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Chapter 24: Problem 26

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

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

Step by step solution

01

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 \(d_i\).
02

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: \[\frac{1}{f} = \frac{1}{25.0 \mathrm{cm}} + \frac{1}{1.75 \mathrm{cm}}\]
03

Calculate the focal length

To find the focal length \(f\), first compute the sum of the fractions and then find the inverse: \[\frac{1}{f} = 0.04 \mathrm{cm^{-1}}+0.5714 \mathrm{cm^{-1}}\] \[\frac{1}{f} = 0.6114 \mathrm{cm^{-1}}\] \[f = \frac{1}{0.6114 \mathrm{cm^{-1}}}\]
04

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 \(1.636 \mathrm{cm}\).

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Most popular questions from this chapter

Unless the problem states otherwise, assume that the distance from the comea- lens system to the retina is \(2.0 \mathrm{cm}\) and the normal near point is \(25 \mathrm{cm}.\) If the distance from the lens system (cornea + lens) to the retina is $2.00 \mathrm{cm},$ show that the focal length of the lens system must vary between \(1.85 \mathrm{cm}\) and \(2.00 \mathrm{cm}\) to see objects from $25.0 \mathrm{cm}$ to infinity.
Esperanza uses a 35 -mm camera with a standard lens of focal length $50.0 \mathrm{mm}\( to take a photo of her son Carlos, who is \)1.2 \mathrm{m}$ tall and standing \(3.0 \mathrm{m}\) away. (a) What must be the distance between the lens and the film to get a properly focused picture? (b) What is the magnification of the image? (c) What is the height of the image of Carlos on the film?
Suppose the distance from the lens system of the eye (cornea + lens) to the retina is \(18 \mathrm{mm}\). (a) What must the power of the lens be when looking at distant objects? (b) What must the power of the lens be when looking at an object \(20.0 \mathrm{cm}\) from the eye? (c) Suppose that the eye is farsighted; the person cannot see clearly objects that are closer than $1.0 \mathrm{m}$. Find the power of the contact lens you would prescribe so that objects as close as \(20.0 \mathrm{cm}\) can be seen clearly.
Keesha is looking at a beetle with a magnifying glass. She wants to see an upright, enlarged image at a distance of \(25 \mathrm{cm} .\) The focal length of the magnifying glass is \(+5.0 \mathrm{cm} .\) Assume that Keesha's eye is close to the magnifying glass. (a) What should be the distance between the magnifying glass and the beetle? (b) What is the angular magnification? (tutorial: magnifying glass II).
A camera lens has a fixed focal length of magnitude \(50.0 \mathrm{mm} .\) The camera is focused on a 1.0 -m-tall child who is standing \(3.0 \mathrm{m}\) from the lens. (a) Should the image formed be real or virtual? Why? (b) Is the lens converging or diverging? Why? (c) What is the distance from the lens to the film? (d) How tall is the image on the film? (e) To focus the camera, the lens is moved away from or closer to the film. What is the total distance the lens must be able to move if the camera can take clear pictures of objects at distances anywhere from \(1.00 \mathrm{m}\) to infinity?
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