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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

The eyepiece of a microscope has a focal length of \(1.25 \mathrm{cm}\) and the objective lens focal length is \(1.44 \mathrm{cm} .\) (a) If the tube length is \(18.0 \mathrm{cm},\) what is the angular magnification of the microscope? (b) What objective focal length would be required to double this magnification?
Two lenses, separated by a distance of \(21.0 \mathrm{cm},\) are used in combination. The first lens has a focal length of \(+30.0 \mathrm{cm} ;\) the second has a focal length of \(-15.0 \mathrm{cm} .\) An object, $2.0 \mathrm{mm}\( long, is placed \)1.8 \mathrm{cm}$ before the first lens. (a) What are the intermediate and final image distances relative to the corresponding lenses? (b) What is the total magnification? (c) What is the height of the final image?
Show that if two thin lenses are close together \((s,\) the distance between the lenses, is negligibly small), the two lenses can be replaced by a single equivalent lens with focal length \(f_{\mathrm{eq}} .\) Find the value of \(f_{\mathrm{eq}}\) in terms of \(f_{1}\) and \(f_{2}.\)

The uncorrected far point of Colin's eye is \(2.0 \mathrm{m} .\) What refractive power contact lens enables him to clearly distinguish objects at large distances?
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.
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