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Chapter 2: Problem 142

Find the angular magnification of an image by a magnifying glass of \(f=5.0 \mathrm{cm}\) if the object is placed \(d_{\mathrm{o}}=4.0 \mathrm{cm}\) from the lens and the lens is close to the eye.

Short Answer

Expert verified
The angular magnification (M) of the image formed by the magnifying glass is calculated using the lens formula and the formula for angular magnification. First, we find the image distance (d_i) using the lens formula: \[\frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{4.0}\] which gives us \(d_i = 20.0 \mathrm{cm}\). Next, we calculate the angular magnification using the formula \(M = \frac{D}{d_i} + 1\) with D = 25 cm (near point) and \(d_i = 20 \mathrm{cm}\). This gives us an angular magnification of \(M = 2.25\), meaning the image is magnified by 2.25 times.

Step by step solution

01

Write down the lens formula

The lens formula is given by: \[\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}\] Here, we are given the focal length (f) as 5.0 cm and the object distance (d_o) as 4.0 cm. We need to find the image distance (d_i).
02

Calculate the image distance (d_i)

Using the lens formula, we can substitute the given values for f and d_o to find d_i: \[\frac{1}{5.0}=\frac{1}{4.0}+\frac{1}{d_i}\] Now, solve for d_i: \[\frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{4.0}\] \[\Rightarrow d_i = \frac{1}{\frac{1}{5.0} - \frac{1}{4.0}}\] After calculating this expression, we get: \[d_i = 20.0 \mathrm{cm}\]
03

Write down the formula for angular magnification

The angular magnification (M) is defined as the ratio of the angle subtended by the image (α_i) to the angle subtended by the object (α_o) at the near point (D) of the eye. The formula for angular magnification is: \[M = \frac{\alpha_i}{\alpha_o} = \frac{D}{d_i} + 1\] The near point for a standard eye is generally taken to be 25 cm. So, D = 25 cm, and we have already calculated d_i = 20 cm.
04

Calculate the angular magnification (M)

By substituting the values of D and d_i in the formula for angular magnification, we get: \[M = \frac{25}{20} + 1\] After calculating this expression, we get: \[M = 2.25\] So, the angular magnification of the image formed by the magnifying glass is 2.25 times.

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Most popular questions from this chapter

Show that, for a flat mirror, \(h_{i}=h_{\mathrm{o}},\) given that the image is the same distance behind the mirror as the distance of the object from the mirror.

You can argue that a flat piece of glass, such as in a window, is like a lens with an infinite focal length. If so, where does it form an image? That is, how are \(d_{\mathrm{i}}\) and \(d_{\mathrm{o}}\) related?

Two convex lenses of focal lengths \(20 \mathrm{cm}\) and 10 \(\mathrm{cm}\) are placed \(30 \mathrm{cm}\) apart, with the lens with the longer focal length on the right. An object of height \(2.0 \mathrm{cm}\) is placed midway between them and observed through each lens from the left and from the right. Describe what you will see, such as where the image(s) will appear, whether they will be upright or inverted and their magnifications.

Unreasonable Results Your friends show you an image through a microscope. They tell you that the microscope has an objective with a 0.500 -cm focal length and an eyepiece with a \(5.00-\mathrm{cm}\) focal length. The resulting overall magnification is 250,000 . Are these viable values for a microscope? Unless otherwise stated, the lens-to-retina distance is 2.00 \(\mathrm{cm}\)

If the lens of a person's eye is removed because of cataracts (as has been done since ancient times), why would you expect an eyeglass lens of about 16 D to be prescribed?
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