Question

A single lens with two convex surfaces made of sapphire with index of refraction \(n=1.77\) has surfaces with radii of curvature \(R_{1}=27.0 \mathrm{~cm}\) and \(R_{2}=-27.0 \mathrm{~cm} .\) What is the focal length of this lens in air? a) \(17.5 \mathrm{~cm}\) c) \(30.7 \mathrm{~cm}\) e) \(54.0 \mathrm{~cm}\) b) \(27.0 \mathrm{~cm}\) d) \(40.8 \mathrm{~cm}\)

Short answer

Answer: The focal length of the lens in air is approximately 17.5 cm.

Step-by-step solution

Write down the lensmaker's formula in air

The lensmaker's formula for a lens in air is given by: \(\frac{1}{f} = (n - 1) \cdot (\frac{1}{R_1} - \frac{1}{R_2})\) Where f is the focal length, n is the index of refraction, and R1 and R2 are the radii of curvature of the two surfaces.

Plug in the given values

We are given the values of index of refraction, n = 1.77 and radii of curvature R1 = 27.0 cm, R2 = -27.0 cm. We will substitute these values in the lensmaker's formula: \(\frac{1}{f} = (1.77 - 1) \cdot (\frac{1}{27} - \frac{1}{-27})\)

Simplify the equation and solve for the focal length

Now, we'll simplify the equation and solve for the focal length (f): \(\frac{1}{f} = 0.77 \cdot (\frac{1}{27} + \frac{1}{27})\) \(\frac{1}{f} = 0.77 \cdot \frac{2}{27}\) \(\frac{1}{f} = \frac{2 \cdot 0.77}{27}\) Multiply both sides by f and divide by the right side to find the focal length: \(f = \frac{27}{2 \cdot 0.77}\) \(f \approx 17.5 cm\) So, the correct answer for the focal length of the lens in air is: a) \(17.5 \mathrm{~cm}\)