1
Light and Laws of Reflection
Properties of Light
- Light is a form of energy that enables us to see things; light itself is not visible
- Speed of light in vacuum: c = 3×10⁸ m/s; does NOT depend on speed of source or observer
- Frequency and sense of colour do NOT depend on medium
- Intensity, wavelength, and speed of light depend on medium
- At crossing point, direction of ray does NOT change
- Formation of image does NOT depend on size of mirror; visibility depends on size of mirror
- μν = c = constant; μ₁ν₁ = μ₂ν₂ (Refractive index × speed = constant)
- VIBGYOR wavelength range: 400 nm (violet) to 700 nm (red)
- Monochromatic light = single wavelength; White light = combination of all wavelengths
Laws of Reflection: (i) ∠i = ∠r (measured from normal) (ii) IR, normal, RR are in same plane
Deviation by plane mirror: δ = 180° − 2i
By two inclined mirrors (angle θ between them): δ = 360° − 2θ (does NOT depend on angle of incidence)
Mirror rotated by θ → reflected ray rotates by 2θ (in same sense)
Image by Plane Mirror
- Virtual image, same size, erect, laterally inverted
- Image distance = Object distance (behind mirror)
- Real object → virtual diverging image; Virtual object → real converging image
- MT = HI/HO = +1 (same size, erect); MT = +ve → erect; MT = −ve → inverted
Height of Mirror Required
(A) To see full height of object: Hmirror = Hobject/2 (always; regardless of distance)
(B) To see full wall behind object (observer at centre): Hmirror = Hwall/3
Number of Images by Two Inclined Mirrors (angle θ)
- If m = 360°/θ is even: n = m − 1
- If m = 360°/θ is odd and object is on bisector: n = m − 1
- If m = 360°/θ is odd and object is NOT on bisector: n = m
- Two parallel mirrors (θ = 0°): infinite images
- Clock system: HI:MI:SI = (11−HC):(59−MC):(60−SC)
| Property | Concave Mirror | Convex Mirror |
|---|
| Nature | Converging | Diverging |
| Focal length f | f = −ve | f = +ve |
| Object distance u | u = −ve (real object) | u = −ve (real object) |
| Real image | Possible (v = −ve) | Not possible (v always +ve) |
Mirror Formula: 1/v + 1/u = 1/f; f = R/2
MT = HI/HO = −v/u = f/(f−u) = (f−v)/f
ML = −dv/du = −MT² (only valid for small object along axis)
Sign convention: distances measured from pole; direction of IR = +ve
MT = +ve → erect/virtual image; MT = −ve → inverted/real image
| Object Position | Image (Concave) | Image (Convex) |
|---|
| At ∞ | At F, real, diminished, inverted | At F, virtual, erect |
| blw C and ∞ | blw C and F, real, small, M < −1, inverted | blw pole and focus |
| At C | At C, real, M = −1, inverted | — |
| blw C and F | Beyond C, real, large, M ≥ −1, inverted | — |
| At F | At ∞, real, very large, inverted | — |
| blw F and pole | Behind mirror, virtual, large, M = +ve, erect | — |
Velocity of Image in Spherical Mirror
- Object moving ⊥ to principal axis: VI = MT × VO
- Object moving ∥ to principal axis: VI = MT² × VO
- Object at C: MT = 1, VI = VO
- Object blw C and F: MT > 1, VI > VO
- Object blw C and ∞: MT < 1, VI < VO
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Refraction and Snell's Law
Laws of Refraction
