Proof of the "Magnification Formula"

The magnification formula states that:

m= H*h^-1 = -v*u^-1

A note: I’m using the negative 1 exponent to indicate that the certain quantity is in the denominators place. 

Here, 

m stands for magnification, H stands for Image height, h stands for object height, v stands for image distance and u stands for object distance. 

In order to see that this relation holds true, I’m going to use the concept of similar triangles and a ray diagram of a concave mirror. Also, a pre-requisite for this proof is the knowledge of sign-convention.

Firstly, let us describe the essential parts of the figure. The circle represents the hollow sphere from which the concave mirror was made. B is the pole of the mirror, C is the Focus and A is the Center of Curvature. DE is the object, DB is the object distance (u), JI is the image and BJ is the image distance (v).

We know,

Angle JBI = Angle DBE (Law of reflection; a light ray incident on the pole gets reflected at the same angle)

Angle JIB = Angle DEB (Both objects are 90 degrees with the principal axis)

Thus,

JIB ~ DEB (by AA criterion)

So,

JI*DE^-1 = JB*DB^-1 (From similar triangles)

Applying sign convention,

-JI*DE^-1 = -v*-u ^-1

So,

JI*DE^-1 = -v*u ^-1 

Thus,

H*h^-1 = -v*u ^-1 

The Magnification formula is verified.