Std 9 Science and Technology · Chapter 11

11. Reflection of Light

Complete solutions to the Exercises. Maharashtra State Board (Balbharati), Standard Nine Science and Technology.

Exercises

1Answer the following questions.

(a) Explain the difference between a plane mirror, a concave mirror and a convex mirror with respect to the type and size of the images produced.

Plane mirrorImage is always virtual and erect, formed behind the mirror at the same distance as the object. It is of the same size as the object and laterally inverted.

Concave mirrorImage may be real and inverted (object beyond F) or virtual and erect (object between P and F). Its size may be larger, equal or smaller than the object depending on the object’s position; a point image forms at F for a very distant object.

Convex mirrorImage is always virtual, erect and diminished, formed behind the mirror between P and F, whatever the position of the object.

(b) Describe the positions of the source of light with respect to a concave mirror in 1. Torch light 2. Projector lamp 3. Floodlight.

Torch light: the bulb is placed at the principal focus of the concave reflector. Rays from the focus become parallel to the principal axis after reflection, giving a strong parallel beam that travels far.
Projector lamp: the lamp is placed at the centre of curvature of the concave mirror behind it. Light going backward from the lamp strikes the mirror and is reflected back along the same path through the lamp towards the film and lenses, so the light that would have been wasted is also used and the picture is brighter.
Floodlight: the source is placed a little beyond the centre of curvature. The reflected rays converge between F and C and then spread out, giving a bright, wide beam over a large area.

A concave mirror is a converging (focusing) mirror. When sunlight, whose rays are practically parallel, falls on it, all the reflected rays meet at the focus. A large amount of solar energy is thus concentrated in a small region, producing a high temperature. So concave mirrors are used in solar cookers, solar water heaters and solar furnaces, with the vessel or tube placed at the focus.

(d) Why are the mirrors fitted on the outside of cars convex?

A convex mirror always forms an erect image, so the driver sees vehicles behind the right way up.
The image is diminished, so a much larger area behind the car can be seen in a small mirror; the field of view is wide.
The image is formed whatever the distance of the vehicle behind.

So convex mirrors help the driver see the traffic behind and drive safely. (Because the image is small, vehicles appear farther away than they actually are.)

(e) Why does obtaining the image of the sun on a paper with the help of a concave mirror burn the paper?

The sun is very far away, so its rays reaching the mirror are parallel to one another. A concave mirror converges these parallel rays to a point at its focus, where a small, very bright image of the sun is formed. All the light and heat energy falling on the mirror is concentrated on that small spot of paper. Its temperature rises above the ignition temperature of paper, and the paper catches fire.

(f) If a spherical mirror breaks, what type of mirrors are the individual pieces?

Each piece is still a spherical mirror of the same type — pieces of a concave mirror are concave mirrors, and pieces of a convex mirror are convex mirrors. Every piece is part of the same sphere, so each has the same radius of curvature and focal length as the original mirror. Only the size (aperture) is smaller, so the images are less bright.

2What sign conventions are used for reflection from a spherical mirror?

The Cartesian sign convention is used. The pole of the mirror is taken as the origin and the principal axis as the X-axis.

The object is always kept on the left of the mirror. All distances parallel to the principal axis are measured from the pole.
Distances measured to the right of the pole (in the direction of the incident light) are positive; distances measured to the left are negative.
Distances measured vertically upward from the principal axis are positive.
Distances measured vertically downward from the principal axis are negative.
So the focal length of a concave mirror is negative and that of a convex mirror is positive. The object distance (u) is always negative, and the height of the object is always positive. The height of a real image is negative and that of a virtual image is positive.

Mirror formula: 1/v + 1/u = 1/f Magnification: M = h₂/h₁ = −v/u

3Draw ray diagrams for the cases of images obtained in concave mirrors as described in the table on page 122.

In each diagram, C is the centre of curvature, F the principal focus and P the pole. Two rays are drawn from the top of the object: one parallel to the principal axis, which after reflection passes through F; and one through F (or along the line from C), which after reflection becomes parallel to the axis (or returns along the same path). The image is formed where the reflected rays meet. Dashed lines show reflected rays extended behind the mirror.

(1) Object between the pole and focus — image behind the mirror, virtual, erect, magnified.

(2) Object at the focus — image at infinity, real, inverted, very large.

(3) Object between focus and centre of curvature — image beyond C, real, inverted, magnified.

(4) Object at the centre of curvature — image at C, real, inverted, same size.

(5) Object beyond the centre of curvature — image between C and F, real, inverted, diminished.

(6) Object at a very large (infinite) distance — image at F, real, inverted, point-sized.

