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State Board · Class 11 · Physics

Lesson 9: Optics

7 concepts · 56 practice questions with answers · Free to read

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Reflection and spherical mirrors

From daily life

Look into a spoon's inside and your face is upside down; its outside shows a smaller erect face. Link to earlier classes : reflection, refraction and lenses of classes 9-10 are now handled with sign convention and formulas.

Simple definition

Reflection : i = r; a spherical mirror is a part of a sphere.

Concave mirrorPFCOI1/v + 1/u = 1/f ; f = R/2

In the figure : Concave mirror : real, inverted image.

Live figure

Reflection and spherical mirrors

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Rule 1: Sign convention — Pole P is the origin; distances along the incident light are +, opposite −. Object on the left ⇒ u is −. Concave : f −, convex : f +.

  • Easy : Concave mirror, R = 20 cm ⇒ f = −10 cm.
  • Medium : Convex R = 30 cm ⇒ f = +15 cm.
  • Exam (CET) : f of a concave mirror? (1) − (2) + (3) 0 (4) infinite → answer 1)

Rule 2: Mirror formula — 1/v + 1/u = 1/f; m = − v/u = h'/h; f = R/2.

  • Easy : f = −10, u = −30 ⇒ v = −15 cm; m = −0.5 (real, inverted, smaller).
  • Medium : Convex f = +15, u = −30 ⇒ v = +10 cm; m = +1/3.
  • Exam (CET) : u = 2f (concave) ⇒ m? (1) −1 (2) +1 (3) 0 (4) 2 → answer 1) Trick : object at C ⇒ image at C, same size.

Rule 3: Nature of images — Concave : u > f ⇒ real, inverted; u < f ⇒ virtual, erect, magnified. Convex : always virtual, erect, diminished.

  • Easy : Shaving mirror : object inside F ⇒ magnified virtual image.
  • Medium : Vehicle mirror : convex ⇒ wide field.
  • Exam (CET) : Image in a convex mirror? (1) virtual (2) real (3) inverted (4) magnified → answer 1)

Derivation : Mirror formula (small aperture)

  • Step 1 : From similar triangles : m = −v/u.
  • Step 2 : From triangles at F : h'/h = (v − f)/f.
  • Step 3 : −v/u = (v − f)/f ⇒ −v f = u v − u f.
  • Step 4 : Divide by uvf : 1/v + 1/u = 1/f.
Remember

1/v + 1/u = 1/f; m = −v/u; f = R/2.

Common mistake / CET trap

u is always −; f − for concave; negative m = inverted.

Shortcut

Memory clue : "1/v plus 1/u equals 1/f".

Spot the difference

MirrorfImage
Concave−real/virtual
Convex+virtual, diminished
Plane∞virtual, same size

Practice questions

Easy
  1. 1Mirror formula?
  2. 2Relation between f and R?
  3. 3Which mirror is a shaving mirror?
Medium
  1. 4f = −12, u = −24 : v?
  2. 5f = +20, u = −40 : v, m?
  3. 6R = 40 concave : f?
Challenging
  1. 7Concave f = −10, m = −2 : u?
  2. 8Derive the mirror formula.
Show answers (try them yourself first)
  1. 11/v + 1/u = 1/f
  2. 2f = R/2
  3. 3Concave
  4. 4v = −24 cm
  5. 5v = 40/3 = 13.3 cm; m = +1/3
  6. 6−20 cm
  7. 7u = −15 cm (v = −30)
  8. 8Using similar triangles

Refraction and refractive index

From daily life

A stick in water looks bent; a pond looks shallower.

Simple definition

Refraction is the change of direction of light on entering another medium.

Refractionirn₁n₂n₁ sin i = n₂ sin rn = c ÷ v

In the figure : i and r are measured from the normal.

Rule 1: Snell's law — n₁ sin i = n₂ sin r; n = c ÷ v = sin i ÷ sin r; water 1.33, glass 1.5, diamond 2.42.

