Q1. Match the columns given below and explain each term
Column Matching:
| Term | Explanation |
|---|---|
| Farsightedness (Hypermetropia) |
|
| Nearsightedness (Myopia) |
|
Detailed Explanations:
(i) Farsightedness (Hypermetropia)
- Medical term: Hypermetropia or Hyperopia
- Cause: Eyeball too short or cornea/lens too flat
- Effect: Light rays focus behind the retina instead of on it
- Symptoms: Difficulty seeing near objects, eye strain, headaches
- Correction: Convex (converging) lenses that converge light before it enters the eye
- Typical prescription: +0.25 D to +4.00 D or more
- Age factor: Often develops with age (presbyopia after 40)
(ii) Nearsightedness (Myopia)
- Medical term: Myopia
- Cause: Eyeball too long or cornea/lens too curved
- Effect: Light rays focus in front of the retina
- Symptoms: Blurry distance vision, squinting, eye strain
- Correction: Concave (diverging) lenses that diverge light before it enters the eye
- Typical prescription: -0.25 D to -20.00 D or more
- Prevalence: Affects about 30% of global population, increasing
- Risk factors: Genetics, excessive near work, limited outdoor time
[Diagram comparing normal, myopic, and hypermetropic eyes]
Q2. Draw a neat diagram and explain the various terms related to a lens
[Detailed diagram of convex lens showing all important points and lines]
i. Optical Centre (C)
The point located at the centre of the lens on the principal axis is known as the optical centre.
- Represented by symbol: C
- Light rays passing through C do not deviate (continue straight)
- Acts as a reference point for measurements
- For thin lenses, optical centre is at geometrical centre
ii. Principal Axis (P)
An imaginary straight line passing through the optical centre and the centres of curvature of both lens surfaces.
- Denoted by: P
- Perpendicular to lens surfaces at optical centre
- Used as reference line for ray diagrams
- All measurements (u, v, f) are taken along this axis
iii. Principal Focus (F)
For convex lens: Point where parallel rays converge after refraction.
For concave lens: Point from which parallel rays appear to diverge after refraction.
- Represented by: F
- Each lens has two principal foci (F₁ and F₂) on either side
- Distance from optical centre to F is focal length (f)
- Real focus for convex lens, virtual focus for concave lens
iv. Focal Length (f)
The distance between the optical centre (C) and the principal focus (F) of a lens.
- Denoted by: f
- Determines lens power: P = 1/f (in meters)
- Positive for convex lenses, negative for concave lenses
- Shorter focal length = greater converging/diverging power
Additional Important Terms:
| Term | Symbol | Description |
|---|---|---|
| Centre of Curvature | R₁, R₂ | Centres of spherical surfaces forming the lens |
| Aperture | - | Effective diameter of lens through which light passes |
| 2F Point | 2F₁, 2F₂ | Points at twice the focal length from optical centre |
| Pole | P | Point where principal axis meets lens surface |
Q3. At what position should an object be placed in front of a convex lens to obtain a real image of the same size? Draw a diagram.
Answer: To obtain a real image of the same size as the object using a convex lens, the object must be placed at 2F₁ (twice the focal length from the lens).
[Ray diagram showing object at 2F₁, image at 2F₂ with same size]
Detailed Explanation:
- Consider a convex lens with focal length f and optical centre C.
- When object AB is placed at 2F₁ (distance = 2f from lens):
- Ray 1: From A parallel to principal axis → refracts through focus F₂
- Ray 2: From A through optical centre C → continues undeviated
- These two refracted rays meet at point A' on the opposite side
- Image A'B' is formed at 2F₂ (distance = 2f from lens on other side)
Characteristics of the image:
- Real: Can be obtained on a screen
- Inverted: Upside down relative to object
- Same size: Magnification m = -1 (negative sign indicates inversion)
- Located at 2F₂: Exactly opposite to object position
Mathematical proof using lens formula:
Given: u = -2f (object at 2F₁)
Magnification:
Negative sign confirms inversion, magnitude 1 confirms same size.
Applications: This position is used in photocopiers, projection systems where 1:1 reproduction is needed.
Q4. Give scientific reasons for the following statements
a. A simple microscope is used for repairing watches.
Reason: A simple microscope (magnifying glass) used in watch repair typically consists of a convex lens with a very small focal length (5-10 cm) providing magnification of 5× to 20×.
Why it's ideal for watch repair:
- High Magnification: Watch components are extremely small (gears, springs, screws often < 1mm). A simple microscope provides the necessary magnification to see these details clearly.
- Working Distance: The focal length allows comfortable working distance (5-10 cm) between lens and watch.
- Hands-free Operation: Often mounted on stands or headbands, leaving both hands free for delicate work.
- No Eye Strain: Brings image to near point (25 cm) or beyond, reducing accommodation strain.
