Mirror Formula

Establish the relation amongst object distance, image distance and focal length in a concave mirror.

Answer

Consider about a concave mirror MPN. PFC is the the the principal axis. P is the pole of the mirror, F is the main focus of the mirror and C is the centre of curvature.

AB is an object placed on the principal axis beyond the centre of curvature C.

A'B' is the image of the object AB on the principal axis between C and F due to reflection of light through the concave mirror MPN.

The ray diagram of image formation is given above the figure.

Mathematical Calculation.

From figure it is clear that

PF = f (Focal length)

PB = u (Object distance)

PB' = v (Image distance)

PC = 2f (Radius of curvature)

Consider about the similar ∆ABC and ∆A'B'C.

AB/A'B' = BC/B'C

From figure it is clear that

BC = PB - PC = u - 2f.

B'C = PC - PB' = 2f - v

PF = f

B'F = PB' - PF = v - f

Therefore,

AB/A'B' = (u - 2f)/(2f - v) …… (i)

When AM is very very close to the principal axis.

AB = MP

Consider about the similar ∆MPF and A'B'F.

MP/A'B' = PF/B'F

Therefore,

AB/A'B' = f/(v - f) ……… (ii)

Now, from (i) and (ii)

(u - 2f)/(2f - v) = f/(v - f)

or, (u - 2f)(v - f) = f(2f - v)

or, uv - uf - 2vf + 2f² = 2f² - vf

or, uv = 2f² - vf + uf + 2vf - 2f²

or, uv = uf + vf

Now, dividing both sides by uvf.

uv/uvf = uf/uvf + vf/uvf

1/f = 1/v + 1/u

This is exact relation amongst u, v and f.

Lens Formula

Consider about a convex lens. O is the optical centre. AB is an object placed on the principle axis. The image A'B' is formed on the principle axis due to refraction of light. 

Mathematical Calculation 

Consider about the ∆ABO and ∆A'B'O.

Both triangles are similar.

AB/A'B' = OB/OB'   ....... (I)

Again, consider about the ∆MOF₂ and A'B'F₂

Both triangles are similar.

MO/A'B' = OF₂/B'F₂ ....... (ii)

From figure it is clear that

AB = MO

OB = -u

OB' = v

OF₂ = f

Form of equation no (ii) will be

AB/A'B' = OF₂/B'F₂ ...... (iii)

Now, from equation no (i) and (iii)

OB/OB' = OF₂/B'F₂

Now, putting the values of        OB = -u OB' = v OF₂ = f B'F₂ = v - f

-u/v = f/v - f

or, -uv + uf = vf

or, uf - vf = uv

Now, dividing both sides by uvf.

1/v - 1/u = 1/f

This is the lens formula.