No of Images Formed by Two Plane Mirrors

 

 

No of Images Formed by Two Plane Mirrors depends on the position of mirrors with respect to each others and also on the position of object between the mirrors. Let us discuses different positions of mirrors:-

Case (I) – Image Formation by Parallel Plane Mirrors (θ=180º)

(No of Images Formed by Two Plane Mirrors)

If two plane mirrors are placed parallel to each other and an object is placed anywhere between them, then:-

• The number of images formed is infinite.
• The images are created by the successive reflections of light between the two mirrors.
• The images are formed at regular intervals, and their brightness decreases with each reflection.

It’s important to note that the images formed by parallel mirrors are virtual images, meaning they cannot be projected onto a screen. These virtual images exist only in the apparent location created by the reflection of light rays.

Special case :- If two plane mirrors are placed parallel to each other and the object is placed vertically above the ends of both of the mirrors as shown in figure below then number of images formed will be 2.

Case (II) – Image formation by Perpendicular Plane Mirrors (θ=90º)

(No of Images Formed by Two Plane Mirrors)

 

Here 3 images are formed :

1. Virtual image I₁ of real object O
2. Virtual image I₂ of real object O
3. Virtual image I₃ of image I₁ (image I₁ acts as virtual object for I₃) or image I₂ (image I₂ acts as virtual object for I₃)

Special case :- If two plane mirrors are placed perpendicular to each other but are of insufficient length and the object is placed vertically above the ends of both of the mirrors as shown in figure below then number of images formed will be 2.

 

Case (III) – Image formation by Inclined Plane Mirrors (For any angle θ)

(No of Images Formed by Two Plane Mirrors)

 

Consider two plane mirrors M₁ and M₂ inclined at an angle (θ₁+θ₂) and an object O placed between them making an angle θ₁ with M₁ and θ₂ with M₂.

Here in triangles AI₁B and AOB :-

AB = AB (common)

OB (distance of object) = I₁B (distance of image)

∠B = ∠B = 90°

Hence from side-angle-side theorem, triangles AI₁B and AOB are congruent. Hence

AO = AI₁

Similarly, in triangles AI₂C and AOC :-

AC = AC (common)

OC (distance of object) = I₂C (distance of image)

∠C = ∠C = 90°

So from side-angle-side theorem, triangles AI₂C and AOC are congruent. Hence

AO = AI₂

In the same way,

AO = AI₁ = AI₂ = AI₃ = AI₄ = AI₅ = AI₆

Here I₁ , I₂ , I₃ , I₄ , I₅ , I₆ are nothing but the multiple images of object O formed by the mirrors M₁ and M₂ and we can see that they all lie on a circle of radius AO.

I₁ = image of object O formed by mirror M₁ at an angle θ₁, anticlockwise from mirror M₁

I₂ = image of object O formed by mirror M₂ at an angle θ₂, clockwise from mirror M₂

I₃ = image of image I₁ formed by mirror M₂ at an angle (2θ₁ + θ₂), clockwise from mirror M₂

I₄ = image of image I₂ formed by mirror M₁ at an angle (2θ₂ + θ₁), anticlockwise from mirror M₁

I₅ = image of image I₄ formed by mirror M₂ at an angle (2θ₁ + 3θ₂), clockwise from mirror M₂

I₆ = image of image I₃ formed by mirror M₁ at an angle (2θ₂ + 3θ₁), anticlockwise from mirror M₁

Now here is a circular method to find out total number of images formed by two inclined plane mirrors :-

 

Example 1.

Two plane mirrors are inclined at an angle of 72º. Find the number of images of a point object placed between.

Solution.

Here

Now

If the object is placed symmetrically between the mirrors, then no. of images = 5 – 1 = 4 and

If the object is placed asymmetrically between the mirrors, then no. of images = 5

 

Example 2.

Two plane mirrors are at 45º to each other. If an object is placed between them, then the number of images will be

(a)   5                                         (b)   9                                         (c)   7                                         (d)   8

Solution.

As 8 is even number, hence no. of images = 8 – 1 = 7

 

Example 3.

Two plane mirrors are inclined at an angle of (a) θ = 112.5º (b) θ = 75º . Find the number of images of a point object placed between.

Solution.

(a)

Now 3 is an odd number, so :-

If object is symmetrically placed between the mirrors then number of images = 3.2 – 1 = 2.2

Number of complete images = 2

If object is asymmetrically placed between the mirrors then number of images = 3.2

Number of complete images = 3

(b)

Now 4 is an even number so no matter how the object is placed(symmetrically or asymmetrically) between the mirrors,

Number of images = 4.8 – 1 = 3.8

Number of complete images = 3