Chapter 34: Problem 93
A convex spherical mirror with a focal length of magnitude 24.0 cm is placed 20.0 cm to the left of a plane mirror. An object 0.250 cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closest to the spherical mirror, and how tall is each image?
Short Answer
Step by step solution
Identify the Setup
Use the Mirror Equation for Convex Mirror
Calculate First Image by Convex Mirror
Calculate Image Height for First Image
Calculate Second Image by Plane Mirror
Reflected Image by Convex Mirror
Calculate Image Height for Second Image
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convex Mirror
- Convex mirrors always form virtual images because reflected light from a convex mirror diverges.
- The images are usually upright and smaller than the actual object.
This is key in the exercise, where the convex mirror's properties help form the images of the object in unique ways.
Mirror Equation
- \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
- \( f \) is the focal length of the mirror (negative for convex mirrors),
- \( d_o \) is the distance of the object from the mirror, and
- \( d_i \) is the image distance from the mirror.
The sign of each term tells us about the image's nature: real or virtual, upright or inverted. In the example problem, we used this equation twice. First, to find out where the first image forms relative to the convex mirror, and again for the reflected image from the plane mirror.
Image Magnification
- \( m = -\frac{d_i}{d_o} \)
- \( m \) is magnification,
- \( d_i \) is the image distance, and
- \( d_o \) is the object distance.
Negative magnification implies that the image is inverted, though in a convex mirror scenario, the image will still be upright due to the virtual nature. This was used in the example to calculate how tall the images are that the convex mirror formed, revealing that both images are smaller than the object placed between the mirrors.
Virtual Image
- Always upright
- Found behind the mirror in the mirror's operational space
In the given exercise, the convex mirror creates virtual images. These images seem to be on the opposite side of the mirror from where the object actually is, reflecting the divergent nature of the rays. The image location is also reflected as a negative distance in calculations, marking its virtuality.
Plane Mirror
- The distance between the object and the mirror equals the distance between the image and the mirror. - Plane mirrors always produce virtual images, which appear reversed horizontally.
In the problem, the image created by the convex mirror interacts with the plane mirror. It enables the formation of another image, which, interestingly, acts as a new object for further reflection by the convex mirror. This interplay between the types of mirrors helps in understanding complex image formations.