Question

A double convex lens has focal length 25 cm. The radius of curvature of one of the surfaces is double of the other. Find the radii if the refractive index of the material of the lens is 1.5 :

• A. 100 cm, 50 cm
• B. 25 cm, 50 cm
• C. 18.75 cm, 37.5 cm
• D. 50 cm, 100 cm

Answer 1

The Correct Option is C

 The problem involves calculating the radii of curvature for a double convex lens given the focal length and refractive index. Let's solve it step by step using the Lens Maker's formula:

The Lens Maker's formula is given by:

\(\frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)

Where:

• \(f\) is the focal length of the lens,
• \(\mu\) is the refractive index of the lens material,
• \(R_1\) and \(R_2\) are the radii of curvature of the two surfaces of the lens.

We are given:

• \(f = 25 \, \text{cm}\)
• \(\mu = 1.5\)
• \(R_2 = 2R_1\) (since one radius is double the other)

Now substituting the values into the Lens Maker's formula:

\(\frac{1}{25} = (1.5 - 1) \left( \frac{1}{R_1} - \frac{1}{2R_1} \right)\)

Simplify the equation:

\(\frac{1}{25} = 0.5 \left( \frac{2 - 1}{2R_1} \right)\)

\(\frac{1}{25} = \frac{0.5}{2R_1}\)

Solve for \(R_1\):

\(2R_1 = 0.5 \times 25\)
\(2R_1 = 12.5\)
\(R_1 = 12.5 \div 2\)
\(R_1 = 18.75 \, \text{cm}\)

Since \(R_2 = 2R_1\), we find:

\(R_2 = 2 \times 18.75 = 37.5 \, \text{cm}\)

Therefore, the radii of curvature are 18.75 cm and 37.5 cm. The correct option is:

18.75 cm, 37.5 cm

 

This solution is consistent with the given answer, confirming the calculations are accurate.