Lens Formula

There are two types of lenses, convex and concave. Lenses play an important role in the study of optics. 

The lens formula is used for both convex and concave lenses. The lens Formula is used for finding the image distance, determining the type of image formed and the focal length of the lens. 

The Lens Formula 

(1 ÷ v ) – (1 ÷ u) = 1 ÷ f 

Derivation Of The Lens Formula:

The above image is of a convex lens with an optical centre of O. F is the point of principal focus and ‘f’ is the point of focal length. An object of length ‘AB’ is held perpendicular to the principal axis beyond the focal length of the lens. An inverted magnified image A’B’ is formed as shown. 

In the above image, we can see that triangles ABO and A’B’O are similar

Hence, 

A’B’ ÷ AB = OB’ ÷ OB 

Similarly, triangles A’B’F and OCF are similar 

Hence, 

A’B’ ÷ OC = FB’ ÷ OF 

But, OC = AB 

Therefore, 

A’B’ ÷ AB = FB’ ÷ OF 

By equating the above equations, we get: 

OB’ ÷ OB = FB’ ÷ OF 

= (OB’ – OF) ÷ OF 

By substituting the signs we get, 

OB = -u 

OB’ = v 

OF = f 

v ÷ (-u) = (v-f) ÷ f 

vf = (-uv) + uf = uf – vf 

By dividing both sides by uvf – 

(uv ÷ uvf) = (uf ÷ uvf) = (vf ÷ uvf) 

Therefore, 

(1 ÷ f)  = (1 ÷ v) – (1 ÷ u) 

Solved Example

Question- the length of a concave lens is 15 cm. Determine the distance of the object to be placed, so that the image formed is 10cm away from the lens. 

Answer- 

f = (-15) cm 

v = (-10) cm 

Let the object distance be ‘u’ 

The lens formula:

(1 ÷ f) = (1 ÷ v) – (1 ÷ u) 

(1 ÷ f) – (1 ÷ v) = (-1 ÷ u) 

(-1 ÷ u) = (1 ÷ f) – (1 ÷ v) 

(1 ÷ u) = (1 ÷ v) – (1 ÷ f) 

(1 ÷ u) = {1 ÷ (-10)} – {1 ÷ (-15)} 

(1 ÷ u) = (-1 ÷ 10) + (1 ÷ 15) 

(1 ÷ u) = (-3 + 2) ÷ 30 

(1 ÷ u) = (-1 ÷ 30) 

u = (-30) cm 

The object is placed at a distance of 30 cm. The negative sign is an indication of the object’s location. It is placed in front of the lens.