Problem 67 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) 0.200 meters in front. (b) 0.390 meters radius of curvature.

Step 1: Understand the Problem

We have a concave mirror forming an image of a filament. We want to find the distance from the mirror to the filament (the object distance) and the radius of curvature of the mirror. We know the image distance, the real height of the object, and the image height.

Step 2: Use the Magnification Formula

The magnification formula for mirrors is \( M = \frac{h'}{h} = -\frac{d_i}{d_o} \), where \( h' \) is the image height, \( h \) is the object height, \( d_i \) is the image distance, and \( d_o \) is the object distance. Substitute the values: \( h' = 24.0 \text{ cm} = 240 \text{ mm} \), \( h = 6.0 \text{ mm} \), and \( d_i = 8.00 \text{ m} = 8000 \text{ mm} \) to find \( d_o \).

Step 3: Calculate Object Distance

Using the magnification equation \( \frac{240}{6} = \frac{8000}{d_o} \), solve for \( d_o \). This yields \( d_o = \frac{8000 \times 6}{240} = 200 \text{ mm} = 0.200 \text{ m} \). Hence, the filament should be placed 0.200 meters in front of the mirror.

Step 4: Use the Mirror Equation

Use the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to find the focal length \( f \). With \( d_o = 200 \text{ mm} \) and \( d_i = 8000 \text{ mm} \), \( \frac{1}{f} = \frac{1}{200} + \frac{1}{8000} = \frac{40+1}{8000} = \frac{41}{8000} \). Thus, \( f \approx 195.12 \text{ mm} = 0.195 \text{ m} \).

Step 5: Calculate Radius of Curvature

The radius of curvature \( R \) of the mirror is \( R = 2f \). From the previous calculation, \( f = 0.195 \text{ m} \), so \( R = 2 \times 0.195 = 0.390 \text{ m} \).

Step 6: Final Step: Solution Summary

(a) The filament should be placed 0.200 meters in front of the mirror. (b) The radius of curvature of the mirror should be 0.390 meters.

Key Concepts

Mirror Equation

When working with concave mirrors, the mirror equation helps to relate the focal length \( f \) of the mirror to the object distance \( d_o \) and the image distance \( d_i \). The equation is given by: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This formula allows us to calculate one of these distances if the other two are known.

• For a concave mirror, any real object that is placed in front of it will have its image formed at a certain spot, either real or virtual.
• The object and image distances are measured from the mirror's vertex, which is the point on the mirror's surface at its axis.
• The focal length is positive for concave mirrors, as light converges after reflecting.

By rearranging this equation, you can determine the missing variable, such as when solving for the focal length to subsequently find the radius of curvature.

Magnification Formula

The magnification formula describes how the size of an image relates to the size of the object. For mirrors, it is expressed as:\[ M = \frac{h'}{h} = -\frac{d_i}{d_o} \]Here, \( h \) is the height of the object and \( h' \) is the height of the image. The negative sign indicates that if the image is real, it is inverted.

• If the magnitude of the magnification \( M \) is greater than 1, the image is larger than the object.
• If \( M \) is positive, the image is upright; if negative, the image is inverted.
• Using known values of image and object heights along with image distance, the object distance can be solved.

This equation, combined with measurements, allows one to derive unknown variables like image or object distance, helping in practical applications like the described exercise.

Radius of Curvature

The radius of curvature \( R \) of a mirror is crucial for understanding its geometry. This distance is the radius of the sphere from which the mirror segment is derived. It relates to the focal length \( f \) through the formula:\[ R = 2f \]This simple relationship shows how the size of the curve influences how the light is focused.

• The radius of curvature is always the same on either side of the mirror's optic pole.
• It's crucial in defining the mirror's focal point - a guide for designing optical systems.
• In practical terms, calculating \( R \) ensures precise focal setups, like in headlights.

By understanding and applying this relationship, you can determine the radius given the focal length to ensure effective focusing of light.

Object Distance

In optics, the object distance \( d_o \) refers to the distance between the object and the mirror. It plays a significant role in determining where the image will appear using the mirror and magnification formulas.

• Object distance is always considered positive for real objects located in front of the mirror.
• The mirror and magnification formulas are used to calculate the precise required object distance.
• In this exercise, the given variables and the corresponding formula calculations determine how far the filament should be placed from the mirror to achieve the desired image height and placement.

In practice, understanding object distance helps ensure that mirrors are placed correctly to project the intended image size and position, as needed in applications like vehicle headlights.