Question

Derive Lens Maker's formula for a thin lens. Mention the effect of the refractive index and radius of curvature of the curved surfaces on the focal length of the lens.

Hint:

For a converging lens (double convex), \( R_1 \) is positive, and for a diverging lens (double concave), \( R_1 \) is negative.

Answer 1

The Lens Maker's formula relates the focal length \(f\) of a lens to the refractive index \(n\) of the lens material and the radii of curvature \(R_1\) and \(R_2\) of the two curved surfaces of the lens.
For a thin lens, the Lens Maker's formula is given by: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] where:
- \(f\) is the focal length of the lens,
- \(n\) is the refractive index of the material of the lens,
- \(R_1\) is the radius of curvature of the first surface (convex or concave),
- \(R_2\) is the radius of curvature of the second surface (convex or concave).
Effect of Refractive Index (\(n\)):
- As the refractive index \(n\) increases, the focal length of the lens decreases. A higher refractive index material bends light more sharply, thus focusing light at a shorter distance.
Effect of Radii of Curvature (\(R_1\) and \(R_2\)):
- The radii of curvature of the lens surfaces also affect the focal length. For a convex surface, \(R_1\) is positive, and for a concave surface, \(R_2\) is negative. The curvature of the surfaces influences the converging or diverging behavior of the lens.