# Numerical Problems : Ray optics & Wave optics

#### LEVEL – I

Q:1. A thin converging lens forms a magnified image (magnification :p) of an object. The magnification factor becomes q when the lens is moved a distance ‘ a ‘ towards the object. Find the focal length of the lens.

Ans: $\displaystyle \frac{a p q}{q – p}$

Q:2. A parallel beam of light is incident normally onto a solid glass sphere of radius R (μ = 1.5). Find the distance of the image from the outer edge of the glass sphere.

Ans: $\displaystyle \frac{R(2-n)}{2(n-1)}$

Q:3. A point object is placed in front of a silvered plano-convex lens of refractive index n, radius of curvature R, so that its image is formed on itself. Calculate the object distance.

Ans: R/n

Q:4. A convex lens focuses a distant object on a screen placed 10 cm away from it. A glass plate (n = 1.5) of thickness 1.5 is inserted between the lens and the screen. Where should the object be placed so that its image is again focused on the screen?

Ans: 190 cm, right of the lens

Q:5. A parallel beam of light travelling in water (refractive index = 4/3) is refracted by a spherical air bubble of radius 2 mm situated in water. Assuming the light rays to be paraxial
(i) find the position of image due to refraction at first surface and position of final image.
(ii) draw a ray diagram showing the position of both images.

Ans: – 5 mm from left of 2nd surface

Q:6. Find the focal length of the lens shown in the figure. The radii of curvature of both the surfaces are equal to R.

Ans: $\displaystyle \frac{\mu_3 R}{\mu_3 – \mu_1}$

Q:7. A converging lens which has a focal length of 20 cm is placed 60 cm to the left of a concave mirror of focal length 30 cm. An object is placed 40 cm to the left of lens. Find the position, nature and magnification of the final image.

Ans: 60 cm behind the mirror, virtual & inverted, 3

Q:8. A cylindrical glass rod has its two coaxial ends of spherical form bulging outward. The front end has a radius of curvature 5 cm and the back end which is silvered has a radius of curvature 8 cm. The thickness of the rod along the axis is 10 cm. Calculate the position of the image of a point object at the axis 50 cm from front face (ang = 1.5)

Ans: 9.365 cm

Q:9. A thin bi-convex lens of refractive index 3/2 and radius of curvature 50 cm is placed on a reflecting convex surface of radius of curvature 100 cm. A point object is placed on the principal axis of the system such that its final image coincides with itself. Now few drops of a transparent liquid is placed between the mirror and lens such that final image of the object is at infinity. Find refractive index of the liquid used. And also find position of the object.

Ans: 7/6 and 100 cm

Q:10. An object of height 2.5 cm is placed at a 1.5 f from a concave mirror where f is the magnitude of the focal length of the mirror. The object is placed perpendicular to the principal axis. Find the height of the image. Is the image erect or inverted ?

Ans: 5 cm and inverted

Q:11. In Young’s double slit experiment the fringe width obtained is 0.6 cm, when light of wavelength 4800 A0 is used. If the distance between the screen and the slit is reduced to half, what should be the wavelength of light used to obtain fringes 0.0045-m width?

Ans: 72 × 10-7 m

Q:12. In a Young’s double slit experiment, the slits are 1.5 mm apart. When the slits are illuminated by a monochromatic light source and the screen is kept 1 m apart from the slits, width of 10 fringes is measured as 3.93 mm. Calculate the wavelength of light used. What will be the width of 10 fringes when the distance between the slits and the screen is increased by 0.5 m. The source of light used remains the same.

Ans: 5.9 × 10-7m , 5.9 × 10-3 m

Q:13. A beam of light consisting of two wavelengths 6500 A° and 5200 A° is used to obtain interference fringes in a Young’s double slit experiment. Find the distance of the third fringe on the screen from the central maximum for the wavelength 6500 A°.

Ans: 0.117 cm

Q:14. In a two-slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance from the slits. If the screen is moved by 5 x 10-2 m towards the slits, the change in fringe width is 3 × 10-5 m. If the distance between the slits is 10-3 m, calculate the wave length of the light used.

Ans: 6000 A°

Q:15. At a certain point on a screen the path difference for the two interfering rays is (1/8)th of a wavelength. Find the ratio of the intensity at this point to that at the centre of a bright fringe.

Ans: 0.853

Q:16. Figure shows three equidistant slits being illuminated by a monochromatic parallel beam of light. Let BP0 – AP0 = λ/3 and D >> λ .
(a) Show that in this case d = √ 2λD/3 .
(b) Show that the intensity at P0 is three times the intensity due to any of the three slits individually.

Q:17. Two sources S1 and S2 emitting light of wave lengths 600 nm are placed at a distance of 1.0 x 10-2cm. A detector can be moved on the line S1P which is perpendicular to S1S2 . Find out the position of first minimum detected.

Ans:1.7 cm

Q:18. White light may be considered to have λ from 4000 A0 to 7500 A0 . If an oil film has thickness 10-6m, deduce the wavelengths in the visible region for which the reflection among the normal direction will be (i) weak, (ii) strong. Take μ of the oil as 1.40.

Ans: For weak reflection : 7000 , 4667 & 4000 A0

Q:19. Find the maximum intensity in case of interference of n identical waves each of intensity I0 if the interference is (a) coherent (b) incoherent

Ans: n2I0 , n I0

Q:20. A monochromatic light of λ = 5000 A0 is incident on two identical slits separated by a distance of 5 x 10-4 m. The interference pattern is seen on a screen placed at a distance of 1m from the plane of slits. A thin glass plate of thickness 1.5 x 10-6 m and refractive index μ = 1.5 is placed between one of the slits and screen. Find the intensity at the centre of the screen if the intensity there is I0 in the absence of the plate. Also find the lateral shift of the central maxima.

Ans: Zero , 1.5 mm