Image formation by inclined plane mirrors, practice problems, FAQs

When you stand in front of a mirror there is one image formed in the mirror. Now do an experiment at your home, take two mirrors and place them at an angle. You will find that there are a number of images formed in the mirrors. As you change the angle between them, the number of images also changes. Let’s discuss this!

Table of content

Image formation by inclined mirrors
Formula to find the number of images
Overlapping of images
Practice problem
FAQs

Image formation by inclined mirrors

Let us take two mirrors placed at angle θ. The object is making an angle of 𝛼 with the first mirror, M₁, and an angle of 𝛽 with the second mirror.

The images, I₁ and I₂, are formed at angle 𝛼 and 𝛽 by mirror M₁ and _M2 respectively,

Now, the image formed by mirror M₁ will act as an object for mirror M₂ and vice versa. This process will go on till any one of the angles becomes 180^o or greater.

The method to obtain the angles for the images formed is as follows:

It is the same as the multiple reflection method for the linear position of an object. Here, angular position is used instead of linear position.

By using this method, we can find the angular position of the images formed by the both mirrors.

Formula to find the number of images

If two plane mirrors, M₁ and M₂, are inclined to each other at an angle θ and if an object is placed between the two mirrors, then to find the number of images formed by the system, we have to check two things: 36⁰0 and whether the position of the object is symmetrical or asymmetrical to the system. Based on these two criteria, we have two cases:

Case 1: If is even

Number of images formed, (n) = -1

If 3600 turns out to be an even integer, the position of the object does not matter.

Case 2: If is odd

If the value is odd, then we have to check if the object is placed symmetrically or asymmetrically to the system.

If the object is placed symmetrically to the system, then the number of images formed by this system, (n) = -1
If the object is placed asymmetrically to the system, then the number of images formed by this system, (n) =

Overlapping of images

Let mirrors M₁ and M₂ and object O be placed as shown in the following figure. Initially, few images formed by the mirrors will not overlap but after some images, the images formed will overlap each other.

The first image is formed by mirrors M₁ and M₂ being at angles α and β. After some images are formed, suppose that the images are formed at angles θ1 and θ2 with mirrors M₁ and M₂ , respectively.

If the sum of the angle between the mirrors and the angles made by the images with their respective mirrors is 360° or greater than 360°, i.e., θ + θ₁ + θ₂ = or more than 360°, then those images overlap each other.
If the angle made by any image with their respective mirrors is 180° or greater than 180°, then that image overlaps with another image.

Practice problem

Q 1. Two plane mirrors are inclined at an angle of 45°. Find the number of images formed of a point object placed symmetrically between them.

a. 7
b. 5
c. 8
d. 6

Answer: Given,

Angle between the mirrors, θ = 45o

Here, , which is even.

Therefore, the number of images formed, (n) = -1

⇒ n = 8 - 1 = 7

Thus, option (A) is the correct answer.

Q 2. If an object is placed asymmetrically between two plane mirrors inclined at an angle of 60o, then find the total number of images formed.

a. 5
b. 4
c. 2
d. Infinite

Answer: Given,

θ = 60^o

Therefore,=, which is even.

Therefore, the number of images formed,

⇒ n = 6 - 1 = 5

Thus, option (A) is the correct answer.

Q 3. Three plane mirrors of equal length are kept as shown. A point object O is kept at the center of the triangle. How many images will be formed?

a. 10
b. 13
c. 12
d. Infinite

Answer: To solve this kind of a problem, break the system into three systems of two mirrors each.

For each pair of mirrors, as all three systems are the same, we get the following:

It is even. Therefore, the number of images formed,

⇒ n = 6 - 1 = 5

Total number of images formed = (3 × n) - Number of mirrors

⇒ (3 × 5) - 3 = 12

Thus, option (C) is the correct answer.

Q 4. An object is placed between two plane mirrors such that it is making angles of 20° and 30° with mirrors M₁ and M₂. Find the number of images formed.

a. 6
b. 7
c. 8
d. 9

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

At 180o, the image formed will overlap with other images. So, we do not count that image.

Number of images formed (n) = 7

Thus, option (B) is the correct answer

FAQs

Q 1. What happens to the number of images when we decrease the angle between mirrors?
Answer: As the angle between the mirrors decreases, the number of images will increase.

Q 2.What is the substitute of the mirrors in past times?

Answer: We used water surfaces for reflection in the past.

Q 3. What are the various uses of a plane mirror?
Answer: Plane mirrors are used in Periscopes and Kaleidoscopes, Automobiles,Torchlights,Shaving Mirrors, Solar Cookers etc.

Q 4. How many images will form when mirrors are at 1200?
Answer: Here is even.

So, the number of images formed by the mirror,