4Which type of mirrors are used in the following? (Periscope, floodlights, shaving mirror, kaleidoscope, street lights, head lamps of a car)

PeriscopePlane mirrors — two mirrors at 45° turn the light twice to show objects above an obstacle.

FloodlightsConcave mirror — source a little beyond C gives a bright, wide beam.

Shaving mirrorConcave mirror — face between P and F gives an erect, magnified image.

KaleidoscopePlane mirrors — placed at an angle to form multiple images and patterns.

Street lightsConvex mirror (reflector) — diverges light to spread it over a wide area.

Head lamps of a carConcave mirror — bulb at F gives a strong parallel beam.

5Solve the following examples.

(a) An object of height 7 cm is kept at a distance of 25 cm in front of a concave mirror. The focal length of the mirror is 15 cm. At what distance from the mirror should a screen be kept so as to get a clear image? What will be the size and nature of the image?

Given: h₁ = 7 cm, u = −25 cm, f = −15 cm; v = ?, h₂ = ?
1/v = 1/f − 1/u = −1/15 + 1/25 = (−5 + 3)/75 = −2/75
v = −37.5 cm
M = −v/u = −(−37.5)/(−25) = −1.5
h₂ = M × h₁ = −1.5 × 7 = −10.5 cm

Answer: the screen should be kept 37.5 cm in front of the mirror. The image is 10.5 cm high, real, inverted and magnified (the negative signs show it is on the object’s side and below the axis).

(b) A convex mirror has a focal length of 18 cm. The image of an object kept in front of the mirror is half the height of the object. What is the distance of the object from the mirror?

Given: f = +18 cm; a convex mirror forms an erect, virtual image, so M = +1/2
M = −v/u = 1/2 ⇒ v = −u/2
1/v + 1/u = 1/f ⇒ −2/u + 1/u = 1/18 ⇒ −1/u = 1/18
u = −18 cm (and v = +9 cm)

Answer: the object is at a distance of 18 cm in front of the mirror. The image is formed 9 cm behind the mirror.

(c) A 10 cm long stick is kept in front of a concave mirror having focal length of 10 cm in such a way that the end of the stick closest to the pole is at a distance of 20 cm. What will be the length of the image?

The stick lies along the principal axis, with its near end at 20 cm and its far end at 20 + 10 = 30 cm from the pole. We find the image of each end.

f = −10 cm
Near end: u = −20 cm (this is the centre of curvature, since R = 2f = 20 cm)
1/v = −1/10 + 1/20 = −1/20 ⇒ v₁ = −20 cm
Far end: u = −30 cm
1/v = −1/10 + 1/30 = −2/30 ⇒ v₂ = −15 cm
Length of image = 20 − 15 = 5 cm

Answer: the image of the stick lies between 15 cm and 20 cm in front of the mirror, and its length is 5 cm. It is real, inverted and diminished.

The textbook’s printed answer is 10 cm. That is the result if the stick stands perpendicular to the principal axis with its foot at 20 cm, the centre of curvature: an object at C gives an image of the same size at C, so the image is 10 cm long. But the wording — “the end of the stick closest to the pole” — describes a stick lying along the axis, which gives 5 cm. Write the answer your teacher expects, showing the case you have assumed.

6Three mirrors are created from a single sphere. Which of the following — pole, centre of curvature, radius of curvature, principal axis — will be common to them and which will not be common?

Common: centre of curvature and radius of curvature. All three mirrors are parts of the same sphere, so the centre of that sphere is the centre of curvature of each, and its radius is the radius of curvature of each. (So their focal length, R/2, is also the same.)

Not common: pole and principal axis. The pole is the centre of each mirror’s own surface, so the three mirrors, cut from different parts of the sphere, have three different poles. The principal axis is the line joining the pole and the centre of curvature; since the poles are different, the three principal axes are different lines, though all pass through the common centre of curvature.

PProject: Make a kaleidoscope and make a presentation in the class about how it works.

Making it: take three rectangular plane mirror strips of equal size (about 15 cm × 4 cm). Join them with tape into a triangular prism with the shiny sides facing inward. Cover one end with a circle of transparent plastic, put a few small pieces of coloured glass or bangle, beads and bits of coloured paper on it, and cover them with a sheet of translucent (ground) paper. Cover the other end with dark paper having a small hole in the centre to look through. Wrap the whole tube in chart paper.

How it works: when you look through the hole towards the light, each coloured piece is reflected again and again by the mirrors kept at 60° to one another. Two plane mirrors at an angle A form 360°/A − 1 images; for 60° that is 5 images, so every piece and its images make a beautiful symmetrical pattern. When the tube is rotated, the pieces shift and a new pattern appears each time.

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