm in front). m = −v/u = −(−48)/(−24) = −2 . Height h' = m · h = −6 cm → Real, inverted, and magnified (2×) .

Q: With a neat ray diagram, derive the Lens Maker's Formula for a thin double convex lens of refractive index n₂ placed in a medium of refractive index n₁.

Step 1 (First surface R₁ ): Virtual image I₁ at v₁ satisfies n₂/v₁ − n₁/u = (n₂ − n₁)/R₁ . Step 2 (Second surface R₂ ): I₁ acts as virtual object at v₁ (thin lens) forming final image I at v in medium n₁ : n₁/v − n₂/v₁ = (n₁ − n₂)/R₂ . Step 3 (Add both): n₁(1/v − 1/u) = (n₂ − n₁)(1/R₁ − 1/R₂) . Setting u = ∞ ⇒ v = f gives 1/f = (n₂/n₁ − 1)(1/R₁ − 1/R₂) .

Q: Draw the i–δ curve for a triangular glass prism and derive the expression for its refractive index n in terms of prism angle A and minimum deviation D_m.

From prism quadrilateral geometry: r₁ + r₂ = A , and total deviation δ = i + e − A . Plot δ vs i (asymmetric U-shaped curve dipping to unique minimum D m ). At δ = D m (symmetric ray): i = e and r₁ = r₂ = r . Thus 2r = A ⇒ r = A/2 , and D m = 2i − A ⇒ i = (A + D m )/2 . By Snell's Law: n = sin[(A + D m )/2] / sin(A/2) .

Q: A compound microscope uses an objective of focal length 1.2 cm and eyepiece of focal length 3.0 cm. An object sits 1.25 cm from the objective, and final image forms at infinity. Calculate magnifying power (take D = 25 cm).

Objective image: 1/v o = 1/1.2 − 1/1.25 = 5/6 − 4/5 = 1/30 ⇒ v o = 30 cm . Objective magnification: m o = v o / |u o | = 30 / 1.25 = 24 . Eyepiece (image at ∞ ): m e = D / f e = 25 / 3 = 8.33 . Total magnification: m = m o · m e = 24 × 8.33 = 200 .

Q: Draw a schematic diagram of a Cassegrain reflecting telescope and state two advantages of reflecting telescopes over refracting telescopes.

Schematic: Parallel starlight hits large parabolic concave primary mirror → converges toward convex secondary mirror → reflects through a central hole in primary mirror into the eyepiece. Two Advantages: (1) Completely free from chromatic aberration . (2) Primary mirror can be supported across its entire back, making giant apertures much lighter and cheaper than giant glass lenses.

Cartesian sign convention & mirror formula, Snell's law & apparent depth, total internal reflection & optical fibres, lens maker's formula, prism minimum deviation & microscopes/telescopes.

📄 Official NCERT Textbook PDF

Topic 9.2: Cartesian Sign Convention, f = R/2 & Mirror Formula

Pick up a shiny stainless-steel spoon on your dining table! Look into the outer bulging back and you see the whole kitchen shrunk tiny and upright (a convex mirror, just like a KSRTC bus side-mirror). Flip to the inner curved bowl: far away your face flips upside-down, yet close up your eyelashes zoom huge and upright (a concave mirror)!

PARAXIAL FOCAL LENGTHf = R2 (Concave: f < 0, Convex: f > 0)

MIRROR FORMULA1v + 1u = 1f (PLUS in middle!)

MAGNIFICATIONm = h'h = −vu = ff − u

🧠 Memory Trick: Put Pole P at origin (0,0) and shine light Left → Right! Against light = minus (−), along light = plus (+); Concave = dark cave (−), Convex = (+)!

💡 Simple Kid-Check: Light always obeys i = r at the local tangent. Curve a mirror inward (concave) and rays bounce inward to meet at focus F; bulge it outward (convex) and rays spray out as if coming from a virtual F behind the mirror!

