Question

The radii of curvature for a thin convex lens are 10 cm and 15 cm respectively. The focal length of the lens is 12 cm. The refractive index of the lens material is

• A. 1.2
• B. 1.4
• C. 1.5
• D. 1.8

Hint:

Use the lensmaker's formula to relate the focal length, radii of curvature, and refractive index of the lens material.

Answer 1

The Correct Option is C

To find the refractive index of the lens material, we can use the Lens Maker's Formula, which relates the focal length of a lens to the refractive index of the material and the radii of curvature:

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

where:

• \(f\)is the focal length of the lens.
• \(n\)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, with the convention that the radius is positive if the surface is convex and negative if it is concave with respect to the refracted light.

Given:

• \(R_1 = +10 \text{ cm}\)(since it's convex)
• \(R_2 = -15 \text{ cm}\)(since it's on the opposite side and concave w.r.t. incident light)
• \(f = 12 \text{ cm}\)

Now, substitute the values into the lens maker's formula:

\(\frac{1}{12} = \left(n - 1 \right) \left(\frac{1}{10} - \frac{1}{-15}\right)\)

Simplifying the terms inside the parentheses:

\(\frac{1}{10} - \left(-\frac{1}{15}\right) = \frac{1}{10} + \frac{1}{15}\)

To combine these fractions, find a common denominator (30):

\(\frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}\)

Substitute back into the formula:

\(\frac{1}{12} = \left(n - 1\right) \frac{1}{6}\)

Multiplying both sides by 6 to solve for \(n - 1\):

\(\frac{6}{12} = n - 1\)

\(\frac{1}{2} = n - 1\)

Add 1 to both sides:

\(n = \frac{1}{2} + 1 = \frac{3}{2} = 1.5\)

Therefore, the refractive index of the lens material is 1.5. This matches with Option 3.

Answer 2

\( \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \) \( \frac{1}{12} = (\mu - 1) \left( \frac{1}{10} - \frac{1}{-15} \right) \) \( \frac{1}{12} = (\mu - 1) \left( \frac{3+2}{30} \right) \) \( \mu = \frac{3}{2} \)