Ray Optics and Optical Instruments Class 12 Physics Notes

A ray incident at angle of incidence $i$ deviates through $\delta =180^{\circ }-2i$. For a plane mirror, an incident ray and its reflection are symmetric about the normal; the image of an object is virtual, erect, of the same size, as far behind the mirror as the object is in front, and laterally inverted.

MCQ Questions

1 Mark Questions

1. 1
   State the two laws of reflection of light.
2. 2
   Define the angle of deviation for reflection from a plane mirror.
3. 3
   Write the expression for deviation produced when a ray is reflected by a plane mirror.

2 Mark Questions

1. 1
   A ray of light makes an angle of 30° with a plane mirror. Find the angle of deviation produced on reflection.
2. 2
   A plane mirror is rotated by 20°. Through what angle does the reflected ray rotate? Justify.
3. 3
   Two plane mirrors are inclined at 60°. How many images of an object placed between them will be formed?
4. 4
   Why does a plane mirror form a laterally inverted image?

3 Mark Questions

1. 1
   Derive an expression for the number of images formed by two plane mirrors inclined at an angle θ with each other.
2. 2
   A point object is placed between two plane mirrors inclined at 40°. Find the number of images formed and explain the special case that arises.
3. 3
   Show with ray diagrams how the deviation of a ray reflected from a plane mirror depends on the angle of incidence.

• Pole (P): the centre of the reflecting surface.
• Centre of curvature (C): the centre of the sphere of which the mirror is a part.
• Radius of curvature (R): the distance PC;
• $R=2f$
• .
• Principal axis: the line joining P and C.
• Focus (F): the point where rays parallel to the principal axis converge (concave) or appear to diverge from (convex).
• Radius of the circular aperture: the radius of the reflecting surface measured along the principal axis aperture.

MCQ Questions

1 Mark Questions

1. 1
   Define pole, centre of curvature and radius of curvature of a spherical mirror.
2. 2
   State the relation between the focal length and the radius of curvature of a spherical mirror.
3. 3
   State the New Cartesian sign convention for measuring distances in mirror problems.

2 Mark Questions

1. 1
   A concave mirror has a radius of curvature of 30 cm. Where is its focus?
2. 2
   Distinguish between concave and convex mirrors on the basis of the sign of their focal length.
3. 3
   Why can a convex mirror never form a real image of a real object?

3 Mark Questions

1. 1
   With a neat ray diagram, define the focus of (a) a concave mirror and (b) a convex mirror.
2. 2
   State the new Cartesian sign convention for reflection by spherical mirrors and apply it to write the value of u and R for a concave mirror with the object on the principal axis.

For a real image h₂ is negative; for a virtual image it is positive. A magnification m > 1 means the image is magnified, m < 1 diminished, m = 1 of the same size, and the magnification of a virtual erect image is positive.

MCQ Questions

1 Mark Questions

1. 1
   State the mirror formula for a spherical mirror.
2. 2
   Write the expression for the linear magnification of a spherical mirror.
3. 3
   What is the relation between image velocity and object velocity for a plane mirror?

2 Mark Questions

1. 1
   An object is placed 30 cm in front of a concave mirror of focal length 15 cm. Find the position, nature and size of the image.
2. 2
   A convex mirror has focal length 15 cm. A needle 12 cm from the mirror gives an image at what distance, and what is the magnification?
3. 3
   Use the mirror formula to show that a concave mirror forms a real image beyond 2f when the object lies between f and 2f.

3 Mark Questions

1. 1
   Derive the mirror formula 1/v + 1/u = 1/f for a concave mirror using a ray diagram and the Cartesian sign convention.
2. 2
   An object moves with speed 2 cm/s towards a concave mirror of focal length 20 cm along its principal axis. At the instant when the object is 40 cm from the mirror, find the velocity of the image.
3. 3
   Derive the relation for the velocity of the image of an object moving along the principal axis of a spherical mirror.

Real and apparent depth: a point object at the bottom of a tank appears raised. For normal viewing, $n=\text{real depth}/\text{apparent depth}$. A glass slab of thickness t raises an object by the lateral shift $\Delta =t(1-1/n)$.

MCQ Questions

1 Mark Questions

1. 1
   State the laws of refraction of light.
2. 2
   State Snell’s law and define relative refractive index.
3. 3
   A glass slab of thickness t is placed over a table. By how much does a point on the table appear raised when viewed normally? (Write the expression.)

