Question

The distance from the lens system (cornea + lens) of a particular eye to the retina is \(1.75 \mathrm{cm} .\) What is the focal length of the lens system when the eye produces a clear image of an object \(25.0 \mathrm{cm}\) away?

Short answer

Answer: The focal length is approximately \(1.636 \mathrm{cm}\).

Step-by-step solution

Understand the Lens Formula

Using the lens formula, we can determine the focal length of the lens system. The formula is: \[\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\] where: - \(f\) is the focal length of the lens system, - \(d_o\) is the distance from the object to the lens system, and - \(d_i\) is the distance from the lens system to the image (or the distance from the lens system to the retina). Given that the image is formed exactly on the retina, we can use the distance from the lens system to the retina as \(d_i\).

Substitute the given values

We know that the distance from the object to the eye (\(d_o\)) is \(25.0 \mathrm{cm}\), and the distance from the lens system to the retina (\(d_i\)) is \(1.75 \mathrm{cm}\). Substitute these values into the lens formula: \[\frac{1}{f} = \frac{1}{25.0 \mathrm{cm}} + \frac{1}{1.75 \mathrm{cm}}\]

Calculate the focal length

To find the focal length \(f\), first compute the sum of the fractions and then find the inverse: \[\frac{1}{f} = 0.04 \mathrm{cm^{-1}}+0.5714 \mathrm{cm^{-1}}\] \[\frac{1}{f} = 0.6114 \mathrm{cm^{-1}}\] \[f = \frac{1}{0.6114 \mathrm{cm^{-1}}}\]

Find the final answer

Calculate the focal length of the lens system: \[f \approx 1.636 \mathrm{cm}\] The focal length of the lens system when the eye produces a clear image of an object \(25.0 \mathrm{cm}\) away is approximately \(1.636 \mathrm{cm}\).