  • Easy : Air → glass, i = 30°, n = 1.5 ⇒ sin r = 0.333 ⇒ r = 19.5°.
  • Medium : v = 2.25 × 10⁸ in the medium ⇒ n = 1.33.
  • Exam (CET) : In a denser medium the ray? (1) bends toward the normal (2) away (3) straight (4) reflects → answer 1) Trick : 'dense ⇒ toward normal'.

Rule 2: Real and apparent depth — Viewed normally : n = real depth ÷ apparent depth. Apparent = real ÷ n.

  • Easy : 12 cm of water, n = 1.33 ⇒ apparent = 9.02 cm.
  • Medium : Glass slab t, n ⇒ object shifts by t (1 − 1/n) = 6 × (1/3) = 2 cm (t = 6, n = 1.5).
  • Exam (CET) : A pond looks? (1) shallower (2) deeper (3) same (4) zero → answer 1)

Rule 3: Glass slab — The emergent ray is parallel to the incident ray; lateral shift = t sin(i − r) ÷ cos r.

  • Easy : t = 4 cm, i = 30°, r = 19.5° ⇒ shift = 0.78 cm.
  • Medium : i = 0 ⇒ shift 0.
  • Exam (CET) : Emergent ray? (1) parallel (2) perpendicular (3) deviated (4) none → answer 1)

Derivation : Apparent depth

  • Step 1 : Object O at depth h; ray meets the surface at i, r (small angles).
  • Step 2 : In water sin ≈ x/h; in air ≈ x/h'.
  • Step 3 : n sin i = sin r ⇒ n (x/h) = x/h'.
  • Step 4 : h' = h ÷ n (apparent depth).
Remember

n₁ sin i = n₂ sin r; n = c/v; apparent = real/n.

Common mistake / CET trap

Angles from the normal, not the surface; f unchanged, λ changes.

Shortcut

Memory clue : "Dense = toward the normal".

Spot the difference

Mediumnv
Air13 × 10⁸
Water1.332.25 × 10⁸
Glass1.52 × 10⁸
Diamond2.421.24 × 10⁸

Practice questions

Easy
  1. 1State Snell's law.
  2. 2Formula for n?
  3. 3Formula for apparent depth?
Medium
  1. 4i = 45°, n = 1.5 : r?
  2. 5v = 2 × 10⁸ in glass : n?
  3. 6Depth 8 cm, n = 1.33 : apparent?
Challenging
  1. 7t = 9 cm, n = 1.5 : shift of the object?
  2. 8Derive apparent depth.
Show answers (try them yourself first)
  1. 1n₁ sin i = n₂ sin r
  2. 2c/v
  3. 3real ÷ n
  4. 428.1°
  5. 51.5
  6. 66.02 cm
  7. 73 cm
  8. 8Small angles; n = h/h'

Total internal reflection

From daily life

Seen from under water, the surface can shine like a mirror.

Simple definition

TIR: light from a dense medium at i > C returns completely.

Total internal reflectionair (rare)water (dense)i = Ci > Csin C = 1/n

In the figure : Green : refraction; orange : i = C; red : TIR.

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Total internal reflection

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Rule 1: Critical angle — Dense ⇒ rare; sin C = 1 ÷ n (n = dense/rare). i > C ⇒ total reflection.

  • Easy : Glass n = 1.5 ⇒ C = 41.8°.
  • Medium : Water 1.33 ⇒ 48.8°; diamond 2.42 ⇒ 24.4°.
  • Exam (CET) : Larger n ⇒ C? (1) smaller (2) larger (3) same (4) 90° → answer 1) Trick : diamond sparkles because C is small.

Rule 2: Conditions — (1) dense ⇒ rare, (2) i > C. At i = C the ray grazes the surface (r = 90°).

  • Easy : Air ⇒ water : no TIR.
  • Medium : Water ⇒ air, i = 60° > 48.8° ⇒ TIR.
  • Exam (CET) : TIR when? (1) dense→rare, i > C (2) rare→dense (3) any (4) i < C → answer 1)

Rule 3: Applications — Optical fibre (core n₁ > cladding n₂), sparkle of diamond, total-reflecting prisms, mirage.