2 Mark Questions

1. 1
   The apparent depth of a needle at the bottom of a tank measured with a microscope is 9.4 cm for a water depth of 12.5 cm. Find the refractive index of water and the new apparent depth if the liquid is replaced by one with n = 1.63.
2. 2
   A glass slab of thickness 15 cm and refractive index 1.5 is held parallel to a table. A pin fixed on the table is viewed from above. By how much does it appear to rise? Does the answer depend on where the slab is kept?
3. 3
   Explain why a fish in water appears closer to the surface than it actually is.

3 Mark Questions

1. 1
   Derive the expression for the apparent shift of an object viewed through a glass slab of thickness t and refractive index n.
2. 2
   Light travels from air into a medium of refractive index 1.5 at an angle of incidence 60°. Find the angle of refraction and the angle of deviation.

• Optical fibres: light is guided down a thin core by repeated total internal reflection at the core–cladding interface (used in communication and endoscopy).
• Mirages: in hot air near the ground the refractive index of air decreases, so light bends up and total internal reflection creates the illusion of water.
• Diamond brilliance: diamond has a small critical angle (≈ 24.4°), so light undergoes many total internal reflections and sparkles.
• Prisms: right-angled (45°–45°–90°) prisms are used to turn rays by 90° or 180° where reflection is 100% efficient, unlike mirrors.

Area of emergence from a point source: a point source at a depth h in water (n = 1.33) lets light emerge through a circular patch of radius $r=h\tan C$, where $C$ is the critical angle. For h = 0.8 m this gives r ≈ 0.91 m and area ≈ 2.6 m².

MCQ Questions

1 Mark Questions

1. 1
   Define critical angle.
2. 2
   State the two necessary conditions for total internal reflection.
3. 3
   Why does a diamond sparkle more than a glass piece of the same shape?

2 Mark Questions

1. 1
   A light pipe is made of a glass fibre of refractive index 1.68 clad with a material of refractive index 1.44. Find the range of angles of incidence with the axis for which total internal reflection guides light along the pipe.
2. 2
   The refractive index of water is 1.33. Find the critical angle for the water–air interface.
3. 3
   A small bulb is placed at the bottom of a tank of water 80 cm deep. Find the radius of the circular patch on the surface through which light emerges.

3 Mark Questions

1. 1
   Derive the condition for total internal reflection and obtain the expression for the critical angle in terms of the refractive indices.
2. 2
   Explain with a diagram the propagation of light in an optical fibre. A fibre with core index 1.68 and no cladding accepts light incident at any angle up to what value with its axis?

If a converging beam falls on a lens, u is effectively positive (virtual object). For a convex lens of f = 20 cm with the virtual object 12 cm beyond the lens, the lens formula gives the beam converging at v = 7.5 cm; for a concave lens of f = 16 cm, v = 48 cm.

MCQ Questions

1 Mark Questions

1. 1
   State the lens formula for a thin lens.
2. 2
   Define optical centre of a lens.
3. 3
   Distinguish between concave and convex lenses by the sign of the focal length.

2 Mark Questions

1. 1
   An object 3.0 cm tall is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image and what happens as the object is moved farther away.
2. 2
   A convex lens of focal length 30 cm and a concave lens of focal length 20 cm are placed in contact. Is the combination converging or diverging?
3. 3
   A converging beam falls on a convex lens of focal length 20 cm placed 12 cm from the point of convergence. Where does the beam converge now?

3 Mark Questions

1. 1
   Derive the refraction formula at a spherical surface for light going from rarer to denser medium at a convex refracting surface.
2. 2
   Obtain the lens formula 1/v − 1/u = 1/f for a thin convex lens using the refraction formula for a single spherical surface, stating clearly the assumptions used.

MCQ Questions

1 Mark Questions

1. 1
   Write the lens maker’s formula and name each symbol.
2. 2
   Define the power of a lens. State its SI unit.
3. 3
   State the formula for the effective power of two thin lenses placed in contact.

2 Mark Questions

1. 1
   A double-convex lens of n = 1.55 is wanted with f = 20 cm and both faces of equal radius. Find the required radius of curvature.
2. 2
   A convex lens of f = 30 cm and a concave lens of f = 20 cm are in contact. Find the focal length and nature of the combination.
3. 3
   Object and screen are 90 cm apart. A convex lens gives two sharp images separated by 20 cm. Find the focal length of the lens.

