Question

Diameter of a plano-convex lens is $6 \,cm$ and thickness at the centre is 3 mm. If speed of light in material of lens is $2 \times 10^8$ m/s, the focal length of the lens is

• A. $15\,cm$
• B. $20 \,cm$
• C. $30\, cm$
• D. $10\, cm$

Answer 1

The Correct Option is C

To find the focal length of the plano-convex lens, we need to use the lens maker's formula, which specifically adapts since one surface of the lens is flat (plano). The formula for focal length \((f)\) of a plano-convex lens is given by:

\(f = \frac{R}{(n-1)}\)

Where:

• \(R\) is the radius of curvature of the lens.
• \(n\) is the refractive index of the lens material.

First, we need to calculate the refractive index \((n)\) using the speed of light in vacuum \((c = 3 \times 10^8 \, \text{m/s})\) and the speed of light in the lens material:

\(n = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5\)

The diameter of the lens is \(6 \, \text{cm}\), so the radius \((a)\) is \(3 \, \text{cm}\). The thickness at the center is \(0.3 \, \text{cm (3 mm)}\). The formula for calculating the radius of curvature \((R)\) of a plano-convex lens is given by:

\(R = \frac{a^2}{2h}\)

Substituting the known values:

\(R = \frac{(3)^2}{2 \times 0.3} = \frac{9}{0.6} = 15 \, \text{cm}\)

Finally, substituting \(R\) and \(n\) into the lens maker's formula:

\(f = \frac{15}{1.5 - 1} = \frac{15}{0.5} = 30 \, \text{cm}\)

Therefore, the focal length of the lens is \(30 \, \text{cm}\), thus the correct answer is \(30 \, \text{cm}\).