Image formation due to parallel and perpendicular mirrors, rotation of plane mirror, practice problems, FAQs

Definitely you visit the Barber shop once in a month or two for your hair cut. The shop has mirrors on opposite walls. When you see in the mirror you will find that an infinite number of images form in both the mirrors. This is due to multiple reflections by a combination of plane mirrors. If we put the mirror at different angles and position then we will find the different number of images. Let’s see different cases!

Table of content

• Image formation by two parallel plane mirrors
• Image formation by two parallel plane mirrors but not on the same level
• Image formation by two perpendicular plane mirrors
• Image formation by two perpendicular plane mirrors with their ends being separated
• Rotation of mirror
• Practice problem
• FAQs

Image formation by two parallel plane mirrors

Let us consider that two plane mirrors, M1 and M2, are placed in front of each other with a distance of 2x between them, and that an object, O, is placed at the middle of both the mirrors as shown in the following figure:

Now, both the mirrors will form the image of the object at distance x behind the mirrors as I1 and I2, respectively, and the images, I1 and I2, will act as objects for M2 and M1, respectively. The image of I1  will be formed at a distance of 3x behind the second mirror and the image of I2 will be formed at a distance of 3x behind the first mirror.

In such a way, an infinite number of images can be formed.

Now to find the image distance, let us consider the distance between the mirrors be l and let the respective distance of the object from mirrors M1 and M2 be l1 and l2 as shown in the following figure:

The image distances can be calculated by multiple reflections method as:

From the given flow chart, the following can be deduced.

• The first image of mirror M1 is formed at a distance of l1 behind mirror M1, and this image acts as an object for M2 at a distance of l1 + l from M2. Then, its image will be formed at the distance of l1 + l behind mirror M2, and this image will act as an object for M1 at a distance of l1 + 2l from M1. Again, its image will be formed at the distance of l1 + 2l behind M1 and so on.
• The first image of M2 is formed at a distance of l2 behind M2, and this image acts as an object for M1 at a distance of l2 + l from M1. Then, its image will be formed at l2 + l behind M1, and this image will act as an object for M2 at a distance of l2 + 2l from M2 . Again, its image will be formed at l2 + 2l behind M2 and so on.

Image formation by two parallel plane mirrors but not on the same level

In this case, after the first reflection from both the mirrors, the reflected rays go away from the mirrors and do not meet the mirrors again. Due to this, no more reflections take place. Hence, no more images are formed.

Hence, the number of images formed (n) = 2
The two images are I₁ and I₂ by mirrors M₁ and M₂, respectively. 

Image formation by two perpendicular plane mirrors

In this case, the mirrors are placed perpendicular to each other. Images formed in this case are shown as follows:

• Mirror M₁ will form the image of the object at a distance of x behind mirror M₁.
• Mirror M₂ will form the image of the object at a distance of y behind mirror M₂.
• By the rays that get reflected from both the mirrors, they will form overlapping images as shown in the following figure

Hence, the total number of images formed (n) = 3

If the object is placed at an angle of 45^o with respect to the mirror combination

By placing object O at an angle of 45^o, it makes the object symmetrical to both mirrors. Therefore, all the images are formed at the same distances from the joining of the mirrors and they form images as shown in the given figure. The figure below shows a reflection at 45^o

Image formation by two perpendicular plane mirrors with their ends being separated

In this case, after the first reflection from both the mirrors, the reflected rays go away from the mirrors and do not meet the mirrors again. Due to this, no more reflections take place. Hence, no more images are formed. 

Hence, the number of images formed (n) = 2
The two images are I₁ and I₁ by mirrors M₁ and M₁, respectively.

Rotation of mirror

Let us consider that a ray of light falls on a plane mirror at an angle of i and gets reflected back at an angle of r . From the law of reflection, i = r

Now, let us rotate the mirror by angle θ by keeping the incident ray fixed as shown in the figure:

If we rotate the mirror by angle θ for a fixed incident ray A, then the normal of the mirror also rotates through the same angle θ in the same direction, and it is marked as N2 . Now, the incident ray makes an angle of i +θ with N2 and its corresponding reflected ray R2 rotates through an angle of r from its initial path R1 and makes an angle of (r+r)- with N2 as shown in the given figure.