3 Mark Questions

1. 1
   Derive the lens maker’s formula for a thin double-convex lens stating all assumptions.
2. 2
   Show that two thin lenses in contact behave as a single lens of focal length F with 1/F = 1/f₁ + 1/f₂.

MCQ Questions

1 Mark Questions

1. 1
   Define the refracting angle of a prism and the angle of deviation.
2. 2
   State the condition for minimum deviation in a prism.
3. 3
   Write the thin-prism expression for deviation.

2 Mark Questions

1. 1
   A prism of refracting angle 60° gives minimum deviation 40° for sodium light. Find its refractive index.
2. 2
   If the same prism is placed in water (n = 1.33), estimate the new angle of minimum deviation.
3. 3
   A ray is incident on a prism of angle 60° with refractive index 1.524. Find the angle of incidence at which it just suffers total internal reflection at the other face.

3 Mark Questions

1. 1
   Derive the prism formula n = sin((A + δ_m)/2) / sin(A/2).
2. 2
   Show that the deviation is minimum when the ray passes symmetrically through the prism, and obtain the relation between the angle of incidence and the angle of emergence at minimum deviation.

• Angular dispersion:
• $\theta =(n_v-n_r)A$
• for a thin prism — the angular separation between violet and red rays.
• Dispersive power:
• $\omega =\frac{n_v-n_r}{n_y-1}$
• , where n_y is the refractive index for the mean (yellow) colour. It is a pure number describing how strongly a material separates colours.

MCQ Questions

1 Mark Questions

1. 1
   Define dispersion of light.
2. 2
   Define the dispersive power of a material.
3. 3
   Which colour has the highest refractive index in glass?

2 Mark Questions

1. 1
   Why is violet light deviated more than red light when passing through a prism?
2. 2
   A thin prism gives angular dispersion (n_v − n_r)A. What does this quantity represent?
3. 3
   How would you combine a crown and a flint glass prism so that white light is deviated without dispersion?

3 Mark Questions

1. 1
   Derive the expression for the dispersive power of a prism material in terms of the refractive indices of two colours.
2. 2
   Explain the formation of a rainbow, mentioning the roles of refraction, dispersion and total internal reflection in water droplets.

MCQ Questions

1 Mark Questions

1. 1
   Define the near point and far point of a normal human eye.
2. 2
   State the formula for the magnifying power of a simple microscope with the final image at the near point.
3. 3
   Write the expression for the magnifying power of an astronomical telescope in normal adjustment.

2 Mark Questions

1. 1
   A telescope has objective f₀ = 144 cm and eyepiece f_e = 6 cm. Find its magnifying power and the separation between the lenses.
2. 2
   Explain the correction of myopia and hypermetropia with suitable lenses.
3. 3
   A person with normal near point 25 cm uses a magnifying glass of focal length 5 cm. Find the closest and farthest object distances for clear reading and the corresponding magnifications.

3 Mark Questions

1. 1
   Draw a labelled ray diagram of a compound microscope and derive an expression for its magnifying power with the final image at infinity.
2. 2
   A giant telescope has an objective of focal length 15 m and an eyepiece of focal length 1.0 cm. Find the angular magnification and, for the moon of actual diameter 3.48 × 10⁶ m at a distance 3.8 × 10⁸ m, the diameter of the image formed by the objective.

• Angular size of the primary rainbow: about 42° (red) to 40° (violet) from the antisolar point.
• Secondary rainbow: about 50° (violet) to 53° (red) — colours are reversed.
• Scattering of sunlight by air molecules makes the sky look blue (Rayleigh scattering) and reddening occurs at sunrise and sunset when sunlight travels through a longer atmospheric path.

MCQ Questions

1 Mark Questions

1. 1
   Name the optical phenomenon responsible for the formation of a rainbow.
2. 2
   Why is the sky blue?
3. 3
   In which order do the colours appear in the primary rainbow?

2 Mark Questions

1. 1
   Using refraction and total internal reflection inside a water drop, explain the formation of a primary rainbow.
2. 2
   Why does the red colour appear on the outer edge of the primary rainbow?

3 Mark Questions

1. 1
   Explain with a diagram the formation of the primary rainbow and the physical reason for the order of colours.
2. 2
   Why is the secondary rainbow fainter and how are its colours arranged?

