Final answer:
The Lens Maker's Formula relates the focal length of a thin lens to the radii of curvature of its surfaces and the refractive index of the lens material. For a double convex lens, the derived formula is 1/f = (n - 1)(1/R1 - 1/R2), assuming the lens is thin and placed in air.
Step-by-step explanation:
Derivation of the Lens Maker's Formula
The Lens Maker's Formula is a fundamental equation in optics that relates the focal length (f) of a thin lens to the radii of curvature of its two surfaces (R1 and R2) and the refractive index (n) of the lens material. For a thin double convex lens, the assumptions made are that the thickness of the lens is much less than the radii of curvature, and the lens is placed in air where the refractive index (n1) is 1.0.
Let's derive the expression for the Lens Maker's Formula step by step:
- Consider a lens with radii of curvature R1 and R2, and refractive index n.
- Using ray tracing, determine the image formed by the first surface. Let the object distance be do and the image distance be di.
- Under the thin-lens approximation, apply the refraction formula n/do - n'/di = (n' - n)/R1, where n' is the refractive index on the other side of the first surface.
- For the second surface, use the image formed by the first surface as the object. Apply the refraction formula again with the values for the second surface to find the final image position q.
- Combine the equations from both surfaces to eliminate the intermediate variables and obtain the final formula for the thin lens: 1/f = (n - 1)(1/R1 - 1/R2).
Thus, for a lens in air, the Lens Maker's Formula simplifies to: 1/f = (n - 1)(1/R1 - 1/R2), where f is the focal length, R1 is the radius of curvature of the first surface, R2 is the radius of curvature of the second surface, and n is the refractive index of the lens material.
This formula is widely used for designing lenses and analyzing their optical properties.