Chapter 6 — Exercise Solutions
Ray optics is sign-convention heavy. Draw a diagram and mark object/image position before substituting values.
How to Use This Page
- Start with the Key Concepts table to quickly refresh definitions before attempting questions.
- Work through sections in order — MCQs and Fill in the Blanks test recall; True or False tests conceptual clarity; Match the Following links terms to formulas.
- Use the Formula Recap as a cheat sheet while solving numericals, but try each problem without it first.
- Check Common Mistakes after every worked example to catch sign-convention and unit errors early.
- Tick off the Exam Checklist before your test to ensure you have covered every frequently asked topic.
Key Concepts (Quick Recap)
| Term | Definition |
|---|
| Reflection | Bouncing back of light into the same medium when it strikes a polished surface. |
| Refraction | Bending of light at the interface of two media due to change in speed. |
| Snell's law | n₁ sin i = n₂ sin r — relates angles of incidence and refraction to refractive indices. |
| Total Internal Reflection (TIR) | Complete reflection of light back into a denser medium when the angle of incidence exceeds the critical angle. |
| Critical angle | Angle of incidence in the denser medium for which the refracted ray grazes the surface (angle of refraction = 90°). |
| Mirror formula | 1/f = 1/v − 1/u — relates focal length, image distance, and object distance for spherical mirrors. |
| Lens formula | 1/f = 1/v − 1/u — same relation as mirror formula but with Cartesian sign convention for lenses. |
| Magnification | m = v/u — ratio of image height to object height; negative value means inverted image. |
| Power of lens | P = 1/f (in metres) — measured in dioptres (D); positive for convex, negative for concave. |
| Refractive index | n = c/v — ratio of speed of light in vacuum to speed in the medium; also n = sin i / sin r. |
| Prism | Transparent medium with two plane refracting surfaces inclined at an angle (angle of prism A). |
| Dispersion | Splitting of white light into constituent colours due to wavelength-dependent refractive index. |
| Microscope | Optical instrument that produces a magnified image of a near, small object (compound microscope uses two convex lenses). |
| Telescope | Optical instrument for viewing distant objects; astronomical telescope uses an objective and eyepiece lens. |
| Lens maker formula | 1/f = (n − 1)(1/R₁ − 1/R₂) — determines focal length from refractive index and radii of curvature. |
Formula Recap
| Concept | Formula |
|---|
| Mirror/lens formula | 1/f = 1/v - 1/u |
| Magnification | m = v/u |
| Refractive index | n = c/v |
| Snell law | n1 sin i = n2 sin r |
| Critical angle | sin C = 1/n |
| Lens maker formula | 1/f = (n - 1)(1/R1 - 1/R2) |
I. Choose the Best Answer (MCQs)
1. When light travels from a denser to a rarer medium, the angle of incidence for which the refracted ray just grazes the surface is called:
Answer
(a) Angle of deviation (b) Angle of polarisation **(c) Critical angle (d) Angle of minimum deviation
Explanation: The critical angle is the specific angle of incidence in the denser medium at which the angle of refraction becomes 90°.
2. A convex lens has a focal length of 20 cm. Its power is:
Answer
(a) +0.05 D **(b) +5 D (c) −5 D (d) +20 D
Explanation: Power P = 1/f (in metres) = 1/0.20 = +5 D. Positive because the lens is convex.
3. According to Snell's law, if light passes from air (n = 1) into a medium of refractive index 1.5 at an angle of 45°, the refracted angle will be:
Answer
(a) Greater than 45° **(b) Less than 45° (c) Equal to 45° (d) 90°
Explanation: When light enters a denser medium (n₂ > n₁), it bends towards the normal, so the refracted angle is smaller than the incident angle.
4. In a compound microscope, the objective lens has:
Answer
(a) Large focal length and large aperture **(b) Small focal length and small aperture (c) Large focal length and small aperture (d) Small focal length and large aperture
Explanation: The objective of a compound microscope needs a small focal length to produce a large magnified real image close to the eyepiece.
5. Total internal reflection will NOT occur when light travels from:
Answer
(a) Glass to air (b) Water to air **(c) Air to glass (d) Diamond to air
Explanation: TIR requires light to travel from a denser medium to a rarer medium. Air to glass is rarer to denser, so TIR cannot occur.
6. A concave mirror produces a real image when the object is placed:
Answer
(a) Between F and P **(b) Beyond C (c) At F (d) Between C and F
Explanation: A concave mirror forms a real image when the object is placed beyond the focal point. Beyond C gives a real, inverted, diminished image.