Exercises (NCERT Q9.1–Q9.27)

• 9.1) A small candle, 2.5 cm in size, is placed 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?
• 9.2) A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.
• 9.3) A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom measured by a microscope is 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would the microscope have to be moved to focus on the needle again?
• 9.4) Figures (a) and (b) show refraction of a ray in air incident at 60° with the normal to a glass–air and water–air interface respectively. For (c) predict the angle of refraction in glass when the angle of incidence in water is 45°.
• 9.5) A small bulb is placed at the bottom of a tank containing water to a depth of 80 cm. What is the area of the surface of water through which light from the bulb can emerge? Take the refractive index of water as 1.33.
• 9.6) A prism is made of glass of unknown refractive index. A parallel beam of light is incident on a face and the angle of minimum deviation measured is 40°. What is the refractive index of the prism material? (Refracting angle 60°.) If the prism is placed in water (n = 1.33), predict the new angle of minimum deviation.
• 9.7) Double-convex lenses are to be manufactured from glass of refractive index 1.55, both faces having the same radius of curvature. What radius is required for a focal length of 20 cm?
• 9.8) A beam of light converges at a point P. A lens is placed in its path 12 cm from P. Where does the beam converge if the lens is (a) a convex lens of focal length 20 cm, (b) a concave lens of focal length 16 cm?
• 9.9) An object of size 3.0 cm is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image formed. What happens if the object is moved further away from the lens?
• 9.10) What is the focal length of a convex lens of focal length 30 cm in contact with a concave lens of focal length 20 cm? Is the system converging or diverging?
• 9.11) A compound microscope consists of an objective of focal length 2.0 cm and an eyepiece of focal length 6.25 cm, separated by 15 cm. How far from the objective should the object be placed for the final image at (a) the least distance of distinct vision (25 cm), (b) infinity? What is the magnifying power in each case?
• 9.12) A person with normal near point (25 cm) uses a compound microscope with objective f = 8 mm and eyepiece f = 2.5 cm, object placed 9.0 mm from the objective. What is the separation between the two lenses? Calculate the magnifying power of the microscope.
• 9.13) A small telescope has an objective of focal length 144 cm and an eyepiece of focal length 6.0 cm. What is the magnifying power and the separation between the objective and the eyepiece?
• 9.14) (a) A giant refracting telescope has an objective of focal length 15 m and eyepiece 1.0 cm. What is the angular magnification? (b) For the moon of diameter 3.48 × 10⁶ m at a distance 3.8 × 10⁸ m, what is the diameter of the image formed by the objective?
• 9.15) Use the mirror equation to deduce that: (a) an object placed between f and 2f of a concave mirror produces a real image beyond 2f; (b) a convex mirror always produces a virtual image independent of the object location; (c) the virtual image by a convex mirror is always diminished and lies between the pole and the focus; (d) an object between the pole and focus of a concave mirror produces a virtual and enlarged image.
• 9.16) A small pin fixed on a table top is viewed from above from a distance of 50 cm. By what distance would it appear to be raised if viewed through a 15 cm thick glass slab of n = 1.5 held parallel to the table? Does the answer depend on the location of the slab?
• 9.17) (a) A "light pipe" is a glass fibre of refractive index 1.68 clad with material of index 1.44. What range of angles of incidence with the axis gives total internal reflection inside the pipe? (b) What if there is no outer covering?
• 9.18) Answer: (a) Can plane and convex mirrors produce real images under some circumstances? (b) A virtual image cannot be caught on a screen, yet we see one on the retina. Is there a contradiction? (c) A diver under water looks obliquely at a fisherman on the bank — taller or shorter? (d) Does the apparent depth of a tank change when viewed obliquely? (e) Is the much larger refractive index of diamond of use to a diamond cutter?
• 9.19) The image of a small electric bulb fixed on a wall is to be obtained on the opposite wall 3 m away by a large convex lens. What is the maximum possible focal length of the lens?
• 9.20) A screen is placed 90 cm from an object. The image is formed by a convex lens at two positions separated by 20 cm. Determine the focal length of the lens.
• 9.21) (a) Determine the effective focal length of the two lenses of Q9.10 placed 8.0 cm apart with coincident principal axes. Does the answer depend on which side the parallel beam is incident? (b) An object 1.5 cm tall is placed 40 cm from the convex lens. Find the magnification and size of the final image.
• 9.22) At what angle should a ray be incident on a prism of refracting angle 60° and refractive index 1.524 so that it just suffers total internal reflection at the other face?
• 9.23) You are given prisms of crown and flint glass of a wide variety of angles. Suggest combinations that (a) deviate a pencil of white light without much dispersion, (b) disperse and displace a pencil of white light without much deviation.
• 9.24) For a normal eye: far point at infinity, near point 25 cm, cornea converges with power 40 D and the eye-lens contributes about 20 D at least. Estimate the range of accommodation of the eye-lens.
• 9.25) Does short-sightedness (myopia) or long-sightedness (hypermetropia) necessarily imply partial loss of accommodation? If not, what might cause these defects?
• 9.26) A myopic person wears spectacles of power −1.0 D for distant vision and in old age also needs a reading glass of +2.0 D. Explain what may have happened.
• 9.27) A person sees vertical lines of a shirt pattern more distinctly than horizontal lines. What is this defect due to, and how is it corrected?

