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Chapter 23: Problem 92

A concave mirror has a radius of curvature of \(14 \mathrm{~cm}\). If a pointlike object is placed \(9.0 \mathrm{~cm}\) away from the mirror on its principal axis, where is the image ?

Short Answer

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Question: Given a concave mirror with a radius of curvature of 14 cm, find the position of the image formed when an object is placed 9 cm away from the mirror. Answer: The image is formed at a distance of 31.5 cm from the mirror on its principal axis.

Step by step solution

01

Find the focal length

We are given the radius of curvature \(R = 14 cm\). Since the focal length (\(f\)) of a concave mirror is half of its radius of curvature, we can calculate the focal length easily: \(f = \frac{R}{2} = \frac{14}{2} = 7 cm\).
02

Substitute values in the mirror formula

With the object distance, \(d_o = 9 cm\), we will now use the mirror formula to solve for the image distance (\(d_i\)): \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) Substitute the values: \(\frac{1}{7} = \frac{1}{9} + \frac{1}{d_i}\)
03

Solve for the image distance

To find the image distance (\(d_i\)), we'll rearrange the formula: \(\frac{1}{d_i} = \frac{1}{7} - \frac{1}{9}\) Now, find a common denominator and simplify: \(\frac{1}{d_i} = \frac{9 - 7}{63} = \frac{2}{63}\) Take the reciprocal of both sides: \(d_i = \frac{1}{\frac{2}{63}} = \frac{63}{2} = 31.5 cm\) The image formed by the concave mirror is located \(31.5 cm\) away from the mirror on its principal axis.

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Most popular questions from this chapter

Show that the deviation angle \(\delta\) for a ray striking a thin converging lens at a distance \(d\) from the principal axis is given by \(\delta=d l f .\) Therefore, a ray is bent through an angle \(\delta\) that is proportional to \(d\) and does not depend on the angle of the incident ray (as long as it is paraxial). [Hint: Look at the figure and use the small-angle approximation \(\sin \theta=\tan \theta \approx \theta(\text { in radians) } .]\)
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Sketch a ray diagram to show that when an object is placed between twice the focal length and the focal length from a converging lens, the image formed is inverted, real, and enlarged in size.
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