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A plane mirror and a concave mirror \((f=8.0 \mathrm{cm})\) are facing each other and are separated by a distance of \(20.0 \mathrm{cm} .\) An object is placed between the mirrors and is \(10.0 \mathrm{cm}\) from each mirror. Consider the light from the object that reflects first from the plane mirror and then from the concave mirror. Using a ray diagram drawn to scale, find the location of the image that this light produces in the concave mirror. Specify this distance relative to the concave mirror.

Short Answer

Expert verified
The image formed by the concave mirror is approximately 10.67 cm in front of the mirror.

Step by step solution

01

Understand the Setup

We have a plane mirror and a concave mirror with a focal length \(f = 8.0\, \mathrm{cm}\), separated by \(20.0\, \mathrm{cm}\). An object is positioned \(10.0\, \mathrm{cm}\) from each mirror. This means the object is exactly halfway between the two mirrors.
02

Reflected Image in Plane Mirror

When light reflects back from the plane mirror, the virtual image seems to be behind the mirror at the same distance as the object is in front. Hence, the virtual image is \(10.0\, \mathrm{cm}\) inside the plane mirror, or \(30.0\, \mathrm{cm}\) from the concave mirror.
03

Determine Image Location in Concave Mirror

Consider the virtual image as the object for the concave mirror. The object distance \(d_o\) from the concave mirror is \(30.0\, \mathrm{cm}\). Use the mirror formula \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f = 8.0\, \mathrm{cm}\). Substitute the values to calculate \(d_i\), the image distance from the concave mirror.
04

Solve for Image Distance

Rearrange the mirror formula to find \(d_i\):\[\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}\]Substitute \(f = 8.0\, \mathrm{cm}\) and \(d_o = 30.0\, \mathrm{cm}\):\[\frac{1}{d_i} = \frac{1}{8.0} - \frac{1}{30.0}\]Calculate \(d_i\):\(d_i \approx 10.67\, \mathrm{cm}\)

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Key Concepts

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

Understanding Concave Mirrors
Concave mirrors are curved inward like a bowl. This unique shape causes light rays that strike the mirror to converge, or come together, at a single point known as the focal point. The distance from the mirror to this focal point is called the focal length, which is crucial in determining how images are formed. In our specific example with the concave mirror having a focal length of 8.0 cm, we use this focal point to calculate where the image will appear after reflection.

Images created by concave mirrors can be real or virtual, depending on the position of the object in relation to the focal point:
  • If the object is beyond the focal point, a real and inverted image is formed on the same side of the mirror as the object.
  • If the object is between the focal point and the mirror, a larger virtual image appears behind the mirror.
Therefore, accurate knowledge of the object's position and focal length helps in predicting the nature and position of the image formed by a concave mirror.
Principles of Plane Mirrors
A plane mirror is a flat surface that reflects light. Unlike curved mirrors, the images formed in a plane mirror are always virtual. This means they appear to be behind the actual mirror at the same distance as the object is in front. When you look into a plane mirror, it seems as though the reflection is tangible and exists behind the mirror, but in reality, it cannot be projected onto a screen.

Plane mirrors produce images with the following characteristics:
  • The image is upright
  • It is the same size as the object
  • The left and right are reversed
In the exercise, the object is placed 10 cm from the plane mirror. Due to the symmetry in plane mirror physics, the virtual image forms 10 cm behind the plane mirror, resulting in an image 30 cm from the concave mirror.
Utilizing Ray Diagrams
Ray diagrams are simplified representations that help us understand how light travels and how images form in mirrors and lenses. They show the path of light rays as they interact with reflective surfaces, making it easier to visualize where an image will form.

When creating ray diagrams:
  • Draw the principal axis for reference
  • Mark the focal point and center of curvature (for concave mirrors) on the principal axis
  • Draw at least two light rays from the top of the object:
    • The first ray is drawn parallel to the principal axis and reflects through the focal point
    • The second ray passes through the center of curvature and reflects back along its original path
These steps help determine where the image will appear after the light reflects. In the exercise, this approach helps us discover that the image formed by the concave mirror is approximately 10.67 cm away from its surface.

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Most popular questions from this chapter

A spherical mirror is polished on both sides. When the concave side is used as a mirror, the magnification is \(+2.0 .\) What is the magnification when the convex side is used as a mirror, the object remaining the same distance from the mirror?

A man holds a double-sided spherical mirror so that he is looking directly into its convex surface, \(45 \mathrm{cm}\) from his face. The magnification of the image of his face is \(+0.20 .\) What will be the image distance when he reverses the mirror (looking into its concave surface), maintaining the same distance between the mirror and his face? Be sure to include the algebraic sign (+ or \- ) with your answer.

An object is located \(14.0 \mathrm{cm}\) in front of a convex mirror, the image being \(7.00 \mathrm{cm}\) behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located?

You walk at an angle of \(\theta=50.0^{\circ}\) toward a plane mirror, as in the drawing. Your walking velocity has a magnitude of \(0.90 \mathrm{m} / \mathrm{s}\). What is the velocity of your image relative to you (magnitude and direction)?

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.

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