Proof of Mirror Formula

Proof of Mirror Formula

For any spherical Mirror :-

1. Distance from pole(p) to object(O) is called as object distance and it is denoted by u.
2. Distance from pole (p) to Image(I) is called as image distance and it is denoted by v.
3. Distance from pole (p) to focus of mirror (F) is called as focal length and is denoted by f.
4. Distance from pole (p) to Centre of curvature(C) is called as  radius of curvature and it is donoted by R.

AP=u

A'P=v

PC=R

A'C=R-v

AC=u-R

In similar triangle ABC and  A'B'C

 

∠BAC=∠B'A'C (Right Angle)

 

∠BCA=∠B'CA'(Opposite Angle)

 

so,∠ABC=∠A'B'C

 

Hence, ABC〜A'B'C

 

know, 

 

AB/A'B' =AC/A'C

 

AB/A'B' =(u-R)/(R-v) .........(i)

 

In similar triangle ABP and  A'B'P

 

∠BAP=∠B'A'P(Right Angle)

 

 ∠APB=∠A'PB' (Low of Reflection)

 

so,∠ABP=∠A'B'P

 

Hence, ABP〜A'B'P

 

know, 

 AB/A'B' =AP/A'P

 

 AB/A'B' =u/v ...................(ii)

 

from (i) & (ii)

 

(u-R)/(R-v) =u/v

 

u (R-v) = v(u-R)

 

uR-uv=vu-vR

 

uR+vR=2uv

 

Divide both side by uv

 

uR/uv +vR/uv =2uv/uv

 

R(1/v + 1/u)=2

 

1/v + 1/u=2/R ...........(iii)

 

we know that R=2f  ,put in Eqn(iii)

 

1/v + 1/u=2/2f 

 

1/v + 1/u =1/f  Proved