PhysicsCBSE Class 12 · Sample paper, session 2023-24 · 5 marks · Long answer

CBSE Class 12 Physics sample paper 2023-24: question 31(a)

From the official CBSE sample question paper for 2023-24, section E, question 31(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q31(a) · 5 marks · Long answerRay Optics and Optical Instruments

i. Draw a ray diagram for the formation of image of a point object by a thin double convex lens having radii of curvature $R_{1}$ and $R_{2}$ . Hence derive lens maker's formula.
ii. A converging lens has a focal length of 10 cm in air. It is made of a material of refractive index 1.6. If it is immersed in a liquid of refractive index 1.3, find its new focal length.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

Treat the lens as two refracting spherical surfaces, the image of the first being the object for the second.

Hint 2

Use $\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}$ for each surface and add.

Hint 3

For the numerical, only $(n_{21} - 1)$ changes in the liquid; take the ratio of focal lengths.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Ray diagram (described): A thin double convex lens of refractive index $n_{2}$ is placed in a medium of refractive index $n_{1}$ . Point object O on the principal axis. A ray from O strikes the first surface ABC (centre $C_{1}$ , radius $R_{1}$ ) at A and bends towards the axis, heading to $I_{1}$ (image by the first surface). Before reaching $I_{1}$ it meets the second surface ADC (centre $C_{2}$ , radius $R_{2}$ ) and is refracted to meet the axis at the final image I. B and D are the poles of the two surfaces.
1 mark
Refraction at the first surface (rarer to denser), image at $I_{1}$ :
$\frac{n _{2}}{B I _{1}} + \frac{n _{1}}{O B} = \frac{n _{2} - n _{1}}{B C _{1}}$ ...(i)
½ mark
$I_{1}$ acts as a virtual object for the second surface (denser to rarer), final image at I:
$\frac{n _{1}}{D I} - \frac{n _{2}}{D I _{1}} = \frac{n _{2} - n _{1}}{D C _{2}}$ ...(ii)
½ mark
For a thin lens $B I_{1} \approx D I_{1}$ . Adding (i) and (ii):
$\frac{n _{1}}{O B} + \frac{n _{1}}{D I} = (n_{2} - n_{1}) (\frac{1}{B C _{1}} + \frac{1}{D C _{2}})$ ...(iii)
If the object is at infinity ( $O B \to \infty$ ), the image forms at the focus: $D I = f$ :
$\frac{n _{1}}{f} = (n_{2} - n_{1}) (\frac{1}{B C _{1}} + \frac{1}{D C _{2}})$
½ mark
Sign convention: $B C_{1} = + R_{1}$ , $D C_{2} = - R_{2}$ . Hence
$\frac{1}{f} = (\frac{n _{2}}{n _{1}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) = (n_{21} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$ — the lens maker's formula.
½ mark
(ii) In air: $\frac{1}{f _{a}} = (1.6 - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) = 0.6 (\frac{1}{R _{1}} - \frac{1}{R _{2}})$ ...(1)
1 mark
In liquid: $\frac{1}{f _{l}} = (\frac{1.6}{1.3} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) = \frac{0.3}{1.3} (\frac{1}{R _{1}} - \frac{1}{R _{2}})$ ...(2)
Dividing (1) by (2): $\frac{f _{l}}{f _{a}} = \frac{0.6 \times 1.3}{0.3} = 2.6 \Rightarrow f_{l} = 2.6 \times 10 = 26$ cm.
1 mark

Answer: $\frac{1}{f} = (n_{21} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$ ; new focal length = 26 cm

Where the marks are

Labelled ray diagram of refraction through double convex lens (1 mark)
Refraction equation at first surface (½ mark)
Refraction equation at second surface with $I_{1}$ as virtual object (½ mark)
Adds equations, takes object at infinity so image at focus (½ mark)
Applies sign convention $B C_{1} = R_{1}$ , $D C_{2} = - R_{2}$ to get lens maker's formula (½ mark)
Lens maker's equation in air with 0.6 factor (1 mark)
Equation in liquid and $f_{l} = 26$ cm (1 mark)

Revise the chapter

Chapter: Ray Optics and Optical Instruments (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Ray Optics and Optical Instruments, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Physics sample paper 2023-24: question 31(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Ray Optics and Optical Instruments