  • Easy : Fibre n₁ = 1.5, n₂ = 1.4 ⇒ sin C = 0.933 ⇒ C = 69.0°.
  • Medium : Point source 2 m deep : circle radius = h tan C = 2.28 m.
  • Exam (CET) : Fibre principle? (1) TIR (2) refraction (3) diffraction (4) polarisation → answer 1)

Derivation : sin C = 1/n

  • Step 1 : At i = C, r = 90° : n_dense sin C = n_rare sin 90°.
  • Step 2 : n_rare = 1 (air).
  • Step 3 : n sin C = 1.
  • Step 4 : sin C = 1 ÷ n.
Remember

sin C = 1/n; TIR : dense → rare, i > C.

Common mistake / CET trap

TIR only dense → rare; n is the dense medium's.

Shortcut

Memory clue : "sin C = 1 over n".

Spot the difference

CaseConditionResult
Refractioni < Cpartial reflection
Criticali = Cr = 90°
TIRi > C100 % reflection

Practice questions

Easy
  1. 1What is the critical angle?
  2. 2Conditions for TIR?
  3. 3State one use.
Medium
  1. 4n = 2 : C?
  2. 5n = 1.6 : sin C?
  3. 6n = 1.33, h = 3 m : circle radius?
Challenging
  1. 7C = 30° : n?
  2. 8How does a mirage form?
Show answers (try them yourself first)
  1. 1The i for which r = 90°
  2. 2Dense→rare; i > C
  3. 3Optical fibre
  4. 430°
  5. 50.625
  6. 63.42 m
  7. 72
  8. 8Rare hot air below ⇒ TIR of rays

Refraction at spherical surfaces and lenses

From daily life

A lens can focus the Sun's image on paper and burn it.

Simple definition

A lens is a transparent medium bounded by spherical surfaces.

Convex lensFF'1/v − 1/u = 1/f ; P = 1/f

In the figure : A real image from a convex lens.

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Refraction at spherical surfaces and lenses

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Rule 1: Thin lens — 1/v − 1/u = 1/f; m = v/u; convex f +, concave f −.

  • Easy : f = +20, u = −30 ⇒ v = +60; m = −2 (real, inverted, magnified).
  • Medium : f = −20, u = −30 ⇒ v = −12; m = +0.4.
  • Exam (CET) : Convex, u = −2f ⇒ v? (1) +2f (2) +f (3) −2f (4) infinite → answer 1) Trick : 2f ⇒ 2f.

Rule 2: Lens-maker's formula — 1/f = (n − 1)(1/R₁ − 1/R₂) (in air). P = 1/f (m⁻¹ = D).

  • Easy : n = 1.5, R₁ = +20, R₂ = −20 cm ⇒ f = 20 cm.
  • Medium : f = 40 cm ⇒ P = +2.5 D.
  • Exam (CET) : Unit of P? (1) dioptre (2) m (3) cm (4) J → answer 1)

Rule 3: Combination of lenses — In contact : 1/f = 1/f₁ + 1/f₂, P = P₁ + P₂; m = m₁ × m₂.

  • Easy : +4 D and −1.5 D ⇒ +2.5 D (f = 40 cm).
  • Medium : f₁ = 10, f₂ = 10 ⇒ f = 5 cm.
  • Exam (CET) : P₁ = +3, P₂ = +2 ⇒ P? (1) +5 D (2) +6 D (3) +1 D (4) 0 → answer 1)

Derivation : Thin lens : two refractions

  • Step 1 : First surface : n/v' − 1/u = (n − 1)/R₁.
  • Step 2 : Second surface : 1/v − n/v' = (1 − n)/R₂.
  • Step 3 : Adding : 1/v − 1/u = (n − 1)(1/R₁ − 1/R₂).
  • Step 4 : u = ∞ ⇒ v = f : 1/f = (n − 1)(1/R₁ − 1/R₂).
Remember

1/v − 1/u = 1/f; P = 1/f; P = P₁ + P₂.

Common mistake / CET trap

Lens : 1/v − 1/u (mirror : +); use metres for dioptres.

