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Chapter 24: Problem 41

Repeat Problem \(40(\mathrm{c})\) using a different eyepiece that gives an angular magnification of 5.00 for a final image at the viewer's near point \((25.0 \mathrm{cm})\) instead of at infinity.

Short Answer

Expert verified
Answer: The length of the microscope is 5.47 cm.

Step by step solution

01

Find linear magnification of eyepiece

We will find the linear magnification of the eyepiece, using the total angular magnification and the linear magnification of the objective lens. Formula: \(M_{total} = M_{objective} \times M_{eyepiece}\) Solve for \(M_{eyepiece}\): \(M_{eyepiece} = \frac{M_{total}}{M_{objective}}\)
02

Calculate linear magnification of eyepiece

Now substitute the values for \(M_{total}\) and \(M_{objective}\) in the above equation to find \(M_{eyepiece}\): \(M_{eyepiece} = \frac{5.00}{40} = 0.125\)
03

Find the image distance of the eyepiece

Since we are given that angular magnification is for a final image at viewer's near point, we can find the distance of an object placed \(25\text{ cm}\) away from the eyepiece and is magnified by \(0.125\) times. Formula: \(d_{image} = d_{near} \times M_{eyepiece}\)
04

Calculate the image distance of the eyepiece

Substitute the values for \(d_{near}\) and \(M_{eyepiece}\) in the above equation to find \(d_{image}\): \(d_{image} = 25\text{ cm} \times 0.125 = 3.125\text{ cm}\)
05

Find the object distance of the eyepiece

Now, we will use the lens formula on the eyepiece to determine the object distance of the eyepiece. Lens formula: \(\frac{1}{f_{eyepiece}} = \frac{1}{d_{object}} + \frac{1}{d_{image}}\) Solve for \(d_{object}\): \(d_{object} = \frac{f_{eyepiece} \times d_{image}}{f_{eyepiece} - d_{image}}\)
06

Calculate the object distance of the eyepiece

Substitute the values for \(f_{eyepiece}\) and \(d_{image}\) in the above equation to find \(d_{object}\): \(d_{object} = \frac{10\text{ cm} \times 3.125\text{ cm}}{10\text{ cm} - 3.125\text{ cm}} = 4.47\text{ cm}\)
07

Find the length of the microscope

Now that we have the object distance for the eyepiece and the focal length of the objective lens, we can add them to find the length of the microscope. Formula: \(L = f_{objective} + d_{object}\)
08

Calculate the length of the microscope

Substitute the values for \(f_{objective}\) and \(d_{object}\) in the above equation to find \(L\): \(L = 1.00\text{ cm} + 4.47\text{ cm} = 5.47\text{ cm}\) The length of the microscope is \(5.47\text{ cm}\).

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Most popular questions from this chapter

You have a set of converging lenses with focal lengths $1.00 \mathrm{cm}, 10.0 \mathrm{cm}, 50.0 \mathrm{cm},\( and \)80.0 \mathrm{cm} .$ (a) Which two lenses would you select to make a telescope with the largest magnifying power? What is the angular magnification of the telescope when viewing a distant object? (b) Which lens is used as objective and which as eyepiece? (c) What should be the distance between the objective and eyepiece?
Veronique is nearsighted; she cannot sce clearly anything more than $6.00 \mathrm{m}$ away without her contacts. One day she doesn't wear her contacts; rather, she wears an old pair of glasses prescribed when she could see clearly up to \(8.00 \mathrm{m}\) away. Assume the glasses are \(2.0 \mathrm{cm}\) from her eyes. What is the greatest distance an object can be placed so that she can see it clearly with these glasses?
Callum is examining a square stamp of side \(3.00 \mathrm{cm}\) with a magnifying glass of refractive power \(+40.0 \mathrm{D}\). The magnifier forms an image of the stamp at a distance of \(25.0 \mathrm{cm} .\) Assume that Callum's eye is close to the magnifying glass. (a) What is the distance between the stamp and the magnifier? (b) What is the angular magnification? (c) How large is the image formed by the magnifier?
A photographer wishes to take a photograph of the Eiffel Tower $(300 \mathrm{m}\( tall ) from across the Seine River, a distance of \)300 \mathrm{m}$ from the tower. What focal length lens should she use to get an image that is 20 mm high on the film?
A simple magnifier gives the maximum angular magnification when it forms a virtual image at the near point of the eye instead of at infinity. For simplicity, assume that the magnifier is right up against the eye, so that distances from the magnifier are approximately the same as distances from the eye. (a) For a magnifier with focal length \(f,\) find the object distance \(p\) such that the image is formed at the near point, a distance \(N\) from the lens. (b) Show that the angular size of this image as seen by the eye is $$ \theta=\frac{h(N+f)}{N f} $$ where \(h\) is the height of the object. [Hint: Refer to Fig. 24.15 .1 (c) Now find the angular magnification and compare it to the angular magnification when the virtual image is at infinity.
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