Question

Obtain lens maker’s formula and mention its significance.

Answer 1

1. Let us consider a thin lens made up of a medium of refractive index n₂, is placed in a medium of refractive index n₁. Let R₁ and R₂ be the radii of curvature of two spherical surfaces (1) and (2) respectively and P be the pole.
2. The general equation for the refraction at a spherical surface is given from equation
   `"n"_2/"v" - "n"_1/"u" = ("n"_2 - "n"_1)/"R"_1`
   

   For the refracting surface (1) , the light goes from n₁ to n₂

3. `"n"_2/"v" - "n"_1/"u" = ("n"_2 - "n"_1)/"R"_1`
   

   For the refracting surface (2), the light goes from medium n₂ to n₁.
   `"n"_1/"v" - "n"_2*"v'" = ("n"_1 - "n"_2)/"R"_2`

4. By adding
   `"n"_1/"v" - "n"_1/"u" = ("n"_2 - "n"_1)(1/"R"_1 - 1/"R"_2)`
   

   Further simplifying and rearranging,

   `1/"v" - 1/"u" = (("n"_2 - "n"_1)/"n"_1)(1/"R"_1 - 1/"R"_2)`
   
   `1/"v" - 1/"u" = ["n"_2/"n"_1 - 1] (1/"R"_1 - 1/"R"_2)`
5. If the object is at infinity, the image is formed at the focus of the lens.
   Thus, for u = ∞, υ = f.
   Then the equation becomes,
   `1/"v" - 1/"u" = ["n"_2/"n"_1 - 1] (1/"R"_1 - 1/"R"_2)`
   
   `1/"f" = ("n"_2/"n"_1 - 1) (1/"R"_1 - 1/"R"_2)`
6. If the refractive index of the lensis n₂ and it is placed in air, then n₂ = n and n₁ = 1. So the equation becomes,
   `1/"f" = ("n - 1")(1/"R"_1 - 1/"R"_2)`
   

   The above equation is called the lens maker’s formula,

7. `1/"v" - 1/"u" = 1/"f"`
   

   This is called lens equation.