Ray Optics - Mirror Formula

In a spherical mirror, the distance of the object from its pole is called the object distance (u).

The triangles B′A′F and PMF are similar because they share common angles. Using the property of similar triangles:

`"B'A'"/"PM"`=`"B'F"/"FP"`

Substituting PM=AB into the equation:

`"B'A'"/"BA"`=`"B'F"/"FP"` .............(i)

∠APB = ∠A'PB

△A'B'P' ∼ △AB'P

`"B'A'"/"BA"` = `"B'P"/"BP"` ............(ii)

By comparing equation (i) and (ii), we get

`"B'F"/"FP"`= `"B'P-FP"/"FP"` = `"B'P"/"BP"`

B'P = -ν (image distance),

FP = -f  (focal length),

BP = -u (object distance).

Substituting these values:

`"-ν+f"/"-f"` = `"-ν"/"-u"`

`"ν-f"/"f"` = `"ν"/"u"`

On further solving, we get

`"1"/"v"`+`"1"/"u"`=`"1"/"f"`

The distance of the image from the pole of the mirror is called the image distance (v).

The distance of the principal focus from the pole is called the focal length (f).

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

Magnification (m) = \[\frac{\text{Height of the image (}h'\text{)}}{\text{Height of the object (}h\text{)}}\] = \[\frac {h'}{h}\]

Magnification in terms of object and image distances:

Magnification (m) = \[\frac {h'}{h}\] = -\[\frac {v}{u}\]