Chapter 10: Problem 14
An object \(5.0 \mathrm{~cm}\) in length is placed at a distance of \(20 \mathrm{~cm}\) in front of a convex mirror of radius of curvature \(30 \mathrm{~cm}\). Find the position of the image, its nature and size.
Short Answer
Expert verified
Image is at -60 cm, virtual, erect, and 15 cm tall.
Step by step solution
01
- Understand and identify given values
Identify and write down the given values. Object length, object distance, and radius of curvature are provided.Given:Object height, \( h_o = 5.0 \rm{cm} \)Object distance, \( u = -20 \rm{cm} \) (Object distance is taken as negative for mirrors)Radius of curvature, \( R= 30 \rm{cm} \)Find the mirror's focal length using \(f = \frac{R}{2} \).
02
- Calculate the focal length
Calculate the focal length from the radius of curvature.\[ f = \frac{R}{2} \]Substitute the value of \(R = 30 \rm{cm}\).\[ f = \frac{30 \rm{cm}}{2} = 15 \rm{cm} \]Note that the focal length is also negative for a convex mirror.\(f = -15 \rm{cm}\)
03
- Use the mirror equation
Apply the mirror equation \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u} \) to find image distance \( v \).\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-20} \]Rearrange to solve for \( \frac{1}{v} \).
04
- Solve for image distance
Combine the terms to solve for \( \frac{1}{v} \):\[ \frac{1}{v} = \frac{1}{-15} - \frac{1}{20} \]Find a common denominator and solve:\[ \frac{1}{v} = -\frac{4}{60} + \frac{3}{60} = -\frac{1}{60} \]So, \( v = -60 \rm{cm} \)
05
- Determine the magnification
Use the magnification formula \( m = -\frac{v}{u} \) to find the magnification:\[ m = -\frac{-60 \rm{cm}}{-20 \rm{cm}} = \frac{60}{20} = 3 \]The positive magnification indicates that the image is virtual and erect.
06
- Calculate the image height
Using magnification, find the image height \( h_i \) from \( m = \frac{h_i}{h_o} \):\[ h_i = m \times h_o = 3 \times 5 \rm{cm} = 15 \rm{cm} \]So, the image height is \( h_i = 15 \rm{cm} \).
07
- Summarize the results
Summarize the information gathered to describe the image:The image distance is \( v = -60 \rm{cm} \) (virtual, same side as the object).The image is virtual, erect, and magnified.The image height is \( h_i = 15 \rm{cm} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Image Distance
In the context of a convex mirror, understanding image distance is crucial. The image distance, denoted as \(v\), relates to where the image formed by the mirror appears. To find this distance, we use the mirror equation: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \). Here, \(f\) is the focal length, and \(u\) is the object distance. It's important to note the sign conventions:
- The object distance \(u\) is always negative for mirrors because it is typically measured against the direction of the incident light.
- The image distance \(v\) for a convex mirror is always negative as the image forms behind the mirror.
Magnification Formula
Magnification in mirrors reveals information about the size and orientation of the image relative to the object. The magnification formula for a mirror is: \( m = -\frac{v}{u} \). Here, \(m\) is the magnification, \(v\) is the image distance, and \(u\) is the object distance. Based on the provided values in our exercise:
- Image distance, \(v = -60 \text{ cm}\)
- Object distance, \(u = -20 \text{ cm}\)
Radius of Curvature
The radius of curvature, \(R\), is an essential parameter for understanding how curved mirrors work. It is the radius of the sphere from which the mirror segment is cut. In our problem, the radius of curvature is given as \(R = 30 \text{ cm}\).
The radius plays a pivotal role in determining the focal length of the mirror, which is derived using the following relation: \[ f = \frac{R}{2} \] With the given \(R = 30 \text{ cm}\), we find the focal length: \[ f = \frac{30}{2} = 15 \text{ cm} \] However, for convex mirrors, we assign the focal length a negative value, hence \(f = -15 \text{ cm}\).
The radius plays a pivotal role in determining the focal length of the mirror, which is derived using the following relation: \[ f = \frac{R}{2} \] With the given \(R = 30 \text{ cm}\), we find the focal length: \[ f = \frac{30}{2} = 15 \text{ cm} \] However, for convex mirrors, we assign the focal length a negative value, hence \(f = -15 \text{ cm}\).
Focal Length Calculation
Calculating the focal length of a mirror is a fundamental step in understanding image formation. The focal length \(f\) is the distance from the mirror's surface to its focal point. It is derived from the mirror's radius of curvature \(R\) using the formula: \[ f = \frac{R}{2} \] Given the radius of curvature \(R = 30 \text{ cm}\) in our exercise, we compute the focal length as: \[ f = \frac{30}{2} = 15 \text{ cm} \] For convex mirrors, the focal length is considered negative to reflect the nature of image formation. Therefore, our focal length is \(f = -15 \text{ cm}\). This negative sign indicates that the focal point is virtual and lies on the side opposite to the object.