Proof of Lens Law

Following graphic illustrates a simple lens model:

where,

h= height of the object

h’= height of the object projected in an image

G and C = focal points

f= focal distance

u= Distance between the object and the focal point

O= Centre of the lens

v= Distance between the centre of the lens and image plane

Assumptions

1. Lens is very thin
2. Optical axis is perpendicular to image plane

To Prove

1/f=1/u + 1/v

Proof

In ΔAHO, tanα=h/u

In ΔEDO, tanα=h’/v

∴ tanα=h/u=h’/v

⇒ h’/h=v/u ————- (1)

In ΔBOC, tanβ=h/f

In ΔEDC, tanβ=h’/(v-f)

∴ tanβ=h’/(v-f)=h/f

⇒ h’/h=(v-f)/f ———- (2)

Equating (1) and (2),

v/u=(v-f)/f

⇒ v/u=v/f -1

Dividing both sides by v,

1/u=1/f – 1/v

or,

1/f=1/u + 1/v

Hence proved.

Notes

h’/h is often referred to as the Magnification factor M. If M is negative, the projected image is real but inverted.