7. The lens maker formula is used to determine:
Answer
(a) Magnification of a lens **(b) Focal length of a lens from its material and radii of curvature (c) Power of a mirror (d) Deviation through a prism
Explanation: The lens maker formula relates focal length to the refractive index of the material and the radii of curvature of the two surfaces.
8. In an astronomical telescope, the objective has a focal length of 100 cm and the eyepiece has 5 cm. The magnifying power in normal adjustment is:
Answer
(a) 5 (b) 10 **(c) 20 (d) 500
Explanation: M = f₀/fₑ = 100/5 = 20. The telescope magnifies distant objects by 20 times.
9. Dispersion of light through a prism occurs because:
Answer
(a) Different colours travel at the same speed in glass **(b) Refractive index of prism material depends on wavelength (c) All colours have the same frequency (d) The prism absorbs some colours
Explanation: Dispersion happens because violet light has a higher refractive index than red light in glass, causing different wavelengths to bend by different amounts.
10. A virtual image formed by a convex lens is always:
Answer
(a) Inverted and diminished (b) Erect and diminished **(c) Erect and magnified (d) Inverted and magnified
Explanation: When an object is placed within the focal length of a convex lens, the image is virtual, erect, and magnified — this is how a simple magnifying glass works.
II. Fill in the Blanks
1. The relation 1/f = 1/v − 1/u is called the mirror formula.
2. According to Snell's law, n₁ sin i = n₂ sin r.
3. The angle of incidence for which the refracted ray just grazes the surface is called the critical angle.
4. Total internal reflection takes place only when light travels from a denser medium to a rarer medium.
5. The power of a lens of focal length 25 cm is +4 D.
6. The ratio of the speed of light in vacuum to the speed of light in a medium is called the refractive index of that medium.
7. The magnification produced by a lens is given by m = v/u.
8. The lens maker formula is 1/f = (n − 1)(1/R₁ − 1/R₂).
III. True or False (with Corrections)
1. Total internal reflection occurs when light travels from a rarer medium to a denser medium.
False. Total internal reflection occurs when light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle.
2. A convex lens always produces a real image.
False. A convex lens produces a virtual, erect, and magnified image when the object is placed between the focal point and the lens.
3. The critical angle depends on the wavelength of light used.
True. Since refractive index varies with wavelength (dispersion), the critical angle also varies with wavelength. Violet light has a smaller critical angle than red light.
4. The power of a concave lens is positive.
False. The power of a concave lens is negative because its focal length is negative by the Cartesian sign convention (P = 1/f, f is negative).
5. In a compound microscope, the eyepiece has a larger focal length than the objective.
True. The objective has a very small focal length (to produce a magnified real image), while the eyepiece has a larger focal length (to further magnify that image).
6. A ray passing through the centre of curvature of a concave mirror is reflected back along the same path.
True. A ray directed towards the centre of curvature strikes the mirror normally and reflects back along the same path (angle of incidence = 0°).
7. Dispersion does not occur in a rectangular glass slab.
True. In a rectangular slab, the emergent ray is parallel to the incident ray, so the angular dispersion at the two faces cancels out, though lateral displacement of colours occurs.
IV. Match the Following
| No. | Column A | Column B |
|---|
| 1 | Mirror formula | A. n₁ sin i = n₂ sin r |
| 2 | Snell's law | B. m = v/u |
| 3 | Critical angle | C. 1/f = 1/v − 1/u |
| 4 | Magnification | D. P = 1/f |
| 5 | Power of lens | E. sin C = 1/n |
| 6 | Lens maker formula | F. 1/f = (n − 1)(1/R₁ − 1/R₂) |
Answer1 → C, 2 → A, 3 → E, 4 → B, 5 → D, 6 → F
Numerical Pattern
1. Write sign convention clearly. 2. Identify mirror/lens/refraction/prism question. 3. Substitute values with signs. 4. Solve for unknown. 5. Interpret positive/negative result.
V. Answer Briefly (2-mark)
1. State the laws of reflection.
Answer
The two laws of reflection are: (i) The angle of incidence equals the angle of reflection (i = r). (ii) The incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane. These laws hold for both plane and curved reflecting surfaces.
2. What is total internal reflection? Give one application.
Answer
When light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle, the light is completely reflected back into the denser medium. This phenomenon is called total internal reflection. One important application is optical fibres, which use TIR to transmit light signals over long distances with minimal loss.