Additional Exercises (NCERT Q9.28–Q9.38)

• 9.28) A man with normal near point (25 cm) reads small print using a thin convex lens of focal length 5 cm. (a) Find the closest and farthest distances at which he should keep the lens from the page to read through it. (b) What is the maximum and minimum angular magnification possible?
• 9.29) A card sheet divided into squares of each size 1 mm² is viewed at a distance of 9 cm through a magnifying glass of focal length 10 cm held close to the eye. (a) What is the magnification and the area of each square in the virtual image? (b) What is the angular magnification? (c) Are the two equal? Explain.
• 9.30) (a) At what distance should the lens of the previous question be held from the figure to view the squares distinctly with the maximum magnifying power? (b) What is the magnification then? (c) Is it equal to the magnifying power? Explain.
• 9.31) What should be the distance between the object of the previous question and the magnifier if each square in the virtual image is to have area 6.25 mm²? Would you be able to see the squares distinctly with your eyes close to the magnifier?
• 9.32) Answer: (a) The angle subtended by an object at the eye equals that subtended by the virtual image in a magnifying glass — in what sense then is there angular magnification? (b) If the eye is moved back, does the angular magnification change? (c) What stops us from using lenses of smaller and smaller focal length for greater magnification? (d) Why must both lenses of a compound microscope have short focal lengths? (e) When viewing through a compound microscope, why should the eye be a short distance from the eyepiece, and how much?
• 9.33) A magnification of 30× is desired using an objective of f = 1.25 cm and an eyepiece of f = 5 cm. How will you set up the compound microscope?
• 9.34) A small telescope has an objective of focal length 140 cm and an eyepiece of focal length 5.0 cm. Find the magnifying power for distant objects when (a) the final image is at infinity, (b) the final image is at the least distance of distinct vision.
• 9.35) (a) For the telescope of Q9.34(a), what is the separation between the objective and the eyepiece? (b) If used to view a 100 m tall tower 3 km away, what is the height of the image formed by the objective? (c) What is the height of the final image formed at 25 cm?
• 9.36) A Cassegrain telescope uses two mirrors as shown. Built with mirrors 20 mm apart: the large mirror radius of curvature is 220 mm and the small one 140 mm. Where will the final image of an object at infinity be formed?
• 9.37) Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards. A current deflects the mirror by 3.5°. What is the displacement of the reflected spot on a screen 1.5 m away?
• 9.38) An equiconvex lens (n = 1.50) in contact with a liquid layer on a plane mirror gives an inverted image at the needle position at 45.0 cm. With the liquid removed the distance is 30.0 cm. What is the refractive index of the liquid?