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New Cartesian Sign Convention: Measure all horizontal distances from Pole P. Light travels Left → Right. Distances in the direction of incident light are +ve; opposite to incident light are −ve (so real object u is ALWAYS negative). Upward height is +ve; downward is −ve.
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Relation f = R/2: For narrow paraxial rays close to the principal axis, geometry of triangle CMF gives f = R/2. Concave mirrors focus real light in front (f < 0); convex mirrors diverge light (f > 0).
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Mirror Equation & Magnification: 1/v + 1/u = 1/f and m = −v/u. Negative m ⇒ real & inverted image; positive m ⇒ virtual & erect image! Convex mirrors ALWAYS form diminished virtual erect images (wide field of view for vehicles!).

Concave Image Quick-Map: At ∞ → F (point) • Beyond C → between F & C (real, small) • At C (u = −2f) → at C (m = −1) • Between C & F → beyond C (real, huge) • Inside F (|u| < |f|) → behind mirror (virtual, magnified!)

🧒 Super-Easy Example: A concave shaving mirror has R = −30 cm (so f = −15 cm). Hold your chin at u = −10 cm (inside focus):
1/v = 1/(−15) − 1/(−10) = −1/15 + 1/10 = +1/30 ⇒ v = +30 cm (behind mirror!), m = −(+30)/(−10) = +3 (3× upright & virtual!).

Top 3 DHSE mirror traps: (1) Putting a minus sign in the mirror formula! Remember: Mirrors use PLUS (1/v + 1/u = 1/f) and magnification uses MINUS (m = −v/u). (2) Immersing a concave mirror in water DOES NOT change its focal length f (reflection depends only on mirror curvature R, not medium!). (3) Convex mirrors never produce magnified images of real objects.

Topic 9.3: Refraction, Snell's Law & Apparent Depth Shift

Dip a straight steel spoon into a clear glass of lime juice, or look down at pebbles in a crystal-clear Thrissur pond — the spoon looks broken at the waterline and the bottom looks way shallower than it really is! Light slows down when entering water, forcing the beam to pivot at the boundary.

SNELL'S LAWn₁ sin i = n₂ sin r ⇒ sin isin r = n₂₁ = n₂n₁ = v₁v₂ = λ₁λ₂

REFRACTIVE INDEXn = cv (Air ≈ 1, Water ≈ 1.33, Glass ≈ 1.5, Diamond ≈ 2.42)

APPARENT DEPTHd_app = d_realn | Normal shift Δt = t(1 − 1n)

🧠 Memory Trick: R → D = Toward Normal; D → R = Away from Normal ("DR. Away"!). And across ANY boundary, Frequency NEVER changes!

💡 Simple Kid-Check: Picture a toy car rolling from smooth pavement onto thick mud at an angle: the first front wheel that hits the mud slows down while the other wheel is still fast on pavement, twisting the whole car toward the perpendicular line (normal)!

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Laws of Refraction: Incident ray, refracted ray, and normal lie in one plane, and Snell's Law gives n₁ sin i = n₂ sin r. Higher refractive index n = c/v means optically denser and slower speed v.
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Speed & Wavelength vs Frequency: When light enters glass (n = 1.5) from air, speed shrinks to v = c/n and wavelength shrinks to λ' = λ/n, while frequency ν stays identical (colour energy stays intact!).
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Apparent Depth & Atmospheric Refraction: Looking from air into water (n), rays bend away from normal so a coin at depth d appears raised at d_app = d / n (shift = d(1 − 1/n)). Atmospheric refraction also bends sunlight around Earth's curve, giving us ~2 minutes early sunrise and ~2 minutes delayed sunset (4 extra daylight minutes daily!).

🧒 Super-Easy Example: A clear pool is 4 m deep with water (n = 4/3). Looking vertically down from air:
d_app = 4 / (4/3) = 3 m! The bottom appears raised upward by 4 − 3 = 1 m!

Watch three classic DHSE traps: (1) At normal incidence (i = 0°), the ray does NOT bend (r = 0°), yet its speed and wavelength STILL change! (2) Looking from water up at a bird (denser → rarer), the bird looks HIGHER (h_app = n · h), opposite to looking down at a fish (d/n)! (3) Optical density (n) is NOT mass density (e.g. turpentine is optically denser than water yet floats on water!).