- Portable and Simple: Unlike compound microscopes, simple microscopes are portable, inexpensive, and don't require special lighting.
Optical Principle: When object is placed between F and optical centre of convex lens, virtual, erect, and magnified image is formed at near point (25 cm) or beyond.
Typical specifications: f = 2.5 cm gives M = 10×, perfect for seeing tiny watch parts.
b. Colours can be perceived only in bright light.
Reason: This is due to the different types of photoreceptor cells in the human retina and their sensitivity thresholds.
[Diagram of retina showing rods and cones distribution]
Two types of photoreceptors:
| Feature | Rods | Cones |
|---|---|---|
| Function | Dim light vision (scotopic) | Bright light & color vision (photopic) |
| Sensitivity | High (work in low light) | Low (need bright light) |
| Color Detection | No (monochromatic) | Yes (trichromatic: R, G, B) |
| Acuity | Low (blurry) | High (sharp) |
| Distribution | Peripheral retina | Central retina (fovea) |
Why colors need bright light:
- Cones have higher threshold: Require more photons to activate than rods.
- Three cone types: S-cones (blue), M-cones (green), L-cones (red) need sufficient light to distinguish wavelengths.
- Photopigment regeneration: Cones recover slowly after bleaching in bright light.
- Purkinje shift: In dim light, maximum sensitivity shifts from 555 nm (cones) to 505 nm (rods).
Practical observation: At dusk, colors fade to grey before objects become invisible - cones stop working before rods.
c. Objects placed closer than 25 cm from the eye cannot be seen clearly.
Reason: This limitation is due to the accommodation power of the human eye and the physical constraints of the ciliary muscles and lens.
[Diagram showing eye accommodation for near and far vision]
The Accommodation Mechanism:
- Ciliary muscles control the curvature of the eye lens.
- For distant objects: Muscles relaxed → lens thin → less converging power.
- For near objects: Muscles contract → lens thickens → more converging power.
- Maximum contraction produces maximum lens curvature.
Why 25 cm is the limit (Near Point):
- Physical limit of muscle contraction: Ciliary muscles cannot contract beyond a certain point.
- Lens elasticity decreases with age: Children can see as close as 7-8 cm, adults 25 cm, elderly > 50 cm.
- Minimum focal length achievable: With maximum curvature, minimum focal length is about 2.3 cm.
- Image formation requirement: For objects closer than 25 cm, image would form behind retina even with maximum accommodation.
Mathematical explanation using lens equation:
Eye lens focal length can vary from: f_max = 2.3 cm (fully accommodated) to f_min = 2.5 cm (relaxed)
Using lens formula for image at retina (v ≈ 2.5 cm):
For closest vision (f = 2.3 cm, v = 2.5 cm):
Clinical term: The nearest point of distinct vision is called the near point or punctum proximum.
Age-related change: Presbyopia reduces near point distance requiring reading glasses.
Q5. Explain the working of an astronomical telescope based on refraction of light
Astronomical (Refracting) Telescope
[Detailed ray diagram of astronomical telescope in normal adjustment]
Basic Principle:
An astronomical telescope works on the principle of refraction of light using two convex lenses to provide angular magnification of distant celestial objects.
Construction:
- Objective lens (O): Large aperture convex lens with long focal length (f₀ = 50-100 cm or more)
- Collects maximum light from distant object
- Forms real, inverted, diminished image at its focal plane
- Eyepiece lens (E): Small convex lens with short focal length (fₑ = 1-5 cm)
- Acts as magnifying glass
- Produces virtual, magnified image of the intermediate image
- Tube: Holds lenses at correct separation (f₀ + fₑ in normal adjustment)
Working (Normal Adjustment - Image at Infinity):
- At Objective Lens:
- Parallel rays from distant celestial object (at infinity)
- Objective forms real, inverted, diminished image I₁ at its focal plane F₀'
- This image is called the intermediate image
- At Eyepiece Lens:
- Intermediate image I₁ acts as object for eyepiece
- Eyepiece positioned so I₁ is at its focus Fₑ (just inside focal length)
- Eyepiece produces virtual, inverted, highly magnified image at infinity
- Final image is inverted relative to original object
Ray Path Analysis:
- Ray parallel to axis → passes through F₀' after objective
- Ray through optical centre → goes straight
- These rays enter eyepiece and emerge parallel (for infinity viewing)
Important Formulas:
Example: If f₀ = 100 cm, fₑ = 2 cm:
Characteristics of Final Image:
- Inverted: Upside down relative to object (doesn't matter for celestial bodies)
- Virtual: Cannot be projected on screen
- Magnified: Appears larger than seen with naked eye
- At infinity: Relaxed viewing (eye muscles relaxed)
Advantages of Refracting Telescope:
