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Chapter 25: Problem 23

A mirror produces an image that is located \(34.0 \mathrm{cm}\) behind the mirror when the object is located \(7.50 \mathrm{cm}\) in front of the mirror. What is the focal length of the mirror, and is the mirror concave or convex?

Short Answer

Expert verified
The focal length is approximately \(9.62\, \mathrm{cm}\), and the mirror is convex.

Step by step solution

01

Identify Given Variables

From the problem, we know the object distance (\(d_o\)) and image distance (\(d_i\)). The object is located \(7.50\, \mathrm{cm}\) in front of the mirror \((d_o = +7.50\, \mathrm{cm})\) and the image is \(34.0\, \mathrm{cm}\) behind the mirror \((d_i = -34.0\, \mathrm{cm}\) because images behind the mirror are considered negative in sign convention).
02

Use Mirror Formula

The mirror formula is \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length to be found. Substitute \(d_o = +7.50\, \mathrm{cm}\) and \(d_i = -34.0\, \mathrm{cm}\) into the formula to calculate \(f\).
03

Calculate the Focal Length

Substitute the known values into the formula: \[\frac{1}{f} = \frac{1}{7.50} + \frac{1}{-34.0}.\]Calculate \[\frac{1}{f} = \frac{1}{7.50} - \frac{1}{34.0} = 0.1333 - 0.0294 = 0.1039.\]Then, \(f = \frac{1}{0.1039} \approx 9.62\, \mathrm{cm}\).
04

Determine Type of Mirror

Since the image is virtual and located behind the mirror, the mirror is a convex mirror, as only convex mirrors form virtual images such that the image distance is negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mirror Formula
The mirror formula is a crucial equation in optics. It relates the object distance \(d_o\), the image distance \(d_i\), and the focal length \(f\) of a spherical mirror. The formula is given by:
  • \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
This simple yet powerful formula allows us to calculate the focal length of a mirror if we know the object and image distances.

To find the focal length, you take the reciprocals of the object and image distances, sum them up, and then take the reciprocal of that sum. Remember:
  • Distances are positive in front of the mirror and negative if behind it (according to the sign convention for spherical mirrors).
  • The focal length \(f\) provides information about the focal point of the mirror, which is where light rays parallel to the principal axis either converge (concave mirror) or appear to diverge (convex mirror).
The mirror formula helps you solve many problems related to image formation by spherical mirrors.
Concave Mirror
A concave mirror, also known as a converging mirror, has a reflective surface that curves inward, resembling the interior of a sphere.

Here are key characteristics of a concave mirror:
  • It can form real or virtual images depending on the position of the object relative to the focal point.
  • When an object is placed beyond the focal length of a concave mirror, it produces a real, inverted image. This happens because the reflected rays converge at the image location.
  • If the object is located at the focal point, no image is formed because the reflected rays are parallel.
  • For objects placed between the focal point and the mirror, a virtual, upright, and magnified image appears behind the mirror.
The focal length of a concave mirror is positive according to the sign convention, indicating a real focal point where light converges.

Concave mirrors are widely used in devices like telescopes, flashlights, and makeup mirrors, where magnification or projection is required.
Convex Mirror
A convex mirror is a type of diverging mirror with a reflective surface that bulges outward. It is often used for viewing large areas due to its ability to spread reflected light.

Key features of a convex mirror include:
  • Only virtual, upright, and smaller images are formed, independent of the object's position relative to the mirror.
  • This is due to the light rays appearing to diverge from a point behind the mirror (the virtual focal point).
  • The image formed is always diminished and located closer to the mirror compared to the object.
The focal length of a convex mirror is negative, which aligns with the concept of a virtual focal point.

Convex mirrors are commonly used in car rear-view mirrors, in hallways of buildings, and in security applications due to their wide field of view and ability to provide smaller-sized images.

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Most popular questions from this chapter

On the \(+y\) axis a laser is located at \(y=+3.0 \mathrm{cm} .\) The coordinates of a small target are \(x=+9.0 \mathrm{cm}\) and \(y=+6.0 \mathrm{cm} .\) The \(+x\) axis represents the edge-on view of a plane mirror. At what point on the \(+x\) axis should the laser be aimed in order for the laser light to hit the target after reflection?

An object is placed in front of a convex mirror, and the size of the image is one-fourth that of the object. What is the ratio \(d_{\mathrm{o}} / f\) of the object distance to the focal length of the mirror?

An object is placed in front of a convex mirror. Draw the convex mirror (radius of curvature \(=15 \mathrm{cm}\) ) to scale, and place the object \(25 \mathrm{cm}\) in front of it. Make the object height \(4 \mathrm{cm} .\) Using a ray diagram, locate the image and measure its height. Now move the object closer to the mirror, so the object distance is \(5 \mathrm{cm}\). Again, locate its image using a ray diagram. As the object moves closer to the mirror, (a) does the magnitude of the image distance become larger or smaller, and (b) does the magnitude of the image height become larger or smaller? (c) What is the ratio of the image height when the object distance is \(5 \mathrm{cm}\) to its height when the object distance is \(25 \mathrm{cm} ?\) Give your answer to one significant figure.

When viewed in a spherical mirror, the image of a setting sun is a virtual image. The image lies \(12.0 \mathrm{cm}\) behind the mirror. (a) Is the mirror concave or convex? Why? (b) What is the radius of curvature of the mirror?

The radius of curvature of a mirror is \(24 \mathrm{cm} .\) A diamond ring is placed in front of this mirror. The image is twice the size of the ring. Concepts: (i) Is the mirror concave or convex, or is either type possible? (ii) How many places are there in front of a concave mirror where the ring can be placed and produce an image twice the size of the object? (iii) What are the possible values for the magnification of the image of the ring? Calculation: Find the object distance of the ring.
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