Topic 9.4: Total Internal Reflection (TIR), Mirage, Prisms & Optical Fibres

Driving along a sun-baked Palakkad highway at noon, you see a shimmering puddle of water on the dry tar ahead — a desert Mirage! That phantom water, the fiery sparkle of a cut diamond, and the fibre-optic cable beaming high-speed internet across Kerala all run on Total Internal Reflection (TIR).

2 STRICT CONDITIONS1. Travel Denser → Rarer (n₁ > n₂)
2. Incidence angle i > i_c!

CRITICAL ANGLEsin i_c = n_rarern_denser = 1n (at r = 90°)

OPTICAL FIBREn_core > n_cladding! Guides pulses via repeated TIR along curved bends!

🧠 Memory Trick: Critical angles to love: Water ≈ 49°, Crown Glass ≈ 41°–42°, Diamond ≈ 24.4°! Smaller i_c = rays stay trapped inside = extra sparkle!

💡 Simple Kid-Check: As a diver shines a torch upward from underwater at steeper tilt angles, the beam skims closer to the water surface until at 48.7° it skims right along the top (r = 90°). Tilt just a tiny bit more — the water surface turns into a 100% lossless mirror!

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Deriving Critical Angle i_c: Apply Snell's Law at grazing emergence (i = i_c, r = 90°): n sin i_c = 1 · sin 90° = 1 ⇒ sin i_c = 1 / n. For i > i_c, ZERO refraction occurs — 100% Total Internal Reflection!
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Mirage & Sparkling Diamonds: Hot air touching sunny roads has lower n than cooler upper air; descending sky rays bend away from normal until TIR flips them upward into your eyes like a water reflection! Diamond (n = 2.42) has i_c = 24.4°, trapping light for brilliant multiple bounces.
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TIR Prisms & Optical Fibres: Right-angled isosceles glass prisms (45°–90°–45°, i_c ≈ 42° < 45°) bend light by 90° or 180° in binoculars with ZERO energy loss (silvered mirrors absorb light and tarnish!). Optical fibres wrap a high-n silica core inside a slightly lower-n cladding for telecom & medical endoscopes.

🧒 Super-Easy Example: Glass has refractive index n = √2 ≈ 1.414 in air. Critical angle sin i_c = 1/√2 ⇒ i_c = 45°. Any ray hitting inside at i = 48° (> 45°) suffers 100% TIR!

Top 3 Kerala HSE traps: (1) Can TIR happen going from air to water? NEVER! Must go Denser → Rarer! (2) At i = i_c, angle of refraction is 90° (NOT 0°!). (3) In optical fibres, students frequently swap core and cladding indices — remember n_core > n_cladding!

Topic 9.5: Spherical Surface Refraction, Lens Maker's Formula & Power Combination

Look closely at a clear raindrop sitting on a colocasia leaf or your grandfather's reading glasses — light bends TWICE, once entering the curved front face and once exiting the curved back face! How do eyewear factories predict the exact curve radii needed for a +2 D lens? With the Lens Maker's Formula!

SPHERICAL SURFACEn₂v − n₁u = n₂ − n₁R

LENS MAKER'S FORMULA1f = (n₂n₁ − 1)(1R₁ − 1R₂) | 1v − 1u = 1f

POWER & COMBINATIONP = 1f(m) (Dioptre D) | P = P₁ + P₂, m = m₁ m₂

🧠 Memory Trick: Mirrors vs Lenses signs flip! Lens equation has MINUS (1/v − 1/u = 1/f), Magnification has NO minus (m = +v/u)! Equiconvex in air: R₁ = +R, R₂ = −R!

💡 Simple Kid-Check: A thin lens is just two curved glass bowls glued back-to-back! The first curved face bends incoming light toward a temporary image I₁, which immediately acts as the object for the second curved face — adding both steps magically cancels I₁ to give the Lens Maker's Formula!

