Derivation of Mirror Formula

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What is Derivation of Mirror Formula?

Ibn al-Haitham was a physicist who had described the theory of vision. Scientists used to call him the father of optics.

Mirror Equation Derivation is very common among the students. Many questions are asked from this section of various boards as well as entrance tests. Mirror Formula Proof is very easy. It can be explained as the relation between the distance of an object, the distance of an image, and the focal length of the mirror.


Facts on Spherical Mirror

Some facts are there that you must know about the spherical mirror:

The object distance(u) is the length between the object and the pole of the mirror.

Image distance(v) is the length between the image and the pole of the mirror. 

Focal Length(f) is the distance between the principal focus and the pole of the mirror.

\[\frac{1}{f}\] = \[\frac{1}{V}\] + \[\frac{1}{U}\]  

Where, 

f = focal length of the mirror

V = the distance of the image

U = the distance of the object


Derive Mirror Formula for Convex

From the below diagram, you can get to know the Derivation of Mirror Formula for Convex Mirror.

[Image will be Uploaded Soon]

The above picture shows the following value:

 -u = PB

+V = PB’

+b = PF

+R = PC

In triangle ABC and A’B’C

\[\frac{AB}{A’B}\] = \[\frac{CB}{CB’}\] → 1

In triangle ABP and A/B/P

\[\frac{AB}{A’B}\] = \[\frac{PB}{PB’}\] → 2

From 1 and 2, we get

\[\frac{CB}{CB’}\] = \[\frac{PB}{PB’}\] …. [3]

We also know CB = PB + PC and CB’ = PC – PB’

Putting the above values in eq [3]

\[\frac{PB+PC}{PC-PB’}\] = \[\frac{PB}{PB’}\] = \[\frac{-u+R}{R-v}\] = \[\frac{-u}{v}\]

⇒ -uv + vR = -uR + uv

⇒ uR + vR = 2uv

After dividing, uvR The final form of the equation is

\[\frac{1}{V}\] + \[\frac{1}{U}\] = \[\frac{1}{f}\]  


Derive Mirror Formula for Concave Mirror

The Derivation of Mirror Formula for Concave Mirror is shown hereunder:

[Image will be Uploaded Soon]

From the above image, we get

\[\frac{B’A’}{PM}\] = \[\frac{B’F}{FP}\] or \[\frac{B’A’}{BA}\] = \[\frac{B’F}{FP}\] (∵ PM = AB)....1

 \[\frac{B’A’}{BA}\] = \[\frac{B’P}{BP}\] ( ∵A P B angle = A’ P B’ angle)..... 2  

Equating 1 and 2, we will get,

\[\frac{B’F}{FP}\] = \[\frac{B’P-FP}{FP}\] = \[\frac{B’P}{BP}\]

Also, B'P = -v, FP = -f, BP = -u

\[\frac{-v+f}{-f}\] = \[\frac{-v}{-u}\]

or, \[\frac{v-f}{f}\] =\[\frac{v}{u}\]

or, \[\frac{1}{u}\] + \[\frac{1}{v}\] = \[\frac{1}{f}\]  


Derivation of Mirror Formula for Convex Lens

[Image will be Uploaded Soon]

In the above image, you can notice two similar triangles i.e. △ABO and △A’B’O.

We can write 

\[\frac{A’B’}{AB}\] = \[\frac{OB’}{OB}\]   [1]

Also, △A’B’F and △OCF are similar.

We can write a relation as

\[\frac{A’B’}{OC}\] = \[\frac{FB’}{OF}\]

However, OC = OB

Then, \[\frac{A’B’}{AB}\] = \[\frac{FB’}{OF}\]   [2]

Equating both the equation 1 and 2, we get

\[\frac{OB’}{OB}\] = \[\frac{FB’}{OF}\] = \[\frac{OB’-OF}{OF}\]  

If we conduct some sign convention, we will find

  • OB=-u,  

  • OB’=v 

  • and OF=f

\[\frac{v}{-u}\] = \[\frac{v-f}{f}\]   

vf = -uv + uf or uv = uf - vf 

Dividing uvf into both sides, we get

\[\frac{uv}{uvf}\] = \[\frac{uf}{uvf}\] - \[\frac{vf}{uvf}\]

⇒ \[\frac{1}{f}\] = \[\frac{1}{v}\] - \[\frac{1}{u}\] [ This is the convex lens formula]


Conclusion

This article will help you to know about the mirror formula and its relevant usages. With the use of optics, we can calculate and analyze the behaviour of light and image formation. Thus, this chapter is very helpful.

FAQ (Frequently Asked Questions)

Q1. Calculate the Position of the Image, if the Location of the Bus is 8 Meters from a Convex Mirror. The Radius of Curvature of the Convex Mirror is 5 Meters.

Ans: 

Given Data:

The radius of curvature (R)= +5.00 m

Object distance(u) = -5.00 m

We need to find the image distance(v) =?

We know that f = R/2 = 8/2 = 4 m

The formula of the mirror is 1/u + 1/v = 1/f. [1]

After rearranging the above equation, we get

1/v = 1/f - 1/u .. [2]

Substituting the data in the above equation 2

1/v = 1/f - 1/u = ¼ - 1/(-5) = 9/20 

V = 20/9 = 2.22 meters

Therefore, the image is formed 2.22 meters behind the mirror.

Q2. Define Optics. What are the Types of Optics?

Ans: The learning of the wave properties of light is known as Optics. 3 categories of optics are there which can be grouped into:

  1. Interference

  2. Polarization

  3. Diffraction

Q3. Mention the Categories of the Wavefront.

Ans: Wavefronts can be of 3 types as per the source of light; they are:

  • Spherical wavefront

  • Plane wavefront

  • Cylindrical wavefront