Question

Arrive at lens equation from lens maker’s formula.

Answer 1

From refraction through a double convex lens, the relation between the object distance u, image distance v₁ and radius of curvature R₁ as

`mu_2/"v"_1 - mu_1/"u" = (mu_2 - mu_1)/"R"_1`  ....(1)

The relation between the object distance image distance v₁ and radius of curvature R₂ can be

`mu_1/"v" - mu_2/"v"_1 = (mu_1 - mu_2)/"R"_2`  ....(2)

Adding equation (1) and (2)

`mu_1/"v" - mu_1/"u" = (mu_2 - mu_1) [1/"R"_1 - 1/"R"_2]`

`1/"v" - 1/"u" = ((mu_2 - mu_1)/mu_1) [1/"R"_1 - 1/"R"_2]`

If the object is placed at infinity (µ = ∞), the image will be formed at the focus. i.e.v = ƒ

`1/"f" = ((mu_2 - mu_1)/mu_1) [1/"R"_1 - 1/"R"_2]`

This is len’s maker’s formula. When the lens is placed in air µ₁ = 1 and µ₂ = µ.
Equation (4) becomes,

`1/"f" = (mu - 1) [1/"R"_1 - 1/"R"_2]`

From equation (3) and (4), we have `1/"v" - 1/"u" = 1/"f"`

This is the len’s equation.