Answers at a Glance

• 9.1) Screen at 54 cm from the mirror; image real, inverted, magnified (5 cm); moving the candle closer moves the screen away. | 9.2) Image at 6.7 cm behind the mirror, m ≈ +0.56, erect and diminished; image stays between pole and focus as the needle moves away. | 9.3) n = 1.33; microscope raised by 1.73 cm. | 9.4) About 38.7°. | 9.5) Area ≈ 2.6 m². | 9.6) n ≈ 1.532; in water δ_m ≈ 10.3°. | 9.7) R = 22 cm.
• 9.8) (a) 7.5 cm, (b) 48 cm from the lens. | 9.9) Virtual, erect, at 8.4 cm, m = 0.6; image moves closer to the focus as the object recedes. | 9.10) f = −60 cm, diverging. | 9.11) (a) u₀ = −2.5 cm, m = 20; (b) u₀ ≈ −2.59 cm, m ≈ 13.5. | 9.12) Separation ≈ 9.47 cm; magnifying power ≈ 88. | 9.13) m = 24; L = 150 cm. | 9.14) (a) 1500; (b) ≈ 13.7 cm. | 9.15) Deductions from 1/v + 1/u = 1/f as stated. | 9.16) Apparent rise 5 cm; independent of slab position. | 9.17) (a) 0°–60°; (b) 0°–90°.
• 9.18) (a) Yes, with a converging incident beam (virtual object); (b) no contradiction — the retina is a screen; (c) taller; (d) apparent depth decreases; (e) yes — smaller critical angle, more total internal reflection. | 9.19) f_max = 0.75 m. | 9.20) f ≈ 21.4 cm. | 9.21) (a) Depends on the side (216 cm / 416 cm) — not a useful concept; (b) m ≈ 0.65, size ≈ 0.98 cm. | 9.22) i = 29.75°. | 9.23) (a) Crown + flint, bases opposite: deviation without dispersion; (b) same arrangement sized for dispersion without deviation.
• 9.24) Eyepiece lens range ≈ 20–24 D (total eye power 60–64 D). | 9.25) No — they come from an elongated/shortened eyeball; loss of accommodation itself is presbyopia. | 9.26) Far point was 100 cm (myopia) and the near point receded to about 50 cm in old age — needs −1.0 D for distance, +2.0 D for near. | 9.27) Astigmatism — a non-spherical cornea — corrected with a cylindrical lens. | 9.28) (a) 4.17 cm to 5 cm; (b) m = 6 maximum, 5 minimum. | 9.29) (a) m = 10, area 100 mm² (=1 cm²); (b) ≈ 2.8; (c) not equal.
• 9.30) (a) 7.14 cm; (b) m = 3.5; (c) yes, equal when the image is at the near point. | 9.31) u = 6 cm; no — the image forms inside the near point. | 9.32) (a) The object can be brought closer than 25 cm, subtending a larger angle; (b) it decreases; (c) manufacturing limits and increasing aberrations; (d) to get large magnification; (e) at the emergent-beam (eye-ring) position. | 9.33) Object 1.5 cm from the objective; lenses separated by 11.67 cm. | 9.34) (a) 28; (b) 33.6. | 9.35) (a) 145 cm; (b) 4.7 cm; (c) ≈ 28 cm.
• 9.36) Final image 315 mm from the small mirror. | 9.37) ≈ 18.4 cm. | 9.38) n ≈ 4/3 ≈ 1.33.

MCQ Questions

1 Mark Questions

1. 1
   State the sign convention used in the mirror and lens formulae.
2. 2
   Define the power of a lens and give its SI unit.
3. 3
   State the conditions for total internal reflection.
4. 4
   Write the lens maker's formula.

2 Mark Questions

1. 1
   An object is placed 30 cm in front of a convex lens of focal length 20 cm. Find the position and nature of the image. (Ans: v = 60 cm, real and inverted)
2. 2
   A concave mirror has a focal length of 20 cm. Where should an object be placed to get a real image double the object size? (Ans: 2 = v/u ⇒ u = −30 cm)
3. 3
   The refractive index of glass is 1.5. Find the critical angle for glass–air interface. (Ans: sin ic = 1/1.5 ⇒ ic ≈ 41.8°)

3 Mark Questions

1. 1
   Derive the lens maker's formula for a thin convex lens. (Ans: 1/f = (μ − 1)(1/R₁ − 1/R₂))
2. 2
   Derive the relation between refractive index, prism angle and minimum deviation, μ = sin[(A + δm)/2]/sin(A/2).
3. 3
   Explain the construction and working of a compound microscope and derive its magnifying power at normal adjustment.

5 Mark Questions

1. 1
   Describe the construction and working of a refracting astronomical telescope in normal adjustment, derive its magnifying power, and state the effect of objectives of different apertures on the resolving power.
2. 2
   Explain total internal reflection — conditions, relation with critical angle, and two applications (optical fibre and mirage formation) — with ray diagrams.