Shortcut

Memory clue : "Lens = minus; mirror = plus".

Spot the difference

LensfP
Convex++
Concave−−
Combination1/f₁ + 1/f₂P₁ + P₂

Practice questions

Easy
  1. 1Lens formula?
  2. 2Formula for P?
  3. 3P for a combination?
Medium
  1. 4f = 15, u = −45 : v, m?
  2. 5f = 50 cm : P?
  3. 6R₁ = 15, R₂ = −15, n = 1.5 : f?
Challenging
  1. 7+5 D and −2 D in contact : f?
  2. 8Derive the lens-maker's formula.
Show answers (try them yourself first)
  1. 11/v − 1/u = 1/f
  2. 21/f
  3. 3P₁ + P₂
  4. 4v = 22.5 cm; m = −0.5
  5. 5+2 D
  6. 615 cm
  7. 7+3 D ⇒ f = 33.3 cm
  8. 8Adding the two surface refractions

Dispersion and prisms

From daily life

Sunlight through a glass can paint a coloured band on a wall.

Simple definition

Dispersion is the splitting of white light into its colours.

Dispersion by a prismRVδ = (n − 1) A ; ω = (n_v − n_r)/(n_y − 1)A

In the figure : Red bends least; violet bends most.

Live figure

Dispersion and prisms

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Rule 1: Prism formula — n = sin[(A + δ_m) ÷ 2] ÷ sin(A ÷ 2); for small A, δ = (n − 1) A.

  • Easy : A = 60°, n = 1.5 ⇒ δ_m = 37.2°.
  • Medium : A = 4°, n = 1.5 ⇒ δ = 2°.
  • Exam (CET) : At minimum deviation the ray inside is? (1) parallel to the base (2) perpendicular (3) reflected (4) doubled → answer 1)

Rule 2: Dispersion — n decreases with λ : n_V > n_R; violet deviates more. Dispersive power ω = (n_V − n_R) ÷ (n_Y − 1); angular dispersion = (n_V − n_R) A.

  • Easy : n_V = 1.532, n_R = 1.513, n_Y = 1.519 ⇒ ω = 0.0366.
  • Medium : A = 5° ⇒ angular dispersion = 0.095°.
  • Exam (CET) : Deviates least? (1) red (2) green (3) blue (4) violet → answer 1) Trick : 'red is lazy'.

Rule 3: Achromatic combination — Cancel dispersion, keep deviation : ω₁ δ₁ + ω₂ δ₂ = 0 (two thin prisms placed oppositely). For lenses : ω₁/f₁ + ω₂/f₂ = 0.

  • Easy : Crown ω = 0.02, δ = 5.2°; flint ω = 0.04 ⇒ δ₂ = 2.6°.
  • Medium : Lens : f₁ = 20 (ω₁ = 0.02) ⇒ f₂ = −40 (ω₂ = 0.04).
  • Exam (CET) : Achromatic means? (1) colour-free (2) monochrome (3) thick (4) transparent → answer 1)

Derivation : δ = (n − 1) A (small A)

  • Step 1 : Small angles ⇒ sin x ≈ x.
  • Step 2 : At the first face i = n r₁; at the second e = n r₂.
  • Step 3 : δ = i + e − A = n (r₁ + r₂) − A.
  • Step 4 : r₁ + r₂ = A ⇒ δ = (n − 1) A.
Remember

δ = (n − 1)A; ω = (n_V − n_R)/(n_Y − 1).

Common mistake / CET trap

The small-A formula fails for large A; ω has no unit.

Shortcut

Memory clue : "Red is lazy, violet is quick".

Spot the difference

Colourn (glass)Deviation
Red1.513least
Yellow1.519medium
Violet1.532most

Practice questions

Easy
  1. 1What is dispersion?
  2. 2Formula for ω?
  3. 3Which colour has the largest n?
Medium
  1. 4A = 6°, n = 1.5 : δ?
  2. 5n_V = 1.65, n_R = 1.61, n_Y = 1.63 : ω?
  3. 6A = 60°, δ_m = 30° : n?
Challenging
  1. 7Why ω₁δ₁ + ω₂δ₂ = 0?
  2. 8Derive δ = (n − 1)A.
Show answers (try them yourself first)
  1. 1Splitting white light into colours
  2. 2(n_V − n_R)/(n_Y − 1)
  3. 3Violet
  4. 43°
  5. 50.063
  6. 61.41
  7. 7Equal and opposite angular dispersion
  8. 8Small angles; r₁ + r₂ = A

Natural phenomena and lens defects

From daily life

Some cannot read the board; some hold a book far away to read.