- Sealed tube protects optics from dust
- No central obstruction (unlike reflecting telescopes)
- Good for lunar and planetary observation
Disadvantages:
- Chromatic aberration (color fringing)
- Spherical aberration
- Large lenses are heavy, expensive, and difficult to make
- Size limited by practical constraints
Q6. Distinguish between the following
A. Farsightedness (Hypermetropia) vs Nearsightedness (Myopia)
| Aspect | Farsightedness (Hypermetropia) | Nearsightedness (Myopia) |
|---|---|---|
| Medical Term | Hypermetropia / Hyperopia | Myopia |
| Vision Problem | Difficulty seeing near objects clearly | Difficulty seeing distant objects clearly |
| Image Formation | Image forms behind the retina | Image forms in front of the retina |
| Eye Shape | Eyeball too short (axial) or cornea too flat | Eyeball too long (axial) or cornea too curved |
| Correcting Lens | Convex (converging) lens | Concave (diverging) lens |
| Lens Power Sign | Positive (+D) | Negative (-D) |
| Typical Symptoms | Eye strain, headaches, blurred near vision | Squinting, eye strain, blurred distance vision |
| Age of Onset | Often age-related (presbyopia after 40) | Often develops in childhood/adolescence |
| Prevalence | ~10% of population | ~30% of global population |
| Far Point | Infinity (can see distant objects) | Finite distance (closer than infinity) |
| Near Point | Farther than 25 cm | Closer than 25 cm (sometimes very close) |
B. Concave Lens vs Convex Lens
| Aspect | Concave Lens (Diverging Lens) | Convex Lens (Converging Lens) |
|---|---|---|
| Shape | Thinner at center, thicker at edges | Thicker at center, thinner at edges |
| Effect on Light | Diverges parallel light rays | Converges parallel light rays |
| Principal Focus | Virtual focus (on same side as object) | Real focus (on opposite side from object) |
| Focal Length Sign | Negative (-f) | Positive (+f) |
| Power Sign | Negative (-D) | Positive (+D) |
| Image Type | Always virtual, erect, diminished | Can be real or virtual depending on object position |
| Uses | Correcting myopia, peepholes, Galilean telescope | Correcting hypermetropia, magnifying glasses, cameras |
| Ray Diagram Symbol | Arrow pointing inward at both ends | Arrow pointing outward at both ends |
| Behavior Formula | 1/f = 1/v - 1/u (f negative) | 1/f = 1/v - 1/u (f positive) |
| Common Materials | Glass, plastic (often in combination with convex) | Glass, plastic, crystal (diamond for luxury) |
| Alternative Names | Diverging lens, negative lens | Converging lens, positive lens |
Q7. State the functions of the iris and the muscles attached to the lens in the human eye
Function of Iris
The iris is the colored circular structure behind the cornea that controls the amount of light entering the eye through the pupil.
[Diagram showing iris and pupil dilation/constriction]
Primary Functions:
- Light Regulation: Controls pupil size (2-8 mm diameter)
- Bright light → Iris contracts → Pupil constricts (miosis, 2-3 mm)
- Dim light → Iris relaxes → Pupil dilates (mydriasis, 7-8 mm)
- Depth of Field: Smaller pupil increases depth of field (more in focus)
- Reducing Aberrations: Smaller pupil reduces spherical and chromatic aberrations
- Protection: Limits intense light that could damage retina
- Aesthetic: Gives eye its color (melanin content determines color)
Muscles in Iris:
- Sphincter pupillae: Circular muscles → constrict pupil
- Dilator pupillae: Radial muscles → dilate pupil
Function of Ciliary Muscles
Ciliary muscles are ring-shaped muscles attached to the lens via zonular fibers (suspensory ligaments).
[Diagram showing ciliary muscles and lens accommodation]
Primary Functions:
- Accommodation: Changes lens curvature for focusing
- Distant object → Muscles relaxed → Lens thin → Less power
- Near object → Muscles contract → Lens thick → More power
- Lens Suspension: Hold lens in position via zonular fibers
- Aqueous Humor Flow: Helps regulate flow through trabecular meshwork
- Changing Focal Length: Allows eye to focus from near point (~25 cm) to infinity
Accommodation Range:
Example (young adult): NP = 0.25 m, FP = ∞
Age-related changes (Presbyopia):
- Lens becomes less elastic with age
- Ciliary muscles weaken
- Near point recedes (need reading glasses after ~40 years)
Coordination Between Iris and Ciliary Muscles:
- Pupillary Accommodation Reflex: When eyes converge for near vision, pupils automatically constrict
- Light-Near Response: Pupil constricts both for bright light and when focusing on near objects
- Autonomic Control: Both systems controlled by autonomic nervous system
- Parasympathetic: Constricts pupil, contracts ciliary muscles
- Sympathetic: Dilates pupil, relaxes ciliary muscles
Q8. Solve the following problems
A. A doctor prescribes a lens of power +1.5 D. Calculate the focal length of the lens. Identify the type of lens and the vision defect.
