Image formation by Spherical Mirrors: Types, Solved Examples

We often see ourselves in a plane mirror placed in our bathroom or on our dressing table while getting ready. But what if these mirrors get curved! Will they still form the same kind of image as earlier? Will there be any difference in the size or location of the image formed? We will be able to answer these questions by knowing the concept of Image formation by Spherical Mirrors.

There are two types of spherical mirrors namely, convex and concave mirrors. The spherical mirror has various applications in our day-to-day life. In this article, we will discuss about image formation by spherical mirrors in detail. Continue reading to learn more!

Image Formation by Spherical Mirrors: Overview

Image formation by spherical mirrors can be well understood by first knowing the concept of spherical mirrors.

A spherical mirror is a part of a hollow sphere that is generally made of glass whose one surface is silvered. The surface other than the silvered one is capable of reflecting light. So, this surface is known as the reflecting surface of the spherical mirror. Light rays falling on the reflecting surface will get reflected following the laws of reflection. 

Laws of reflection states that:

1. The incident ray, the reflected ray and the normal to the reflecting surface at the point of incidence lie in the same plane.
2. The angle of incidence is equal to the angle of reflection. \((\therefore \angle i = \angle r)\)

Various light rays from the same point after getting reflected from the spherical mirrors may meet or appear to meet. If they actually meet, then they form a real image, and if they appear to meet, then they form a virtual image. 

Thus, images can be of two types: 

1. Real image: The image that can be obtained on a screen is called a real image. A real image is formed when two or more rays, after reflection or refraction, actually meet at a point.
2. Virtual image: The image that cannot be obtained on a screen is called a virtual image. A virtual image is formed when two or more rays after reflection or refraction appear to meet at a point.

Spherical mirrors form both real images as well as virtual images depending upon the curvature of the reflecting surface of the spherical mirror used and the position of the object with respect to the reflecting surface of the spherical mirror.

Types of Image Formation by Spherical Mirrors

Image formation by spherical mirrors will depend on the curvature of the reflecting surface of the spherical mirror. There are two types of spherical mirrors that are as mentioned below:

1. Concave mirror: The reflecting surface of this spherical mirror is curved inwards. A concave mirror can form both a real image as well as a virtual image depending upon the distance of the object from its reflecting surface. 

The real image formed by a concave mirror may be enlarged, of the same size or diminished as compared to the object. But the virtual image formed by a concave mirror is always enlarged as compared to the object.

Learn About Concave And Convex Mirror

2. Convex mirror: The reflecting surface of this spherical mirror is curved outwards. A convex mirror can form only a virtual image irrespective of the distance of the object from its reflecting surface. The virtual image formed by a convex mirror is always diminished as compared to the object.

Thus, a real image of varying size can be obtained only by using a concave mirror. But the virtual image can be obtained by both a concave mirror as well as a convex mirror.

Interestingly, the virtual image obtained by a concave mirror will always be enlarged, whereas the virtual image obtained by a convex mirror will always be diminished. 

So, if we need a virtual image of the same size as that of the object, then spherical mirrors are not the one to be chosen; rather, we need to choose a plane mirror.

Rules of Image Formation by Spherical Mirrors

Image formation by spherical mirrors is governed by a few rules that are as mentioned below:

1. A ray of light falling on a spherical mirror in a direction parallel to its principal axis passes through or appears to come from the principal focus of the spherical mirror after reflection.
2. A ray of light incident on the spherical mirror after passing through or directed towards the principal focus of the spherical mirror becomes parallel to the principal axis of the spherical mirror after reflection.
3. A ray of light incident on the spherical mirror after passing through or directed towards the centre of curvature of the spherical mirror is reflected back along the same path; that is, such a ray retraces its path in the opposite direction.
4. A ray of light incident obliquely towards the pole of the spherical mirror is reflected obliquely as per the laws of reflection.

