Question

The radius of curvature of each surface of a biconvex lens is 20 cm and the refractive index of the material of the lens is 1.5. The focal length of the lens is

• A. 20 m
• B. 1/20 m
• C. 20 cm
• D. 1/20 cm

Hint:

For a symmetric biconvex lens (\(R_1 = R, R_2 = -R\)) made of a material with \(\mu=1.5\), the focal length is simply equal to the radius of curvature (\(f=R\)). This is a useful shortcut to remember for quick checks.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem requires finding the focal length of a biconvex lens given its radii of curvature and refractive index. This can be solved using the Lens Maker's Formula.
Step 2: Key Formula or Approach:
The Lens Maker's Formula is: \[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] where: \(f\) = focal length of the lens \(\mu\) = refractive index of the lens material with respect to the surrounding medium (air, in this case) \(R_1\) = radius of curvature of the first surface (where light enters) \(R_2\) = radius of curvature of the second surface
Step 3: Detailed Explanation:
Given data: - Type of lens: Biconvex - Refractive index, \(\mu = 1.5\) - Radius of curvature of each surface = 20 cm. We must apply the Cartesian sign convention. Assume light travels from left to right. - For the first surface (left), it is convex towards the incident light. Its center of curvature is on the right side. Thus, \(R_1 = +20\) cm. - For the second surface (right), it is also convex, but its center of curvature is on the left side. Thus, \(R_2 = -20\) cm. Now, substitute these values into the Lens Maker's Formula: \[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{+20} - \frac{1}{-20} \right) \] \[ \frac{1}{f} = (0.5) \left( \frac{1}{20} + \frac{1}{20} \right) \] \[ \frac{1}{f} = (0.5) \left( \frac{2}{20} \right) \] \[ \frac{1}{f} = \left(\frac{1}{2}\right) \left( \frac{1}{10} \right) \] \[ \frac{1}{f} = \frac{1}{20} \] Therefore, the focal length is: \[ f = 20 \text{ cm} \] Step 4: Final Answer:
The focal length of the lens is 20 cm. This corresponds to option (C).