Chapter 2 — Optics: Long Answer Questions
1. State and explain Snell's Law of Refraction. Derive the relationship between refractive index, angle of incidence, and angle of refraction.
Answer
Refraction is the bending of light when it passes from one medium to another due to a change in its speed.
Snell's Law: When a ray of light passes from one medium to another, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the given pair of media and for the given colour of light.
Mathematical statement:
sin i / sin r = constant = n₂₁ = n₂/n₁
Where:
- i = angle of incidence (in medium 1)
- r = angle of refraction (in medium 2)
- n₁ = refractive index of medium 1
- n₂ = refractive index of medium 2
- n₂₁ = refractive index of medium 2 with respect to medium 1
More complete form:
n₁ sin i = n₂ sin r
Refractive index (n) of a medium = speed of light in vacuum (c) / speed of light in that medium (v)
n = c/v
Key observations:
- When light goes from a rarer medium (low n, e.g., air) to a denser medium (high n, e.g., glass): light bends towards the normal (r < i)
- When light goes from a denser to rarer medium: light bends away from the normal (r > i)
- If light strikes the boundary at 90° (along the normal): no bending occurs
Example calculation: Light travels from air (n₁ = 1) into glass (n₂ = 1.5) at angle of incidence 30°.
- n₁ sin i = n₂ sin r
- 1 × sin 30° = 1.5 × sin r
- 0.5 = 1.5 × sin r
- sin r = 0.5/1.5 = 0.333
- r = 19.47° ≈ 19.5°
2. What is Total Internal Reflection (TIR)? State the conditions for TIR and explain its applications.
Answer
Total Internal Reflection occurs when light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle — all the light is reflected back into the denser medium; no refraction occurs.
Critical angle (C): The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is exactly 90°.
Relationship:
sin C = n₂/n₁ = 1/n (when n₂ = 1 for air)
For glass (n = 1.5): sin C = 1/1.5 = 0.667, so C ≈ 42°
Conditions for TIR: 1. Light must travel from a denser medium to a rarer medium (e.g., glass to air, water to air) 2. The angle of incidence must be greater than the critical angle
Applications of TIR:
1. Optical fibres:
Answer
- Glass or plastic fibres with a high refractive index core surrounded by a lower refractive index cladding
- Light entering one end undergoes repeated TIR along the fibre length — virtually no energy loss
- Uses: Internet data transmission (broadband), telephone communications, medical endoscopes (viewing inside the body without surgery), decorative lighting
2. Mirage:
Answer
- In hot deserts, air near the ground is hotter and less dense (lower refractive index)
- Light from the sky undergoes TIR in the hot air layer near the ground
- Appears as if water is reflecting the sky — a mirage
3. Brilliant cut diamonds:
Answer
- Diamond has a very high refractive index (~2.42) → very small critical angle (~24°)
- Most light entering a diamond undergoes TIR multiple times → exits from the top as brilliant sparkle
4. Prism-based periscopes:
Answer
- Right-angle prisms replace mirrors; TIR at the glass–air interface reflects light at exactly 90°
- No silvering needed; more efficient than mirrors
3. Explain the refraction of light through a convex lens and derive the lens formula.
Answer
Refraction through a convex lens: A convex (converging) lens is thicker at the centre than at the edges. When parallel rays of light pass through a convex lens, they converge to a point called the principal focus (F).
Key terms:
- Principal axis: Horizontal line passing through the centre of the lens
- Optical centre (O): Centre of the lens — rays passing through it are undeviated
- Principal focus (F): Point where parallel rays converge after refraction
- Focal length (f): Distance from optical centre to principal focus
- Radius of curvature: Related to focal length by f = R/2 (for thin lenses)
Rules for ray diagrams: 1. A ray parallel to the principal axis passes through the focus F after refraction 2. A ray passing through the optical centre O continues without bending 3. A ray passing through the focus F emerges parallel to the principal axis after refraction
Lens Formula derivation: Consider an object AB at distance u from optical centre O. Image A'B' forms at distance v.
