Derivation of lens Formula with Solved Example

Derivation of lens formula

 Lenses are used widely for various purposes and making various optical instruments. They have different focal length, material, radii and refractive index. Therefore to make lenses for different purposes, lens makers use lens formula so that the focal length is appropriate for the purpose. 

Meaning of lens formula: 

Lens formula refers to a formula which shows the relationship between the focal length, distance of the image, and distance of the object. All these three factors play an important role in making the appropriate lens. The relation can be given by following formula: 

1/v -1/u = 1/f

Here, v represents the distance of the image from the lens. F denotes the focal length and u means the distance of the object from the lens. 

 Lens maker formula: 

Lens maker formula shows the interconnection between the focal length of the lens, refractive index of the material and radii of the lens. It is only because of these three factors that the lens makers are able to make appropriate length with a specific power. 

The formula used is: 1/f= ( μ- 1)(1/R1 – 1/R2)

F represents the focal length (which is the total measurement of divergence or convergence of the light). 

Assumptions before deriving lens maker formula: 

1. While deriving the lens maker formula, the first assumption is that there are two lenses with radii R1 and R2.
2. These lenses are presumed to have refractive indexes of n1 & n2. 

Therefore we first derive the formula for first surface which is given by: n2/v1– n1/u = n2-n1/ R1

Similarly, the formula for second surface will be: n1/v- n2/v1 = n1-n2/ R2

Then add first and second formula, n1/v – n1/u = (n2-n1) (1/R1 – 1/R2)

Thus 1/v – 1/u = (n2/n1 – 1)(1/R1 – 1/R2)

Since f= 1/v – 1/u, 

1/f= (n2/n1 – 1)(1/R1 – 1/R2)

Where, n2/n1 = μ

In this way, we derive the lens maker formula which is 1/f= ( μ- 1)(1/R1 – 1/R2)

Solved examples

Example: How to derive lens maker formula for convex lens? 

Let’s suppose that a convex lens has F as principal focus and f is the focal length. Let a perpendicular AB fall on the principal focus. The image of AB will be A’B’ 

Therefore, A’B’/AB= OB’/OB and A’B’/OC = FB’/OF

Since OC =AB because both triangles are congruent, A’B’/AB= FB’/OF = OB’-OF/ OF

Now, we know that OB=u distance of object, OB’= v distance of image and OF = f focal length. 

Therefore, V/-u = v-f/f 

If we divide both sides with uvf, we get the lens formula: 1/f= 1/v-1/u.