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Refraction at Single Spherical Surface: For paraxial rays crossing from medium n₁ to n₂ across radius R: n₂/v − n₁/u = (n₂ − n₁)/R.
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Lens Maker's & Thin Lens Formula: Combining surface 1 (R₁) and surface 2 (R₂) in medium n₁ gives 1/f = (n₂/n₁ − 1)(1/R₁ − 1/R₂). Notice f depends on BOTH lens refractive index n₂ AND surrounding medium n₁!
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Power & Thin Lenses in Contact: Power P = 1 / f(in metres) measured in dioptre (D) (Convex: +, Concave: −). Touching thin lenses act like a single lens of focal length 1/f = 1/f₁ + 1/f₂ and power P = P₁ + P₂!

🧒 Super-Easy Example: A glass biconvex lens (n₂ = 1.5) has f = +20 cm in air (n₁ = 1). Dip it in water (n₁ = 4/3):
Air factor: (1.5 − 1) = 0.5; Water factor: (1.5/(4/3) − 1) = 1.125 − 1 = 0.125 (4× smaller!). So f_water = 4 × 20 = +80 cm!

Massive Kerala HSE exam favourite: What happens if you immerse a convex glass lens (n_g = 1.5) into carbon disulphide (n_L = 1.63, so n_L > n_g)? The factor (n_g/n_L − 1) turns NEGATIVE — the convex lens flips into a DIVERGING (concave) lens! And if n_L = n_g, 1/f = 0 ⇒ f = ∞ (it disappears!). Also always use f in metres for Dioptres!

Topic 9.6: Refraction through a Prism & Minimum Deviation Formula

Place a triangular glass prism on a sunlit verandah floor in Kerala and watch a crisp white beam shatter into a glowing VIBGYOR rainbow across the wall! Why doesn't a flat windowpane do that? Because a prism's two tilted faces bend light in the same direction twice!

PRISM GEOMETRYr₁ + r₂ = A & Total Deviation δ = i + e − A

MINIMUM DEVIATIONAt δ = D_m: Symmetric ray! i = e & r₁ = r₂ = A/2

PRISM FORMULAn = sin[(A + D_m)/2]sin(A/2) | Thin prism: D_m ≈ (n − 1)A

🧠 Memory Trick: Think "Symmetric Bottom of the U!" — At D_m, ray inside runs parallel to base (i = e, r₁ = r₂ = A/2). And in dispersion, Violet Bends Violently (Most!) because n_V > n_R!

💡 Simple Kid-Check: Inside a triangular prism, light climbs in on the left face and slides out on the right face. Tilt the prism slowly: deviation drops to one lowest sweet spot D_m where left and right angles match symmetrically, then rises back up!

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Two Basic Angle Equations: Normals at the two faces enclose 90° with the glass surfaces inside a quadrilateral, yielding r₁ + r₂ = A. Adding deviations at both faces gives δ = (i − r₁) + (e − r₂) = i + e − A.
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The Asymmetric δ vs i Curve: Two incidence angles (i and e, by reversibility!) produce the exact same deviation δ, EXCEPT at one unique minimum D_m where they merge: i = e and r₁ = r₂ = A/2!
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Deriving the Prism Formula (4-Mark Derivation!): At δ = D_m, D_m = 2i − A ⇒ i = (A + D_m)/2 and r = A/2. By Snell's Law: n = sin i / sin r = sin[(A + D_m)/2] / sin(A/2). For small prism angle A: D_m ≈ (n − 1)A.

🧒 Super-Easy Example: An equilateral glass prism (A = 60°) gives minimum deviation D_m = 30°. Find its refractive index:
n = sin[(60° + 30°)/2] / sin(60°/2) = sin 45° / sin 30° = (1/√2) / (1/2) = √2 ≈ 1.414!

Watch three classic DHSE traps: (1) The δ vs i curve is NOT symmetric — left side (small i) drops steeply while right side rises gently! Always draw it asymmetric for full marks. (2) At minimum deviation, i = e and r₁ = r₂ = A/2, NOT i = r₁! (3) Don't mix up Violet vs Red: shorter wavelength Violet sees higher n, so δ_V > δ_R!

