Question

Answer the following question.

Derive lens makers’ equation. Why is it called so? Under which conditions focal length f and radii of curvature R are numerically equal for a lens?

Answer 1

1. Consider a lens of radii of curvature R₁ and R₂ kept in a medium such that the refractive index of the material of the lens with respect to the medium is denoted as n.
2. Assuming the lens to be thin, P is the common pole for both the surfaces. O is a point object on the principal axis at a distance u from P.
3. The refracting surface facing the object is considered as the first refracting surface with radii R₁.
4. In the absence of the second refracting surface, the paraxial ray OA deviates towards normal and would intersect the axis at I₁. PI₁ = v₁ is the image distance for intermediate image I₁.
5. For a curved surface,
   `(("n"_2-"n"_1))/"R"="n"_2/"v"-"n"_1/"u"`
   Thus, in this case,
   ∴ n₂ = n, n₁ = 1, R = R₁, u = u and v = v_{1
   ∴} `(("n"-1))/"R"_1="n"/"v"_1=1/"u"` ....(1)
6. Before reaching I₁, the incident rays (AB and OP) strike the second refracting surface. In this case, image I₁ acts as a virtual object for the second surface.
7. For second refracting surface,
   n₂ = 1, n₁ = n, R = R₂, u = v₁ and PI = v
   ∴ `((1-"n"))/"R"_2=1/"v"-"n"/"v"_1-(("n"-1))/"R"_2=1/"v"-"n"/"v"_1` .....(2)
8. Adding equations (1) and (2),
   `("n"-1)(1/"R"_1-1/"R"_2)=1/"v"-1/"u"`
   For object at infinity, image is formed at focus, i.e., for u = ∞, v = f. Substituting this in above equation,
   `1/"f"=("n"-1)(1/"R"_1-1/"R"_2)` .....(3)
   This equation is known as the lens makers’ formula.
9. Since the equation can be used to calculate the radii of curvature for the lens, it is called the lens makers’ equation.
10. The numeric value of focal length f and radius of curvature R is the same under the following two conditions:
    Case I: For a thin, symmetric, and double convex lens made of glass (n = 1.5), R₁ is positive and R₂ is negative but, |R₁| = |R₂|.
    In this case,
    `1/"f"=(1.5-1)(1/"R"-1/(-"R"))=0.5(2/"R")=1/"R"`
    ∴ f = R
    Case II: Similarly, for a thin, symmetric and double concave lens made of glass (n = 1.5), R₁ is negative and R₂ is positive but, |R₁| = |R₂|.
    In this case,
    `1/"f"=(1.5-1)(1/(-"R")-1/"R")=0.5(-2/"R")=-1/"R"`
    ∴ f = – R or |f| = |R|