Question

Consider a convex lens made of material of refractive index \( n = \frac{3}{2} \) and radius of curvature \( R \). What is the relation between focal length and radius?

• A. \( f = \frac{R}{2} \)
• B. \( f = \frac{R}{4} \)
• C. \( f = \frac{R}{3} \)
• D. \( f = \frac{2R}{3} \)

Hint:

For a convex lens, the focal length is related to the radius of curvature by the equation \( f = \frac{R}{2} \), which is derived from the lens maker's formula.

Answer 1

The Correct Option is A

Step 1: Use the lens maker's formula.
The lens maker's formula is given by: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] For a convex lens, \( R_1 = R \) (radius of curvature of the first surface) and \( R_2 = -R \) (radius of curvature of the second surface). Substituting these values into the formula: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) \] \[ \frac{1}{f} = (n - 1) \left( \frac{2}{R} \right) \] Substitute \( n = \frac{3}{2} \): \[ \frac{1}{f} = \left( \frac{3}{2} - 1 \right) \frac{2}{R} = \frac{1}{2} \frac{2}{R} = \frac{1}{R} \] Therefore: \[ f = \frac{R}{2} \]
Final Answer: \( f = \frac{R}{2} \).