derive lens maker’s formula

The Lensmaker's Formula is a fundamental equation in optics that relates the focal length of a lens to the radii of curvature of its two surfaces and the refractive index of the material of the lens. Here's how to derive it:

1. Consider a thin lens with two surfaces. The first surface has a radius of curvature R1 and the second surface has a radius of curvature R2. The lens is made of a material with refractive index n and is placed in a medium with refractive index n1.

2. Let's consider a ray of light parallel to the principal axis incident on the lens. After refraction at the first surface, the ray converges to a point on the principal axis. Using the lens formula for refraction at the first surface, we have:

   1/n1 - 1/n = (n-n1)/R1

3. The ray then refracts at the second surface and finally converges to a point F on the principal axis. Using the lens formula for refraction at the second surface, we have:

   1/n - 1/n1 = (n1-n)/R2

4. Adding these two equations, we get:

   1/n1 - 1/n + 1/n - 1/n1 = (n-n1)/R1 + (n1-n)/R2

5. Simplifying, we get:

   2/n1 - 2/n = (n-n1)/R1 + (n1-n)/R2

6. Rearranging terms, we get:

   (n- n1)/n1 = (n-n1)(1/R1 - 1/R2)

7. Simplifying, we get:

   1/f = (n/n1 - 1)(1/R1 - 1/R2)

This is the Lensmaker's Formula, where f is the focal length of the lens. It shows that the focal length of a lens depends on the refractive indices of the lens material and the surrounding medium, and the radii of curvature of the two surfaces of the lens.