Solution:
Given: Power of lens, P = +1.5 D
Formula: Power of lens = 1 / Focal length (in meters)
Calculation:
Convert to cm: f = 66.67 cm ≈ 67 cm
Identification:
- Type of lens: Since power is positive (+1.5 D), it is a convex (converging) lens
- Vision defect: Convex lenses correct Hypermetropia (Farsightedness)
- Additional info: Could also correct Presbyopia (age-related farsightedness)
Physical interpretation: This lens converges light rays to focus them properly on the retina for someone whose eye focuses images behind the retina.
B. An object of height 5 cm is placed at a distance of 25 cm from a converging lens of focal length 10 cm. Find the position, size, and nature of the image formed.
Solution:
Given:
- Object height, h₀ = 5 cm
- Object distance, u = -25 cm (sign convention: object on left, u negative)
- Focal length, f = +10 cm (convex lens, f positive)
Step 1: Find image distance (v) using lens formula:
Step 2: Find magnification (m):
Step 3: Find image height (hᵢ):
Results:
- Image position: v = 16.67 cm on opposite side of lens (positive v)
- Image size: hᵢ = 3.33 cm (magnitude, negative sign indicates inversion)
- Nature of image:
- Real (v positive, can be obtained on screen)
- Inverted (m negative)
- Diminished (|m| = 0.667 < 1)
Verification: Object at 25 cm (beyond 2F = 20 cm), so image should be between F and 2F, which it is (16.67 cm between 10 cm and 20 cm).
C. Three lenses having powers 2 D, 2.5 D, and 1.7 D are kept in contact. Calculate the resultant power of the combination.
Solution:
Given: Powers of three lenses in contact:
- P₁ = 2 D
- P₂ = 2.5 D
- P₃ = 1.7 D
Principle: For thin lenses in contact, resultant power is the algebraic sum of individual powers.
Calculation:
Nature of combination:
- Since P_total = +6.2 D (positive), the combination behaves as a convex (converging) lens
- Focal length of combination: f = 1/P = 1/6.2 ≈ 0.161 m = 16.1 cm
Alternative calculation via focal lengths:
Individual focal lengths:
For lenses in contact:
Application: Such combinations are used in compound lenses (eye glasses with multiple corrections), camera lenses, and optical instruments to correct aberrations or achieve specific optical properties.
D. An object placed 60 cm away from a lens forms a virtual image 20 cm in front of it. Determine the focal length and state the nature of the lens.
Solution:
Given:
- Object distance, u = -60 cm (object on left, u negative)
- Image distance, v = -20 cm (virtual image on same side as object, v negative)
Using lens formula:
Substituting values:
Interpretation:
- Focal length: f = -30 cm (negative)
- Nature of lens: Since focal length is negative, it is a concave (diverging) lens
- Type of image: Virtual, erect (implied by virtual image formation)
- Magnification: m = v/u = (-20)/(-60) = 1/3 ≈ 0.333 (diminished, erect)
Verification with ray diagram rules for concave lens:
- Object anywhere in front of concave lens → always produces virtual, erect, diminished image
- Image is between focus and optical centre on same side as object
- Here: u = 60 cm, f = -30 cm, so object is beyond 2|f| (60 > 60? Actually 60 = 2×30)
- For u = 2f = 60 cm, image should be at v = f/2? Let's check...
Special check: For concave lens with u = 2|f|:
Conclusion: This is a characteristic position for concave lens where object at 2f gives image at f (magnification 0.5, but here m = 1/3? Wait recalc: m = v/u = -20/-60 = 1/3, yes diminished).
Lens Formula, Magnification and Sign Convention
| Quantity | Sign Convention (New Cartesian) | Examples |
|---|---|---|
| Object Distance (u) | Always negative (object on left of lens) | u = -25 cm, -60 cm |
| Image Distance (v) | Positive for real image (right side) Negative for virtual image (left side) |
Real: v = +16.67 cm Virtual: v = -20 cm |
| Focal Length (f) | Positive for convex lens Negative for concave lens |
Convex: f = +10 cm Concave: f = -30 cm |
| Height (h) | Positive upwards, negative downwards | Object: h₀ = +5 cm (erect) Image: hᵢ = -3.33 cm (inverted) |
| Magnification (m) | m = hᵢ/h₀ = v/u Positive: erect image Negative: inverted image |m| > 1: enlarged |m| < 1: diminished |
m = -0.667: inverted, diminished m = +2: erect, enlarged |
Key Formulas:
Remember: Always use consistent units (all in cm or all in m). Convert focal length to meters for power calculation.