From laws of reflection, we get the following:

i+=(r+r)-

Since the angle of incidence is equal to the angle of reflection, i = r

⇒ Δr = 2θ

Hence, for a fixed incident ray, if the plane mirror is rotated by an angle of θ, then the reflected ray gets rotated by 2θ in the same direction.

Practice problem

 Q 1. Two plane mirrors, A and B, are parallel to each other and spaced 20 cm apart. An object is kept in between them at 15 cm from A. Out of the following, at which point does the image not form in mirror A (distances are measured from mirror A)?

a. 15 cm
b. 25 cm
c. 45 cm
d. 55 cm

Answer: Given,

Distance between the two mirrors, l = 20 cm

Distance between the object and mirror A, l1 = 15 cm

Distance between the object and mirror B, l2 = 5 cm

By using the multiple reflections method,

As we know the multiple reflections method, by which we can see the images formed by mirror A are at 15 cm, 25 cm and 55 cm. Therefore from mirror A there is no image at 45 cm.

Thus, option (C) is the correct answer.

Q 2. An object is placed between two plane mirrors such that it is making angles of 30° and 60° with mirrors M1 and M2. Find the number of images formed.

a. 3
b. 4
c. 5
d. 6

Answer: By using the multiple reflections method, we get the following:

As 150° + 120° + 90° = 360°,

Images 3 and 4 will overlap.

The number of images formed (n) = 4 - 1 = 3

Thus, option (A) is the correct answer.

Q 3. The figure shows a point object placed between two parallel mirrors. Its distance from M₁ is 2 cm and that from M₂ is 8 cm. Find the distance of images from the two mirrors by considering reflection on mirror M₁ first.

Answer: First of all, it should be noted that for this case, due to multiple reflections, the number of images formed is infinite (when the mirrors are infinitely long) and the image formed by the first mirror will act as an object for the second mirror and vice versa.
The question can be visualized as follows:

In this figure, I₁₂ is the position of the image formed by the mirror M₂ after reflection from the mirror M₁ and I₁₂₁ is the position of the image formed by the mirror M₁ after consecutive reflection from the mirror M₁ and M₂.

If we consider that the first reflection on the M₁ mirror, then for that case, the object distance is, ON = 2 cm and hence, the first image distance is, NI₁ = 2 cm

It is also given that ON’ = 8 cm and the image I1 serves as the object for mirror M₂

Therefore, the distance of the object for M₂ is,

 N’I₁ = (OI₁ + ON’) = (ON + NI₁ + ON’) = (2 + 2 + 8) = 12 cm

Hence, the first image will form at I12 and the distance is, N’I₁₂ = 12 cm 

Now, this first image of M2 will serve as the second object for M1 and the distance will be,

NI₁₂ = NN’ + N’I₁₂ = 10 + 12 = 22 cm

Hence, the image for this object will form at I₁₂₁ and that distance will be, NI₁₂₁ = 22 cm

This image of M1 will serve as the second object for M2 and this process continues.

Q 4. A ray of light from a source L occurs normally on a plane mirror fixed at a certain distance of x from the source. When the mirror is rotated through a small angle of θ, the spot of the light is found to move through a distance y above the source. Which of the following represents angle θ?

Answer: As we know, for a fixed incident ray, if the plane mirror is rotated by an angle of θ, then the reflected ray gets rotated by 2θ in the same direction.

In triangle POL,

When is very small, tan can be written as

Thus, option (D) is the correct answer.

FAQs

Q 1. How many times can an image be reflected between two facing mirrors?
Answer: Theoretically it can reflect infinite times, but in general there would be ten to twenty images because some light will not be reflected back and will be absorbed in each reflection.

Q 2.How many images will form due to 3 perpendicular mirrors?
Answer: Due to any two mirrors there will be 3 images. Now for the remaining third mirror there will be 4 objects (1 real and 3 virtual), so 4 images will be there. Hence a total of 3+4=7 images will form.

Q 3. Who was the inventor of the mirror?
Answer: German chemist Justus von Liebig first developed the mirror.

Q 4. Mirrors are made of?
Answer: Mirrors are made of glass with a flat surface and have a reflective covering over it.