Simple definition

Eye defects are corrected by lenses; sky phenomena arise from scattering/refraction.

Eye defectsmyopiahypermetropianormalconcaveconvexP = 1/f ; 1/f = 1/v − 1/u

In the figure : Myopia, hypermetropia and a normal eye.

Rule 1: Myopia — Distant objects blurred; image forms before the retina. Concave lens : f = −(far point). Far point x ⇒ P = −1/x.

  • Easy : Far point 50 cm ⇒ f = −50 cm, P = −2 D.
  • Medium : Far point 2 m ⇒ P = −0.5 D.
  • Exam (CET) : Myopia correction? (1) concave (2) convex (3) cylindrical (4) plane → answer 1)

Rule 2: Hypermetropia — Near objects blurred; near point > 25 cm. Convex lens : an object at 25 cm gets a virtual image at the near point : 1/f = 1/v − 1/u.

  • Easy : Near point 100 cm : 1/f = −1/100 + 1/25 = 0.03 ⇒ f = 33.3 cm, P = +3 D.
  • Medium : Near point 50 cm ⇒ P = +2 D.
  • Exam (CET) : For hypermetropia? (1) convex (2) concave (3) plane (4) prism → answer 1)

Rule 3: Natural phenomena — Rainbow : refraction + TIR + dispersion (red ≈ 42°, violet ≈ 40°). Blue sky : Rayleigh : I ∝ 1/λ⁴. Twinkling : changing n of air.

  • Easy : λ_red = 700 nm, λ_blue = 450 nm ⇒ scattering ratio = (700/450)⁴ = 5.9.
  • Medium : Red at sunset : blue is scattered away, red arrives.
  • Exam (CET) : Why is the sky blue? (1) Rayleigh scattering (2) reflection (3) TIR (4) diffraction → answer 1)

Derivation : Lens for hypermetropia

  • Step 1 : Normal object at u = −25 cm.
  • Step 2 : The lens gives a virtual image at v = −near point (e.g. −100 cm).
  • Step 3 : 1/f = 1/v − 1/u = −1/100 + 1/25.
  • Step 4 : 1/f = 0.03 cm⁻¹ ⇒ P = +3 D.
Remember

Myopia : concave; hypermetropia : convex; I ∝ 1/λ⁴.

Common mistake / CET trap

P = 1/f (in metres); P is negative for myopia.

Shortcut

Memory clue : "Near-sighted = concave, far-sighted = convex".

Spot the difference

DefectCauseLens
Myopiaimage before retinaconcave
Hypermetropiaimage behind retinaconvex
Astigmatismuneven curvaturecylindrical
Presbyopiaweak muscles with agebifocal

Practice questions

Easy
  1. 1What is myopia?
  2. 2Which lens for hypermetropia?
  3. 3Rayleigh's law?
Medium
  1. 4Far point 25 cm : P?
  2. 5Near point 40 cm : P (u = −25)?
  3. 6λ halved ⇒ scattering?
Challenging
  1. 7−2 D spectacles : far point?
  2. 8How does a rainbow form?
Show answers (try them yourself first)
  1. 1Far objects blurred, image before the retina
  2. 2Convex
  3. 3I ∝ 1/λ⁴
  4. 4−4 D
  5. 5+1.5 D
  6. 616 times
  7. 750 cm
  8. 8Refraction, dispersion, TIR, refraction

Optical instruments

From daily life

A telescope brings far objects near; a microscope enlarges small ones.

Simple definition

Angular magnification m = β/α.

