Question

A concave lens has two surfaces of equal radii 30 cm and refractive index 1.5. Find its focal length.

Hint:

The focal length of a concave lens is always negative, and the lens maker's formula can be used to determine it based on the radii and refractive index.

Answer 1

Step 1: Understanding the Lens Maker's Formula.
For a concave 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,
- \( n = 1.5 \) is the refractive index,
- \( R_1 = 30 \, \text{cm} \) is the radius of curvature of the first surface,
- \( R_2 = -30 \, \text{cm} \) is the radius of curvature of the second surface (negative for concav.
Step 2: Calculation.
Substituting the values into the formula: \[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{30} - \frac{1}{-30} \right) = 0.5 \times \left( \frac{2}{30} \right) \] \[ \frac{1}{f} = \frac{1}{30} \Rightarrow f = -30 \, \text{cm} \]
Final Answer:
The focal length of the concave lens is \( \boxed{-30 \, \text{cm}} \).