Problem 73 - 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 \(\mathrm{m}\) from the mirror. The filament is 6.00 \(\mathrm{mm}\) tall, and the image is to be 24.0 \(\mathrm{cm}\) 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) Place the filament 0.20 m from the mirror. (b) The radius of curvature is about 0.410 m.

Step 1: Understand the Given Information

We are given a concave mirror and need to find two things: (a) the distance of the filament from the mirror, and (b) the radius of curvature of the mirror. The screen is 8.00 m away from the mirror, the image is 24.0 cm tall, and the filament is 6.00 mm tall.

Step 2: Relate Image and Object Distances

Use the magnification formula for a mirror: \( m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} \), where \( h_i = 24.0 \, \mathrm{cm} \) is the image height, \( h_o = 6.00 \, \mathrm{mm} = 0.60 \, \mathrm{cm} \) is the object height, \( d_i = 8.00 \, \mathrm{m} = 800 \, \mathrm{cm} \) is the image distance, and \( d_o \) is the object distance. Set up the equation \( \frac{24.0}{0.60} = \frac{-800}{d_o} \).

Step 3: Solve for Object Distance

Solve the equation from Step 2: \( \frac{24.0}{0.60} = 40 \), thus \( -\frac{800}{d_o} = 40 \). Solving this gives \( d_o = \frac{-800}{40} = -20 \, \mathrm{cm} \). Convert to meters: \( d_o = -0.20 \, \mathrm{m} \).

Step 4: Use Mirror Equation to Find Focal Length

Apply the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substituting \( d_o = -0.20 \, \mathrm{m} \) and \( d_i = 8.00 \, \mathrm{m} \) gives \( \frac{1}{f} = \frac{1}{-0.20} + \frac{1}{8} \). Solve for \( f \).

Step 5: Solve for Focal Length

Calculate \( \frac{1}{f} = \frac{1}{-0.20} + \frac{1}{8} = -5 + 0.125 = -4.875 \). Thus, \( f = -\frac{1}{4.875} \approx -0.205 \) meters.

Step 6: Find Radius of Curvature

Use the relation between the focal length and the radius of curvature: \( R = 2f \). Substitute \( f \approx -0.205 \) meters into the equation to find \( R = 2 \times -0.205 \approx -0.410 \) meters.

Key Concepts

Mirror Equation

The mirror equation is fundamental when dealing with concave mirrors. It allows us to link the focal length, image distance, and object distance all in one neat formula, making it easy to solve problems involving mirror reflections. The mirror equation is expressed as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Where:- \( f \) is the focal length of the mirror.- \( d_o \) is the object distance.- \( d_i \) is the image distance.In our exercise, you start by substituting known values into the equation. This connection helps us find the missing variable, in this case, the focal length \( f \). You carefully work through each component to understand how the concave mirror bends light to form images at different distances.

Magnification Formula

The magnification formula is a tool to describe how much larger or smaller the image is compared to the actual object. For mirrors, it's denoted by:\[ m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} \]Where:- \( m \) is the magnification of the image.- \( h_i \) is the height of the image.- \( h_o \) is the height of the object.- \( d_i \) is the image distance (negative for concave mirrors if the image is real).- \( d_o \) is the object distance.In the exercise, the problem tells us how tall the image and object are, which allows us to directly use this equation to find the object distance. By rearranging the formula, you solve for \( d_o \), revealing how the mirror's curvature affects the image's size compared to the object's actual dimensions.

Radius of Curvature

Understanding the radius of curvature is key in the design and application of concave mirrors. This radius is the distance from the mirror's surface to its center of curvature. In simpler terms, it indicates how "curved" the mirror is. The relationship between focal length \( f \) and the radius of curvature \( R \) of a spherical mirror is:\[ R = 2f \]This equation shows that the radius of curvature is twice the focal length. By determining \( f \), you can quickly find \( R \), which helps in designing mirrors that need precise focal lengths. In solving the exercise, once the focal length was calculated, calculating the radius of curvature was straightforward using this direct relationship.

Image and Object Distance

Image and object distances are crucial for locating where the image forms in relation to where the object is placed before the mirror. For concave mirrors:- The object distance \( (d_o) \) is usually measured from the object to the mirror.- The image distance \( (d_i) \) indicates where the image appears, and is considered positive if the image is formed on the same side as the object (real image) and negative otherwise.In solving for \( d_o \), you use the mirror equation in relation with the known \( d_i \) — in this case, given in the problem as the distance from the mirror to the screen. Combining this knowledge, you derive where the object (the filament) must be placed to achieve the given image size, illustrating how changing positions affects image formation.