- Incident ray, refracted ray, and normal are in same plane
- R → D (Rare to Dense): bending towards normal (i > r)
- D → R (Dense to Rare): bending away from normal (i < r)
Snell's Law: μ₂₁ = μ₂/μ₁ = sin i/sin r = v₁/v₂ = λ₁/λ₂
Apparent depth: dapp = dreal/μ (object in denser medium, observed from rarer)
Normal shift: dshift = dreal − dapp
Radius of visibility (from below water): R = H/√(μ² − 1)
Refraction Through Glass Slab
- Normal shift (apparent shift): t(1 − 1/μ) [object appears closer by this amount]
- Lateral shift: d = t sin(i − r)/cos r ; for small angle: d ≈ t·i(1 − 1/μ)
- Emergent ray is parallel to incident ray (no angular deviation)
- Only lateral displacement occurs
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Total Internal Reflection (TIR)
Conditions for TIR
- Light must travel from Denser → Rarer medium
- Angle of incidence must be greater than critical angle (i > ic)
sin ic = μ₂/μ₁ = v₁/v₂ = λ₁/λ₂ (critical angle formula)
For glass-air: sin ic = 1/μ
Applications: optical fibre, diamond sparkle, mirage, total reflecting prisms, endoscope
Optical Fibre
- Based on TIR; core (μ₁) > cladding (μ₂)
- sin ic = √(μ₁² − μ₂²)
- For TIR: sin i ≤ √(μ₁² − μ₂²)
- Applications: endoscope, internet cables, communication, decorative lamps
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Refraction at Spherical Surface and Lenses
Refraction at spherical surface: μRR/v − μIR/u = (μRR − μIR)/R
MT = (μIR·v)/(μRR·u) | Concave surface: R = −ve; Convex surface: R = +ve
Lens Maker's Equation: 1/f = (μL/μM − 1)[1/R₁ − 1/R₂]
Lens Equation: 1/v − 1/u = 1/f
Magnification: MT = HI/HO = v/u = f/(f+u) ; ML = MT² × dv/du
Effect of Medium on Lens
- μM = μL → lens behaves as simple glass (f = ∞)
- μM > μL → converging lens becomes diverging (nature reverses)
- μL > μM → lens retains original nature
- Focal length of lens depends on medium; f for mirror does NOT depend on medium
| Object | Image by Convex Lens |
|---|
| At ∞ | At F, small, real, inverted, M = −ve |
| blw 2f and ∞ | blw f and 2f, small, real, inverted, M = −ve |
| blw 2f and f | blw 2f and ∞, large, real, inverted, M = −ve |
| At 2f | At 2f, same size, real, inverted, M = −1 |
| At f | At ∞, very large, real, inverted, M = −∞ |
| blw f and pole | Same side, large, virtual, erect, M = +ve |
Focal Length Formulas for Different Lens Types
- Biconvex lens (R₁ = R₂ = R, in air): f = R/[2(μ−1)]
- Plano-convex lens: f = R/(μ−1)
- Power of lens: P = 1/f(m) = 100/f(cm) ; unit = Dioptre (D)
- Power of mirror: P = −1/f(m) [note negative sign]
- Lenses in contact: P = P₁ + P₂ + P₃; 1/f = 1/f₁ + 1/f₂ + 1/f₃
- Lenses separated by distance d: P = P₁ + P₂ − d·P₁P₂; 1/f = 1/f₁ + 1/f₂ − d/(f₁f₂)
Displacement Method (Lens — finding focal length)
- D = distance between object and screen (fixed); x = shift in position of lens
- f = (D² − x²)/(4D)
- HO = √(I₁·I₂) (size of object = geometric mean of two image sizes)
- |f| = (m₁ − m₂)/x where m₁, m₂ are magnifications at two positions
Prism: A = r₁ + r₂ (refracting angle = sum of refraction angles inside)
Deviation: δ = i + e − A (i = angle of incidence; e = angle of emergence)
