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Chapter 14: Problem 35

A concave mirror is to form an image of the filament of a headlight lamp on a screen \(8.00 \mathrm{~m}\) from the mirror. The filament is \(6.00 \mathrm{~mm} \mathrm{tall}\), and the image is to be \(24.0 \mathrm{~cm}\) tall. (a) How far in front of the vertex of the mimor should the filament be placed? (b) What should be the radius of curvature of the mirror?

Short Answer

Expert verified
(a) 20.0 cm in front of the mirror. (b) Radius of curvature is \(-39.02\) cm.

Step by step solution

01

Establish Variables and Known Values

We have a concave mirror problem with the following known values: Distance from the mirror to the screen (image distance, \( v = 8.00 \, \text{m} = 800 \, \text{cm} \)), height of the filament (object height, \( h_o = 6.00 \, \text{mm} = 0.60 \, \text{cm} \)), and height of the image (image height, \( h_i = 24.0 \, \text{cm} \)). We want to find the object distance \( u \), and the radius of curvature \( R \) of the mirror.
02

Use the Magnification Formula

The magnification formula for mirrors is \( m = \frac{h_i}{h_o} = -\frac{v}{u} \). Plug in the known values: \( \frac{24.0}{0.60} = -\frac{800}{u} \). This simplifies to \( u = -800 \cdot \frac{0.60}{24.0} \). Calculate \( u \).
03

Calculate the Object Distance

From the equation in Step 2, \( u = -800 \times \frac{0.60}{24.0} \), which simplifies to \( u = -20.0 \, \text{cm} \). Hence, the filament should be placed 20.0 cm in front of the mirror.
04

Use the Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f \) is the focal length. We know \( v = 800 \, \text{cm} \) and \( u = -20.0 \, \text{cm} \). Substitute these values to find \( f \): \( \frac{1}{f} = \frac{1}{800} + \frac{1}{(-20.0)} \). Calculate \( f \).
05

Calculate the Focal Length

Solve \( \frac{1}{f} = \frac{1}{800} + \frac{1}{-20.0} \). This becomes \( \frac{1}{f} = \frac{1}{800} - \frac{1}{20} \). Simplify to find \( f = -19.51 \, \text{cm} \). Hence, the focal length is -19.51 cm.
06

Find the Radius of Curvature

The radius of curvature \( R \) is related to the focal length by \( R = 2f \). Therefore, using \( f = -19.51 \, \text{cm} \), we have \( R = 2 \times (-19.51) \). Calculate \( R \).
07

Calculate the Radius of Curvature

\( R = 2 \times (-19.51) = -39.02 \, \text{cm} \). Thus, the radius of curvature of the mirror is -39.02 cm.

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Key Concepts

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

Image Distance
In optics, the image distance is a crucial concept when working with concave mirrors. It refers to the distance between the mirror and the image formed. This is typically denoted by the letter \( v \). When dealing with a concave mirror, the image can be real and inverted depending on the object position. In this exercise, we know the image distance is 800 cm (or 8.00 meters), as the screen is this far from the mirror. This distance allows us to use formulas such as the mirror equation to find unknowns like object distance or focal length.
Magnification Formula
The magnification formula links the sizes of the object and its image as well as the respective distances from the mirror. It is expressed as \( m = \frac{h_i}{h_o} = -\frac{v}{u} \), where \( m \) is the magnification, \( h_i \) is the image height, \( h_o \) is the object height, \( v \) is the image distance, and \( u \) is the object distance.

In the example, we calculate the object distance \( u \) after substituting known values into the magnification formula: \( \frac{24.0}{0.60} = -\frac{800}{u} \). This leads to \( u = -20.0 \, \text{cm} \). The negative sign indicates the location of the object on the opposite side from the real image.
Mirror Equation
The mirror equation is a fundamental relationship in optics that describes how light rays reflect from a mirror. The formula is \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f \) is the focal length of the mirror, \( u \) is the object distance, and \( v \) is the image distance.

For this problem, plugging \( v = 800 \), and \( u = -20.0 \) into the mirror equation, we aim to find \( f \):
\[ \frac{1}{f} = \frac{1}{800} + \frac{1}{(-20.0)} \]
Solving this equation gives us a focal length \( f = -19.51 \, \text{cm} \). This implies the focal point is behind the mirror's vertex.
Radius of Curvature
The radius of curvature \( R \) is an intrinsic characteristic of a mirror that describes the radius of the spherical surface of which the mirror is a part. For a concave mirror, it is related to the focal length with \( R = 2f \).

From our earlier calculation, we found the focal length \( f \) to be \(-19.51 \, \text{cm} \). Using this, the radius of curvature is calculated as follows:
\[ R = 2 \times (-19.51) = -39.02 \, \text{cm} \]
The negative sign indicates that the center of curvature is on the same side of the mirror as the object and the focal point, adhering to the conventions typically used in mirror problems.

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Most popular questions from this chapter

The image of a tree just covers the length of a plane mirror \(4.00 \mathrm{~cm}\) tall when the mirror is held \(35.0 \mathrm{~cm}\) from the eye. The tree is \(28.0 \mathrm{~m}\) from the mirror. What is its height?

An object is \(24.0 \mathrm{~cm}\) from the center of a convex silvered spherical glass Christmas tree ornament \(6.00 \mathrm{~cm}\) in diameter. What are the position and magnification of its image?

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