Why is -v/u = Magnification of Spherical Mirror? ____Proof of Magnification formula

Magnification of mirror =

Most high schoolers, know this formula byheart.

Seeing this formula might take you back to your memories of all the  panic attacks you had just before the board exams trying to remember this formula amongst the various other concepts learnt in the Light - Reflection and Refraction chapter (Yes, been there  😌)  

This formula was and is  very important, both for the boards and real life applications of optical physics.

So we all know this formula byheart , Which is, we just know that m = -v/u, but do you know why they are equal? 

But, first to all those who are wondering What is Magnification!?

Magnification is the ratio of the height of the image and height of the object, basically telling us how large or small the image is from the object

The common formula this is denoted by is - height of image / height of object 

so this is (somehow) related to -v/u ... In this blog we will dive deeper into this "somehow" 

First let us consider a concave mirror , Let the object be places between Focus and Centre of Curvature

👆 That is the diagram we will get. So

The Object -- between Focus and Centre of Curvature

The Image -- Beyond Centre of Curvature

Image height > Object height 

and a "Very Important" point to remember here is that the Magnification will be "negative" as a real image is formed 

(Remember to use the sign conventions when substituting values of distances and heights) 

Now we need to find ratio between height of object and image. For this :-

Let us take two triangles drawn along the incident and reflected ray. The two triangles are 

Triangle ABO and Triangle CDO

In these triangles :-

1.Angle ABO = Angle CDO = 90 degrees (object is always considered perpendicular to principal axis)

2. Angle AOB = Angle COD  ( angle of incidence = angle of reflection )

Hence by AA Criteria of similarity -- Triangle ABO is similar to CDO 

( Corresponding parts of similar triangles)

and  We know that 

and !

and,

Wohoo! We just proved the magnification formula of a concave mirror. The same procedure has to be followed for all other positions of the object and convex lens too.

Now you just have to remind your future self to click this blog address instead of byhearting the formula before the exam 🙃