At minimum deviation (δmin): i = e; r₁ = r₂ = A/2
μ = sin[(A + δmin)/2] / sin(A/2)
For thin prism (small A): δ = (μ − 1)A; placed in medium: δ = A(μP/μM − 1)
Dispersion
- In vacuum, speed of all colours = c = 3×10⁸ m/s (same)
- In medium, different colours travel with different speed (because μ is different)
- Cauchy's equation: μ = A + B/λ² → μ ∝ 1/λ (higher frequency → higher μ)
- VIBGYOR: μV > μI > μB > μG > μY > μO > μR
- Violet deviates most; Red deviates least
- Mean deviation: δmean = A(μavg − 1) = A(μy − 1)
- Mean deviation = A[(μR + μV)/2 − 1]
| Quantity | Formula | Significance |
|---|
| Angular dispersion (θ) | θ = δV − δR = A(μV − μR) | Angular separation between violet and red |
| Dispersive power (ω) | ω = θ/δmean = (μV−μR)/(μy−1) | Dispersion per unit deviation (material property) |
| Dispersion without deviation | δnet = δ₁ + δ₂ = 0; A₁(μ₁−1) = −A₂(μ₂−1) | Achromatic combination: two prisms |
Visual Angle and Near Point
- Visual angle θ₀ = HO/D (where D = 25 cm = distance of distinct vision)
- Magnifying power M = θ/θ₀ (ratio of visual angles with and without instrument)
Simple Microscope (Magnifier)
- Convex lens of small focal length; uses as magnifier
- Forms erect, magnified, virtual image
- Image at infinity (relaxed eye): Mmin = D/fe
- Image at 25 cm (near point): Mmax = 1 + D/fe
- M = D[1/f + 1/v] (general formula)
Compound Microscope
Objective lens: near the object; small focal length; forms real, inverted, magnified image of object
Eyepiece: large focal length; acts as simple microscope; forms large virtual image
Magnification: M = MO × Me = (VO/UO) × Me
Image at 25 cm: M = (VOD/UO)×(1/fe + 1/Ve)
Image at ∞ (far): M = VOD/(UO·fe) = LD/(fO·fe)
Length (near): L = VO + Ue ≈ VO
Length (far): L = VO + fe
Astronomical Telescope (Refracting)
Magnification: MO = fO/Ue = (fO/fe)(1 + fe/Ve)
Image at ∞ (far, normal adjustment): M = fO/fe ; L = fO + fe
Image at D (near): M = (fO/fe)(1 + fe/D) ; L = fO + Ue
Galileo Telescope
- Objective = convex; Eyepiece = concave (diverging)
- Erect final image (unlike astronomical telescope)
- Length: L = fO − fe (shorter than refracting telescope)
- M = fO/fe
| Instrument | M (at ∞) | M (at D=25cm) | Length |
|---|
| Simple Microscope | D/f | 1 + D/f | — |
| Compound Microscope | LD/(fOfe) | (VO/UO)(1+D/fe) | VO+fe |
| Astro Telescope | fO/fe | (fO/fe)(1+fe/D) | fO+fe |
| Galileo Telescope | fO/fe | — | fO−fe |
Q: Object in water (μ = 4/3) at depth 12 cm. Find apparent depth seen from air.
dapp = dreal/μ = 12/(4/3) = 9 cm; Normal shift = 12 − 9 = 3 cm (object appears 3 cm closer)
Ans: Apparent depth = 9 cm ✓
Q: A concave mirror, f = −10 cm, object at u = −30 cm. Find v, MT.
1/v + 1/u = 1/f; 1/v = 1/f − 1/u = 1/(−10) − 1/(−30) = −1/10 + 1/30 = −2/30; v = −15 cm; MT = −v/u = −(−15)/(−30) = −1/2
Ans: v = −15 cm (real); MT = −0.5 (inverted, diminished) ✓
Q: A prism (A = 60°, μ = √2). Find critical angle and minimum deviation.
sin ic = 1/μ = 1/√2; ic = 45°. For δmin: μ = sin[(A+δmin)/2]/sin(A/2) = sin[(60+δmin)/2]/sin30°; √2 = sin[(60+δmin)/2]/0.5; sin[(60+δmin)/2] = √2/2; (60+δmin)/2 = 45°; δmin = 30°
Ans: ic = 45°; δmin = 30° ✓