Ray Diagrams of Image Formation by Spherical Mirrors

Image formation by spherical mirrors can be represented by ray diagrams. The ray diagrams of a concave mirror for different object positions and their corresponding image positions are as mentioned below:

(a) Object Position: At Infinity

Image Position: At the Principal Focus

Object Size: Highly Diminished, Point Sized

Object Nature: Real and Inverted

(b) Object Position: Beyond the Centre of Curvature

Image Position: Between the Centre of Curvature and the Principal Focus

Object Size: Diminished

Object Nature: Real and Inverted

(c) Object Position: At the Centre of Curvature

Image Position: At the Centre of Curvature

Object Size: Same Size

Object Nature: Real and Inverted

(d) Object Position: Between the Centre of Curvature and the Principal Focus

Image Position: Beyond the Centre of Curvature

Object Size: Enlarged

Object Nature: Real and Inverted

(e) Object Position: At the Principal Focus

Image Position: At Infinity

Object Size: Highly Enlarged

Object Nature: Real and Inverted

(f) Object Position: Between the Principal Focus and the Pole

Image Position: Behind the Concave Mirror

Object Size: Enlarged

Object Nature: Virtual and Erect

The ray diagrams of a convex mirror for different object positions and their corresponding image positions are as mentioned below:

(a) Object Position: At Infinity

Image Position: Behind the Convex Mirror at its Principal Focus

Object Size: Highly Diminished, Point Sized

Object Nature: Virtual and Erect

(b) Object Position: Between Infinity and the Pole

Image Position: Behind the Convex Mirror between its Principal Focus and Pole

Object Size: Diminished

Object Nature: Virtual and Erect

Learn Uses Of Convex Mirror & Its Applications

Sign Convention for Image Formation by Spherical Mirrors

Image formation by spherical mirrors follow certain sign conventions that are as follows:

1. The principal axis of the spherical mirror is taken along the X-axis of the rectangular coordinate system, and the pole of the spherical mirror is taken as the origin.
2. The object is taken on the left side of the spherical mirror; that is, light is incident on the spherical mirror from the left-hand side.
3. All the distances parallel to the principal axis of the spherical mirror are measured from the pole of the spherical mirror.
4. The distances measured in the direction of the incident light are taken as positive.
5. The distances measured in the direction opposite to the direction of incident light are taken as negative.
6. The heights measured upwards and perpendicular to the principal axis of the spherical mirror are taken as positive.
7. The heights measured downwards and perpendicular to the principal axis of the spherical mirror are taken as negative.
8. Thus, the focal length of a concave mirror is taken as negative, and the focal length of a convex mirror is taken as positive.

Formula for Image Formation by Spherical Mirrors

Image formation by spherical mirrors uses the mirror formula to establish the relation between the object distance \(u\), image distance \(v\) and focal length \(f\) of the spherical mirror.

The mirror formula is given by the relation,

\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

Here, the radius of curvature \(R\) of the spherical mirror is related to its focal length \(f\) by the relation,

\(f = \frac{R}{2}\)

The linear magnification \(m\) produced by the spherical mirror is given by the relation,

\(m = \frac{{{h_i}}}{{{h_o}}}\)

where,

\({{h_i}}\) is the image height

\({{h_o}}\) is the object height

The linear magnification \(m\) of a spherical mirror is related to the object distance \(u\) and the image distance \(v\) by the relation,

\(m =  – \frac{v}{u}\)

\(\therefore m =  – \frac{v}{u} = \frac{{{h_i}}}{{{h_o}}}\)

Solved Examples on Image Formation by Spherical Mirrors

Example 1: An object is placed at a distance of \(30\;{\rm{cm}}\) from a concave mirror of focal length \(15\;{\rm{cm}}\) Find the position of the image formed.
Solution:
Given that,
The focal length of the concave mirror is \(f = – 15\;{\rm{cm}}\)
The object distance is \(u = – 30\;{\rm{cm}}\)
The image distance is
\(\frac{1}{v} = \frac{1}{f} – \frac{1}{u} = \frac{1}{{ – 15}} – \frac{1}{{ – 30}} = \frac{{ – 2 + 1}}{{30}} = \frac{{ – 1}}{{30}}\)
\(\therefore v = – 30\;{\rm{cm}}\)
Thus, the image is formed at a distance of \(30\;{\rm{cm}}\) on the left side of the concave mirror.

