﻿ Reflection at Curved

# MATHS

## Reflection at Curved

#### Objectives

By the end of the topic you should be able to:

• Describe concave, convex and parabolic reflectors
• Describe using ray diagrams the principal axis, principal focus,�  center of curvature and related terms
• Locate images formed by curved mirrors, using ray construction
• Determine characteristics of images formed by concave mirrors
• Explain applications of curved reflecting surfaces

Introduction
Reflection of light on plane surfaces was learnt in Form One. In this topic, we shall learn about reflection of light on curved surfaces. We shall also discuss images formed by curved surfaces and applications of such surfaces. The picture below shows the image of a city street as seen in a curved mirror.

Introduction

Curved reflectors
Activity 3.1(a): Cutting a hollow sphere to obtain concave and convex reflectors.

1) Click on the button to initiate the process.

2) Describe the results you have obtained

Results
In this activity, we have obtained two curved reflecting surfaces, the shape of a watch glass. In one piece, the reflecting surface curves inwards, while in the other piece it curves outwards.

A reflecting surface which curves inwards is called a concave reflector while that which curves outwards is a convex reflector. Concave and Convex reflectors form part of a sphere; so they are also known as spherical reflectors.

Activity 3.1(b):Cutting a hollow spheroid to obtain a parabolic reflector.On the next panel (page),play the animation to observe how a simple parabolic reflector can be obtained from a hollow egg-shaped object. The inner surface is silvered to act as the reflecting surface. Describe the shape obtained.

Result
The product of our activity is a parabolic reflector. Parabolic reflectors are more sharply curved inwards, and are comparable to an egg cut longitudinally into two halves.

Definition of terms

The terms associated with curved reflectors are as follows.

• Centre of curvature, C, is the centre of sphere of which the curved reflector is part.
• Principal axis, is an imaginary line passing through the center of a curved mirror and is perpedicular to the miror.
• The pole, P, is the geometric centre of a curved reflecting surface.
• Focal point, F is the point at which parallel incident rays converge after reflection on a curved mirror.
• Focal length, f is the distance from the focal point to the pole of a curved mirror.

Images formed by curved mirrors

Images formed by curved mirrors

You will recall that, in plane mirrors, the size of image equals to the size of object. Also, image distance equals object distance. For curved mirrors, however, the size of image formed depends on two factors, namely:distance of the object from the curved mirror and the type of curved mirror; that is, whether concave or convex. The photograph below shows an object and its magnified (larger) image in a curved reflector.

It is possible to determine the nature and position of image formed by curved mirrors.

Rays used to locate images

Rays 1, 2 and 3 described in the following animations are useful in constructing ray diagrams. Observe the paths and behaviour of rays in each case.

RAY 1: Incident rays parallel to the principal axis

PARALLEL RAYS INCIDENT ON A CONCAVE MIRROR

Move the cursor to the word PARALLEL RAY. Describe the path of the ray.

Move the cursor to the word CONVEX MIRROR.

Move the cursor to the word PARALLEL RAY. Describe the path taken by the ray.

Observations: An incident ray that is parallel and close to the principal axis passes through the principal focus, F, after reflection on a concave mirror. For a convex mirror, such a ray appears to emerge from the principal focus, F, after reflection.

RAY 2: An incident ray passing through the principal focus, F

1. Move the cursor to the word CONCAVE MIRROR.
2. Move the cursor to the word RAY.

3.Move the cursor to the word CONVEX MIRROR.

4. Move the cursor to the word RAY.

Observations: An incident ray passing through, F, for concave mirror, or appears to pass through, F, for convex mirror is reflected parallel to the principal axis.

RAY 3: An incident ray passing through the centre of curvature, C

1. Move the cursor to the word CONCAVE MIRROR.
2. Move the cursor the word RAY. Describe the path of the ray.

3. Move the cursor to the word CONVEX MIRROR.
4. Move the cursor to the word RAY.

Observations: A ray that passes or appears to pass through the center of curvature, C, will be reflected along the same path of incidence for a concave mirror and convex mirror respectively.

Summary of observations

1. An incident ray that is parallel and close to the principal axis passes through the principal focus, F, after reflection on a concave mirror

2. An incident ray passing through, F, for concave mirror, or appears to pass through, F, for convex mirror is reflected parallel to the principal axis.

3. A ray that passes or appear to pass through the center of curvature, C, will be reflected along the same path of incidence for a concave mirror and convex mirror respectively.

Images formed by concave mirrors

(a) Image of an object placed behind the centre of curvature, C
Make and record your observations regarding the nature and position of the image in each stage.

1. Move the cursor to the word CONCAVE MIRROR.

2. Move the cursor to the word OBJECT.

What is the position and nature of the image formed?

Results

An image is formed at the point where rays meet after reflection. The image formed has the following characteristics:

• Formed between F and C
• Smaller than the object
• Inverted
• Real (because the rays actually meet).

(b) Object placed at the centre of curvature, C

With the object placed at point C, observe the nature and position of the image formed.

Results

The image is:

• Formed at C
• The same size as the object
• Inverted
• Real and in front of the mirror.

(c) Object between C and F

Results

The image formed is:

• Inverted
• Real
• Bigger than the object (magnified)
• In front of the mirror

(d) Object placed at F

Results

The reflected rays converge very far (infinity). The image formed is:

• Real
• Inverted
• Magnified and far away from the mirror.

Object between F and P

Results

The image formed is:

• Virtual (rays are not meeting but appear to meet)
• Upright
• Behind the mirror and magnified.

Images formed by convex mirrors

Images in a convex mirror are:

• Always formed behind the mirror
• Smaller than the object (diminished)
• Upright
• Virtual (for all positions of the object).

Applications of curved reflectors

(a) Dentist's mirror

Fig. 3.20: Dentist's mirror

Concave mirrors are used by dentists. When the mirror is held behind the teeth, at a distance slightly less than its focal length, a magnified image of the teeth is formed; so details of the damage or infection can be revealed.

(b) Solar concentrators and reflectors
Parabolic mirrors are used as reflectors in car headlamps, torches and solar concentrators. Fig. 3.21 is a simplified diagram for the arrangement used in a solar concentrator. Solar concentrators can be used for domestic purposes such as boiling water and cooking. Click on the play button and observe the effect of a solar concentrator.

Fig. 3.21: Solar heater

(c) Driving mirror
Convex mirrors enable us to see a wider area behind us. That is, they have a wide field of view. In addition, they form upright images. These properties make convex mirrors suitable for use as driving mirrors. See the photograph.

Conclusion

(a) Concave mirror

Characteristics of images formed in a concave mirror depend on the position of object.

For an object placed between the mirror and principal focus, the image is always virtual and upright.

Whenever an object is placed at the principal focus, the image forms at infinity.

(b) Convex mirror

The image in a convex mirror will always be virtual, upright and diminished regardless of the object position.

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