3. Define power of a lens. Write its unit.
Answer
The power of a lens is defined as the reciprocal of its focal length (in metres). Mathematically, P = 1/f, where f is in metres. The SI unit of power is the dioptre (D). A convex lens has positive power and a concave lens has negative power.
4. What is the difference between a real image and a virtual image?
Answer
A real image is formed by the actual convergence of light rays and can be projected on a screen; it is always inverted with respect to the object. A virtual image is formed by the apparent convergence of rays (the rays do not actually meet) and cannot be projected on a screen; it is always erect with respect to the object.
5. Why does a diamond sparkle?
Answer
Diamond has a very high refractive index (n ≈ 2.42), which gives it a very small critical angle (≈ 24.4°). Light entering the diamond undergoes multiple total internal reflections inside the cut faces before emerging. This trapped light exits from only a few faces, creating the brilliant sparkling effect.
Short Answer Points
- Total internal reflection occurs from denser to rarer medium.
- Optical fibre works on total internal reflection.
- Prism produces deviation and dispersion.
- Telescope is used for distant objects; microscope for tiny objects.
VI. Answer in Detail (5-mark)
1. Derive the mirror formula for a concave mirror.
Answer
Consider a concave mirror of focal length f. An object AB is placed beyond C on the principal axis. A ray from point B parallel to the principal axis reflects through the focus F. Another ray from B passing through C reflects back along the same path. The two reflected rays meet at B', forming the image A'B'.
Using similar triangles in the ray diagram:
From triangles A'B'F and A'FP (using the paraxial approximation where PF ≈ PF):
A'B' / AB = A'F / PF = (v − f) / f ... (1)
From similar triangles A'B'P and ABP:
A'B' / AB = A'P / AP = v / (−u) ... (2)
Equating (1) and (2):
(v − f) / f = v / (−u)
−u(v − f) = vf
−uv + uf = vf
uf = vf + uv
1/f = 1/v − 1/u
This is the mirror formula. It holds for concave and convex mirrors under the Cartesian sign convention. Always draw the ray diagram and mark distances with proper signs before substituting values.
2. Explain total internal reflection with conditions and applications.
Answer
Total internal reflection (TIR) is the phenomenon in which a ray of light travelling from a denser medium to a rarer medium is completely reflected back into the denser medium, with no refracted ray emerging into the rarer medium.
Conditions for TIR:
1. The light must travel from a denser medium to a rarer medium (e.g., glass to air, water to air). 2. The angle of incidence in the denser medium must be greater than the critical angle for the pair of media.
The critical angle C is given by sin C = 1/n, where n is the relative refractive index of the denser medium with respect to the rarer medium. For glass to air, C ≈ 42°; for diamond to air, C ≈ 24.4°.
Applications:
- Optical fibres: Light signals undergo repeated TIR within the glass core, enabling data transmission over long distances with minimal loss. Used in telecommunications and endoscopes.
- Diamonds: The high refractive index of diamond gives a small critical angle, causing light to undergo multiple internal reflections before escaping, producing sparkle.
- Mirage: In deserts, hot air near the ground has a lower refractive index than cooler air above. Light from the sky undergoes TIR, creating the illusion of water on the ground.
- Prisms in binoculars: Right-angled prisms use TIR to reflect light through 90° or 180° without silvering, avoiding absorption losses.
3. Derive the lens maker formula.
Answer
Consider a thin lens made of material of refractive index n₂ placed in a medium of refractive index n₁. The lens has two spherical surfaces with radii of curvature R₁ and R₂.
First surface (refraction at surface with radius R₁):
n₂/v₁ − n₁/u = (n₂ − n₁)/R₁
The image formed by the first surface acts as the object for the second surface.
Second surface (refraction at surface with radius R₂):
n₁/v − n₂/v₁ = (n₁ − n₂)/R₂
Adding the two equations:
n₁/v − n₁/u = (n₂ − n₁)(1/R₁ − 1/R₂)
Dividing both sides by n₁:
1/v − 1/u = (n₂/n₁ − 1)(1/R₁ − 1/R₂)
For a lens in air, n₁ = 1, and letting n = n₂/n₁ (the absolute refractive index):
1/f = (n − 1)(1/R₁ − 1/R₂)
This is the lens maker formula. It shows that the focal length of a lens depends on the refractive index of the lens material and the radii of curvature of its two surfaces. For a biconvex lens, R₁ is positive and R₂ is negative, giving a positive focal length.