Using similar triangles (from standard geometry of lens):
1/v − 1/u = 1/f
Sign convention (New Cartesian):
- All distances measured from optical centre O
- Distances in direction of incident light: positive (+)
- Distances opposite to incident light: negative (−)
- For a real object: u is negative
- f is positive for convex lens, negative for concave lens
Magnification:
m = v/u = height of image / height of object
- m > 0: virtual, erect image
- m < 0: real, inverted image
- |m| > 1: magnified; |m| < 1: diminished
4. Describe the defects of vision: Myopia, Hypermetropia, and Presbyopia. Explain how each is corrected.
Answer
The human eye can normally focus objects from 25 cm (near point) to infinity (far point). Defects occur when the eye cannot do this.
Myopia (Near-sightedness)
Definition: The person can see nearby objects clearly but cannot see distant objects clearly.
Cause:
- Eyeball is too long (antero-posterior diameter increased), OR
- Eye lens has too much curvature (too powerful)
ResultImage of distant objects forms in front of the retina
Correction:
- Use a concave (diverging) lens of appropriate focal length
- The concave lens diverges the incoming parallel rays from a distant object
- The diverged rays now appear to come from a point the myopic eye can focus on (its far point)
- If the far point is at distance d: focal length of concave lens needed = f = −d
Example: A person can see clearly only up to 2 m. They need a concave lens of f = −2 m, power = −0.5 D.
Hypermetropia (Far-sightedness)
Definition: The person can see distant objects clearly but cannot see nearby objects clearly.
Cause:
- Eyeball is too short, OR
- Eye lens is too flat (not enough curvature)
ResultImage of nearby objects would form behind the retina
Correction:
- Use a convex (converging) lens of appropriate focal length
- The convex lens converges rays from a near object
- The converged rays appear to come from the hypermetropic eye's near point (farther than 25 cm)
Presbyopia
Definition: Age-related loss of accommodation (ability to change lens shape) causing difficulty in seeing nearby objects.
Cause:
- Progressive hardening and loss of elasticity of the eye lens with age (typically after 40)
- Ciliary muscles also weaken
Correction:
- Bifocal lenses — upper portion is concave (for distance vision), lower portion is convex (for near vision)
- Progressive lenses (no visible dividing line)
Astigmatism
Definition: Blurred/distorted vision at all distances; horizontal and vertical lines cannot both be in focus.
Cause: Unequal curvature of the cornea (or lens) in different planes
Correction: Cylindrical lenses
5. What is a simple microscope? Explain its working and derive the expression for its magnifying power.
Answer
A simple microscope (magnifying glass) is a single convex lens used to view small objects.
Working
- The object is placed between the focus (F) and the optical centre (O) of the convex lens — i.e., within the focal length (u < f)
- The lens produces a virtual, erect, and magnified image on the same side as the object
- The image is formed at the least distance of distinct vision (D = 25 cm) for maximum magnifying power, or at infinity for minimum strain
Magnifying power (M): Magnifying power = angle subtended by image at eye / angle subtended by object when placed at D without lens
When image is at D = 25 cm (near point):
M = 1 + D/f
Where D = 25 cm, f = focal length of the lens in cm
When image is at infinity (relaxed eye):
M = D/f
Example: A convex lens of focal length 5 cm used as a simple microscope:
- Maximum magnifying power (image at D = 25 cm): M = 1 + 25/5 = 6×
- Minimum magnifying power (image at infinity): M = 25/5 = 5×
Uses of simple microscope:
- Watchmakers and jewellers examining fine details
- Reading small print
- Stamp collectors examining stamps
- Forensic scientists examining fingerprints
Limitations:
- Maximum useful magnification is about 10× to 15× — beyond this, image quality degrades
- For higher magnification, compound microscopes are used (multiple lenses)
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