Astronomical telescopeobjectiveeyepiecem = −f₀/f_e ; L = f₀ + f_e

In the figure : A two-lens telescope.

Rule 1: Simple microscope — m = 1 + D/f (image at near point); m = D/f (at infinity). D = 25 cm.

  • Easy : f = 5 cm ⇒ m = 1 + 5 = 6; at infinity 5.
  • Medium : f = 2.5 cm ⇒ m = 11.
  • Exam (CET) : Smaller f ⇒ m? (1) larger (2) smaller (3) same (4) 0 → answer 1)

Rule 2: Compound microscope — m = (L ÷ f₀)(D ÷ f_e) (image at infinity), L = tube length; the objective has a small f₀.

  • Easy : f₀ = 1 cm, f_e = 5 cm, L = 15 cm ⇒ m = 15 × 5 = 75.
  • Medium : f_e halved ⇒ m doubles.
  • Exam (CET) : m = ? (1) (L/f₀)(D/f_e) (2) f₀/f_e (3) L/D (4) 1 + D/f → answer 1)

Rule 3: Telescope — m = f₀ ÷ f_e (normal adjustment); tube L = f₀ + f_e; reflecting telescope : large aperture, no chromatic fault.

  • Easy : f₀ = 100, f_e = 5 ⇒ m = 20; L = 105 cm.
  • Medium : L = 60, m = 9 ⇒ f_e = 6, f₀ = 54 cm.
  • Exam (CET) : Telescope objective f₀? (1) large (2) small (3) zero (4) equal f_e → answer 1) Trick : far-seeing ⇒ large f₀.

Derivation : Telescope : m = f₀/f_e

  • Step 1 : The objective forms an image at f₀; angle subtended α = h/f₀.
  • Step 2 : The angle seen through the eyepiece β = h/f_e.
  • Step 3 : m = β/α.
  • Step 4 : m = f₀ ÷ f_e.
Remember

m = 1 + D/f; m = (L/f₀)(D/f_e); m = f₀/f_e.

Common mistake / CET trap

D = 25 cm; large f₀ in a telescope, small in a microscope.

Shortcut

Memory clue : "Telescope = f₀ / f_e".

Spot the difference

InstrumentmLength
Simple microscope1 + D/f—
Compound(L/f₀)(D/f_e)L
Telescopef₀/f_ef₀ + f_e

Practice questions

Easy
  1. 1m of a simple microscope?
  2. 2m of a telescope?
  3. 3What is D?
Medium
  1. 4f = 10 cm : m (near point)?
  2. 5f₀ = 80, f_e = 4 : m and L?
  3. 6f₀ = 2, f_e = 10, L = 20 : m?
Challenging
  1. 7L = 55, m = 10 : f₀, f_e?
  2. 8Why use a reflecting telescope?
Show answers (try them yourself first)
  1. 11 + D/f
  2. 2f₀/f_e
  3. 325 cm (near point)
  4. 43.5
  5. 5m = 20; L = 84 cm
  6. 6(20/2)(25/10) = 25
  7. 7f_e = 5, f₀ = 50 cm
  8. 8Large aperture, no chromatic fault

Quick revision of the whole lesson

ConceptKey formula / factUnit / dimensions
Mirror1/v + 1/u = 1/f; m = −v/um; —
Snelln₁ sin i = n₂ sin r—
Critical anglesin C = 1/n—
Lens1/v − 1/u = 1/fm
PowerP = 1/fD = m⁻¹
Prismδ = (n − 1)A—
RayleighI ∝ 1/λ⁴—
Telescopem = f₀/f_e—

🎯 Most important points for the exam

  • Mirror : 1/v + 1/u = 1/f; u is always −.
  • Dense = toward the normal; apparent = real/n.
  • TIR : dense → rare, i > C; sin C = 1/n.
  • Lens : 1/v − 1/u = 1/f; P = 1/f.
  • Red bends least, violet most.
  • Myopia : concave; hypermetropia : convex.
  • Telescope m = f₀/f_e; compound microscope m = (L/f₀)(D/f_e).
Study mantra

Mirror → Snell → TIR → lens → prism → eye → instruments.