Example 2: A \(5\;{\rm{cm}}\) tall object is placed at a distance of \(60\;{\rm{cm}}\) from a convex mirror of focal length \(20\;{\rm{cm}}\). Find the position, size and nature of the image formed.
Solution:
Given that,
The focal length of the convex mirror is \(f = 20\;{\rm{cm}}\)
The object distance is \(u = -60\;{\rm{cm}}\)
The object height is \({h_o} = 5\;{\rm{cm}}\)
The image distance is
\(\frac{1}{v} = \frac{1}{f} – \frac{1}{u} = \frac{1}{{20}} – \frac{1}{{ – 60}} = \frac{{3 + 1}}{{60}} = \frac{4}{{60}} = \frac{1}{{15}}\)
\(\therefore v = 15\;{\rm{cm}}\)
The size of the image is:
\({h_i} = – \frac{v}{u} \times {h_0} = – \frac{{15}}{{ – 60}} \times 5 = 1.25\;{\rm{cm}}\)
Thus, a virtual and erect image of size \(1.25\;{\rm{cm}}\) is formed at a distance of \(15\;{\rm{cm}}\) on the right side of the convex mirror.

Applications of Image Formation by Spherical Mirrors

Image formation by spherical mirrors is the deciding factor to determine the usage of a spherical mirror. Some of the applications of spherical mirrors are as under:

1. A concave mirror is used by dentists to get an enlarged image of the teeth for examining it.

1. A concave mirror is used as a reflector in search light, headlights in automobiles, telescopes, solar cookers, etc.
2. A concave mirror is used as a head mirror by ENT specialists.
3. A concave mirror is used as a shaving mirror or make-up mirror as it forms an erect and enlarged image.
4. A convex mirror is used as a rear-view mirror in automobiles.
5. A convex mirror is used as a reflector in street lamps so that the lamp diverges the light over a large area.
6. A convex mirror is used as a vigilance mirror in shopping malls for security purposes as well as outside the streets with sharp turns to enable people to have a wider view of the area, thereby eliminating blind spots.

Hope we could help you understand the concept of image formation by spherical mirrors. We are also positive that this article would have helped you in understanding the types of mirrors, images formed by them, rules as well as sign conventions used, formula pertaining to the image formation by spherical mirrors and their usage in solving numerical problems. 

Next time you get a mirror that is not plane, do check out the image it forms by going near it.

Summary

Image formation by spherical mirrors depends on the curvature of the reflecting surface of the spherical mirror. The mirror formula is given by the relation \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). There are two types of spherical mirrors namely, concave and convex mirrors. Concave Mirrors are defined as the reflecting surface of the mirror is curved inwards. Convex Mirrors are defined as the reflecting surface of the mirror is curved outwards. The spherical mirrors are utilised as a reflector in searchlight, headlights in automobiles, telescopes, solar cookers, etc.

FAQs on Image Formation by Spherical Mirrors

Q.1. How can you distinguish between a concave mirror and a convex mirror just by looking at the image formed by them?
Ans: We can observe the virtual image obtained by the concave mirror and the convex mirror to distinguish between them. If the virtual image obtained from the spherical mirror is enlarged, then it is a concave mirror, whereas if the virtual image obtained is diminished, then it is a convex mirror.

Q.2. Does every part of a spherical mirror, however small it is, form a complete image of an object?
Ans: Yes, every part of a spherical mirror, however small it is, forms a complete image of an object. It is evident from the fact that even if any part of a spherical mirror is painted black, it still will form a full-size image of the object.

Q.3. What are the two types of spherical mirrors?
Ans: The two types of spherical mirrors are concave mirrors and convex mirrors.

Q.4. What is the centre of curvature of a spherical mirror?
Ans: The centre of curvature of a spherical mirror is the centre of the hollow sphere of glass, of which the spherical mirror is a part.

Q.5. What is the pole of a spherical mirror?
Ans: The pole of a spherical mirror is the centre of the spherical mirror. It is also called the vertex of the spherical mirror.

Q.6. What are the two types of images formed by spherical mirrors?
Ans: The two types of images formed by spherical mirrors are real images and virtual images.

Q.7. What is the principal focus of a spherical mirror?
Ans: The principal focus of a spherical mirror is a point on the principal axis of the spherical mirror where the rays of light parallel to the principal axis actually meet or appear to meet after reflection.

Q.8. What is the mirror formula?
Ans: The mirror formula is given by the relation \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

We hope you find this article on Image formation by Spherical Mirrors helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.