Topic 9.7: Simple/Compound Microscopes & Astronomical Refracting/Reflecting Telescopes

How can a biology lab microscope zoom into a single red blood cell while an astronomical telescope atop Mount Abu or Idukki reveals Jupiter's moons? Both pair an Objective and an Eyepiece — yet built with completely flipped focal lengths for tiny close bugs vs giant distant stars!

SIMPLE MICROSCOPEAt Near Point D=25 cm: m = 1 + Df • Relaxed (∞): m = Df

COMPOUND MICROSCOPEm ≈ (Lf_o)(1 + Df_e) or (Lf_o)(Df_e) • Both f_o, f_e SMALL!

ASTRONOMICAL TELESCOPEAt ∞: m = f_of_e, L = f_o + f_e • HUGE f_o, TINY f_e!

🧠 Memory Trick: Microscope = Mini f_o (f_o < f_e, both short!); Telescope = Towering f_o (f_o ≫ f_e + wide aperture!). Cassegrain Mirror Telescope = Zero chromatic aberration + easy back support!

💡 Simple Kid-Check: To look bigger, an object must cast a wider angle on your eye! A microscope pushes a tiny cell super close using a short-focus lens; a telescope collects starlight over a huge lens/mirror and funnels that wide angle straight into your pupil!

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Simple Microscope (Magnifying Glass): Convex lens with object at u ≤ f gives upright virtual image. Maximum magnification at near point (v = −D): m = 1 + D/f; comfortable relaxed eye (v = −∞): m = D/f.
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Compound Microscope: Two short convex lenses (f_o < f_e). Objective forms real inverted magnified image inside tube; Eyepiece acts as simple magnifier: m = m_o · m_e = (v_o/u_o)(1 + D/f_e) ≈ (L/f_o)(D/f_e) (at ∞).
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Astronomical Refracting & Reflecting (Cassegrain) Telescopes: Refracting telescope uses huge f_o and small f_e: m = f_o/f_e, tube length L = f_o + f_e (at ∞); at near point D: m = (f_o/f_e)(1 + f_e/D). Modern observatories use Cassegrain Reflecting Telescopes (parabolic concave mirror + convex secondary mirror) because mirrors eliminate chromatic aberration, avoid heavy sagging edges, and collect huge light cheaply!

Instrument	Objective (`f_o`)	Eyepiece (`f_e`)	Normal Adj. (`∞`) Magnification	Final Image
Compound Microscope	Very small `f_o`, small aperture	Small `f_e` (> `f_o`)	`(L / f_o) · (D / f_e)`	Virtual, inverted, giant
Astronomical Telescope	Very large `f_o`, huge aperture	Small `f_e`	`f_o / f_e` (`L = f_o + f_e`)	Virtual, inverted, magnified angle
Cassegrain Reflector	Parabolic concave mirror + convex 2°	Small `f_e`	`f_o / f_e` (`f_o = R/2`)	Bright, zero colour fringes!

🧒 Super-Easy Example: A telescope has f_o = 140 cm and f_e = 5 cm in normal adjustment (∞):
Magnifying power m = 140 / 5 = 28, and separation between lenses L = 140 + 5 = 145 cm!

Top 3 DHSE traps: (1) Choosing lens pairs from a table: for a Microscope, pick the two smallest focal lengths (smallest as objective!); for a Telescope, pick the largest f & widest aperture as objective and smallest f as eyepiece! (2) Confusing normal adjustment (∞) vs near-point (D) formulas — check whether the question says "relaxed eye / normal adjustment"! (3) Always list no chromatic aberration and lighter support / cheaper grinding for reflecting telescopes!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 1 Marks): A concave mirror of focal length 20 cm in air is completely immersed in water (n = 4/3). What will be its new focal length?

20 cm (unchanged)! Reflection obeys i = r regardless of surrounding medium, so a mirror's focal length f = R/2 does NOT change when dipped in water (unlike a lens!).