4. Describe the working of an astronomical telescope and a compound microscope.
Answer
Astronomical Telescope:
An astronomical telescope consists of two convex lenses: the objective (large focal length f₀) and the eyepiece (small focal length fₑ). The objective collects light from a distant object and forms a real, inverted, diminished image at its focal plane. This image acts as the object for the eyepiece. In normal adjustment, the final image is formed at infinity. The magnifying power is:
M = f₀ / fₑ
The length of the telescope in normal adjustment is L = f₀ + fₑ. The image seen is inverted, which is acceptable for astronomical observation. For terrestrial use, an erecting lens is added between objective and eyepiece.
Compound Microscope:
A compound microscope also uses two convex lenses: the objective (very small focal length f₀) and the eyepiece (small focal length fₑ). The object is placed just beyond the focal point of the objective, which produces a real, inverted, magnified image. This image falls within the focal length of the eyepiece, which acts as a simple magnifier to produce a virtual, erect, highly magnified final image.
The magnifying power when the final image is at the near point (D = 25 cm):
M = (v₀/u₀) × (1 + D/fₑ)
where v₀ is the image distance for the objective and u₀ is the object distance. The compound microscope achieves much higher magnification than a simple magnifier by combining the magnifications of both lenses.
Worked Examples
Example 1 — Lens formula
An object is placed 30 cm from a convex lens of focal length 10 cm. Find the image distance.
1/f = 1/v − 1/u (u = −30 cm, f = +10 cm)
1/10 = 1/v − 1/(−30)
1/v = 1/10 − 1/30 = (3 − 1)/30 = 2/30
v = 15 cm (real image, 15 cm on the other side)
Example 2 — Magnification
For the above, find the magnification.
m = v/u = 15 / (−30) = −0.5
Image is real, inverted, and half the size.
Example 3 — Snell's law
Light passes from air (n=1) into glass (n=1.5) at 30°. Find the refraction angle.
n₁ sinθ₁ = n₂ sinθ₂
1 × sin30° = 1.5 × sinθ₂
sinθ₂ = 0.5 / 1.5 = 0.333
θ₂ = sin⁻¹(0.333) ≈ 19.5°
Common Mistakes to Avoid
1. ❌ Using positive u for objects in the Cartesian sign convention. ✅ Object distance u is always negative (u = −ve) when measured from the mirror/lens towards the object along the principal axis.
2. ❌ Forgetting to convert focal length from cm to m when calculating power of a lens. ✅ Power P = 1/f requires f in metres. If f = 25 cm, then P = 1/0.25 = +4 D, not 1/25.
3. ❌ Confusing the sign conventions for mirrors and lenses. ✅ For mirrors, f is negative for concave and positive for convex. For lenses, f is positive for convex and negative for concave. Always state which convention you are using.
4. ❌ Assuming total internal reflection can occur from rarer to denser medium. ✅ TIR only occurs when light travels from a denser medium to a rarer medium, and the angle of incidence exceeds the critical angle.
5. ❌ Substituting numerical values without sign convention in the mirror/lens formula. ✅ Always assign signs first: mark u, v, f with proper signs based on the Cartesian convention, then substitute. Many students lose marks by substituting all positive values.
6. ❌ Using the telescope formula for a microscope or vice versa. ✅ Telescope magnification M = f₀/fₑ (for distant objects). Microscope magnification M = (v₀/u₀)(1 + D/fₑ) (for near objects). Know which instrument the question is about before choosing the formula.
Exam Checklist
- [ ] I can state and apply the mirror formula (1/f = 1/v − 1/u) with correct sign convention.
- [ ] I can derive and apply the lens maker formula 1/f = (n − 1)(1/R₁ − 1/R₂).
- [ ] I know the conditions for total internal reflection and can calculate the critical angle.
- [ ] I can apply Snell's law (n₁ sin i = n₂ sin r) to refraction problems.
- [ ] I can calculate magnification (m = v/u) and interpret the sign of the result.
- [ ] I can calculate the power of a lens (P = 1/f in metres) and state its unit (dioptre).
- [ ] I can describe the working of an astronomical telescope and state its magnifying power.
- [ ] I can describe the working of a compound microscope and state its magnifying power.
- [ ] I can distinguish between real and virtual images based on sign of v and magnification.
- [ ] I always draw a ray diagram and mark sign conventions before solving numericals.
