Question

Derive the Lens Maker’s Formula for a thin convex lens.

Hint:

Lens Maker’s Formula: \[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] Focal length increases if curvature decreases or refractive index decreases.

Answer 1

Introduction
The lens maker’s formula gives the relationship between the focal length of a lens, the refractive index of the lens material, and the radii of curvature of its two surfaces. It is used in the design and manufacture of lenses for optical instruments such as microscopes, cameras, and telescopes. The derivation is based on the refraction of light at spherical surfaces.

Refraction at the First Surface
Consider a thin convex lens of refractive index \( \mu \) placed in air. Let the radii of curvature of the two spherical surfaces be \(R_1\) and \(R_2\). A light ray from an object at distance \(u\) from the first surface forms an intermediate image at distance \(v_1\) inside the lens.

For refraction at a spherical surface, the formula is
\[ \frac{\mu_2}{v_1} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R_1} \] For the first surface, the light travels from air to glass. Therefore,
\(\mu_1 = 1\) and \(\mu_2 = \mu\).
Substituting these values:
\[ \frac{\mu}{v_1} - \frac{1}{u} = \frac{\mu - 1}{R_1} \] This equation represents refraction at the first surface.

Refraction at the Second Surface
The intermediate image formed by the first surface acts as an object for the second surface. Let the final image be formed at distance \(v\). For refraction at the second surface:
\[ \frac{\mu_3}{v} - \frac{\mu_2}{v_1} = \frac{\mu_3 - \mu_2}{R_2} \] Here light travels from glass to air, so
\(\mu_2 = \mu\) and \(\mu_3 = 1\).
Substituting these values:
\[ \frac{1}{v} - \frac{\mu}{v_1} = \frac{1 - \mu}{R_2} \] This equation represents refraction at the second surface.

Combining the Two Equations
Adding the two equations eliminates \(v_1\):
\[ \frac{1}{v} - \frac{1}{u} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] This is the general formula for a thin lens.

Lens Maker’s Formula
For an object at infinity, the image is formed at the focal point of the lens. Therefore \(u = \infty\) and \(v = f\). Substituting these values:
\[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] This equation is known as the lens maker’s formula.

Conclusion
The focal length of a thin convex lens depends on the refractive index of the lens material and the radii of curvature of its two surfaces. The lens maker’s formula is given by
\[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] This formula helps in determining the focal length and designing lenses for various optical instruments.