Q2 (Kerala HSE Say • 2 Marks): A small ink dot on a paper is viewed through a glass slab of thickness 6 cm and refractive index 1.5. How much does the dot appear raised?

Normal shift Δt = t (1 − 1/n) = 6 × (1 − 1/1.5) = 6 × (1 − 2/3) = 2 cm upward.

Q3 (Kerala HSE March • 2 Marks): State the two necessary conditions for Total Internal Reflection (TIR). In an optical fibre, which has higher refractive index: core or cladding?

Two Conditions: (1) Light must travel from an optically denser medium to a rarer medium. (2) Angle of incidence in the denser medium must exceed the critical angle (i > i_c).
In an optical fibre, the core must have a higher refractive index than the cladding (n_core > n_cladding).

Q4 (Kerala HSE March • 3 Marks): An object 3 cm tall is placed 24 cm in front of a concave mirror of focal length 16 cm. Find the position, nature, and height of the image.

Given u = −24 cm, f = −16 cm, h = +3 cm.
Using 1/v + 1/u = 1/f: 1/v = 1/(−16) − 1/(−24) = −1/16 + 1/24 = −1/48 ⇒ v = −48 cm (48 cm in front).
m = −v/u = −(−48)/(−24) = −2.
Height h' = m · h = −6 cm → Real, inverted, and magnified (2×).

Q5 (Kerala HSE Model • 4 Marks): With a neat ray diagram, derive the Lens Maker's Formula for a thin double convex lens of refractive index n₂ placed in a medium of refractive index n₁.

Step 1 (First surface R₁): Virtual image I₁ at v₁ satisfies n₂/v₁ − n₁/u = (n₂ − n₁)/R₁.
Step 2 (Second surface R₂): I₁ acts as virtual object at v₁ (thin lens) forming final image I at v in medium n₁: n₁/v − n₂/v₁ = (n₁ − n₂)/R₂.
Step 3 (Add both): n₁(1/v − 1/u) = (n₂ − n₁)(1/R₁ − 1/R₂).
Setting u = ∞ ⇒ v = f gives 1/f = (n₂/n₁ − 1)(1/R₁ − 1/R₂).

Q6 (Kerala HSE March • 4 Marks): Draw the i–δ curve for a triangular glass prism and derive the expression for its refractive index n in terms of prism angle A and minimum deviation D_m.

From prism quadrilateral geometry: r₁ + r₂ = A, and total deviation δ = i + e − A.
Plot δ vs i (asymmetric U-shaped curve dipping to unique minimum D_m).
At δ = D_m (symmetric ray): i = e and r₁ = r₂ = r.
Thus 2r = A ⇒ r = A/2, and D_m = 2i − A ⇒ i = (A + D_m)/2.
By Snell's Law: n = sin[(A + D_m)/2] / sin(A/2).

Q7 (Kerala HSE Say • 3 Marks): A compound microscope uses an objective of focal length 1.2 cm and eyepiece of focal length 3.0 cm. An object sits 1.25 cm from the objective, and final image forms at infinity. Calculate magnifying power (take D = 25 cm).

Objective image: 1/v_o = 1/1.2 − 1/1.25 = 5/6 − 4/5 = 1/30 ⇒ v_o = 30 cm.
Objective magnification: m_o = v_o / |u_o| = 30 / 1.25 = 24.
Eyepiece (image at ∞): m_e = D / f_e = 25 / 3 = 8.33.
Total magnification: m = m_o · m_e = 24 × 8.33 = 200.

Q8 (Kerala HSE March • 2 Marks): Draw a schematic diagram of a Cassegrain reflecting telescope and state two advantages of reflecting telescopes over refracting telescopes.

Schematic: Parallel starlight hits large parabolic concave primary mirror → converges toward convex secondary mirror → reflects through a central hole in primary mirror into the eyepiece.
Two Advantages: (1) Completely free from chromatic aberration. (2) Primary mirror can be supported across its entire back, making giant apertures much lighter and cheaper than giant glass lenses.