Question

Derive an expression for the refractive index of a glass prism.
OR
Derive the lens maker formula for thin lenses.

Hint:

For a convex lens, the radii of curvature \( R_1 \) and \( R_2 \) are positive, and for a concave lens, \( R_1 \) is negative and \( R_2 \) is positive.

Answer 1

a. Expression for the Refractive Index of a Glass Prism:
The refractive index \( \mu \) of a glass prism can be derived using the angle of deviation \( \delta \) and the prism angle \( A \).
The formula for the refractive index \( \mu \) of the prism is given as:
\[ \mu = \frac{\sin \left( \frac{A + \delta}{2} \right)}{\sin \left( \frac{A}{2} \right)} \] Where:
- \( A \) is the angle of the prism,
- \( \delta \) is the angle of deviation for a ray passing through the prism.
This formula relates the refractive index of the prism material to the angle of the prism and the deviation of the light passing through it.
The derivation of this formula involves considering the geometry of the light passing through the prism, including the angle of incidence, refraction, and the relationship between these angles for different media.
b. Lens Maker Formula for Thin Lenses:
The lens maker's formula gives the relationship between the focal length \( f \) of a lens and the radii of curvature \( R_1 \) and \( R_2 \) of the lens surfaces. The formula is given as:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] Where:
- \( f \) is the focal length of the lens,
- \( \mu \) is the refractive index of the lens material relative to the surrounding medium,
- \( R_1 \) is the radius of curvature of the first surface of the lens,
- \( R_2 \) is the radius of curvature of the second surface of the lens.
This formula is used to calculate the focal length of a lens based on its curvature and the refractive index of the material from which it is made.
The derivation of the lens maker's formula involves using the refraction laws at each surface of the lens and applying them to the overall geometry of the lens.