Question

Derive the lens maker’s formula in case of a double convex lens. State the assumptions made and the conventions of signs used.

Answer 1

Hint: The basic use of lens maker’s formula is to make lenses of desired focal length. Here, we will use the expression for the refraction of light on a spherical surface to find the required lens maker formula.

Complete answer:
While deriving the formula, various sign conventions are used. Sign convention is basically a set of rules to set signs for image distance, object distance, focal length, etc for mathematical analysis of image formation. The following are the sign conventions:
(A) The object is always kept on the left side of the lens.
(B) All the distances are measured from the optical centre.
(C) Distances measured in the direction of the incident ray are positive and the distances measured in the direction opposite to that of the incident rays are negative.
(D) Distances measured along the y-axis above the principal axis are positive and those measured along the y-axis below the principal axis are negative.
The following assumptions are taken for the derivation of lens maker formula:
(A) The object must be point sized.
(B) The object must be placed at the principal axis
(C) Aperture of the lens which is used should be very small.
(D) The thickness of the lens should be less.
Let us consider a thin lens with two refracting surfaces having radii of curvatures \[{R_1}\] and $ {R_2} $ . Let us also consider the refractive index of the surrounding area be $ {n_1} $ and that of the lens be $ {n_2} $ .
Using the formula for refraction at a single spherical surface we can say that,
For the first surface,
 $ \dfrac{{{n_2}}}{{{v_1}}} - \dfrac{{{n_1}}}{u} = \dfrac{{{n_2} - {n_1}}}{{{R_1}}}.......(1) $
Similarly, for the second surface,
 $ \dfrac{{{n_1}}}{v} - \dfrac{{{n_2}}}{{{v_1}}} = \dfrac{{{n_1} - {n_2}}}{{{R_2}}}.......(2) $
Now, by adding the above two equations, we get,
 $ \dfrac{{{n_1}}}{v} - \dfrac{{{n_1}}}{u} = ({n_2} - {n_1})\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right] $
On further simplifying, we get,
 $ \dfrac{1}{v} - \dfrac{1}{u} = \left( {\dfrac{{{n_2}}}{{{n_1}}} - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right] $
When $ u = \infty $ and $ v = f $
 $ \dfrac{1}{f} = \left( {\dfrac{{{n_2}}}{{{n_1}}} - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right] $
 $ \dfrac{1}{f} = \left( {\mu - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right] $
Where, $ \mu $ is the refractive index of the medium.
This is the expression of the lens maker’s formula.

Note:
In any derivation of Geometrical optics, we use the sign convention twice: once while deriving it and next while using it for general cases. But while deriving lens makers formula, we don't consider negative and positive values of the radius of curvature while solving for both spherical surfaces.