Problem 60 - College 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 36.0 \(\mathrm{cm}\) tall. (a) How far in front of the vertex of the mirror should the filament be placed? (b) To what radius of curvature should you grind the mirror?

Step-by-Step Solution

Answer

(a) Place the filament 0.133 m in front of the mirror. (b) The radius of curvature should be 0.267 m.

Step 1: Understand the Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. Since the screen is placed 8.00 m from the mirror, \( d_i = 8.00 \text{ m} \).

Step 2: Relate Magnification

The magnification \( m \) is given by \( m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \). We have the image height \( h_i = 36.0 \text{ cm} \) and the object height \( h_o = 6.00 \text{ mm} = 0.600 \text{ cm} \). Calculate magnification: \( m = \frac{36.0}{0.600} = 60 \).

Step 3: Substitute Magnification in Distance Relation

Using \( m = -\frac{d_i}{d_o} \), we find \( 60 = -\frac{8.00}{d_o} \). Solving for \( d_o \): \( d_o = -\frac{8.00}{60} \), \( d_o = -0.133 \text{ m} \). But considering the situation in this context, \( d_o = 0.133 \text{ m} \).

Step 4: Calculate the Focal Length

Using the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), substitute \( d_o = 0.133 \text{ m} \) and \( d_i = 8.00 \text{ m} \). We get \( \frac{1}{f} = \frac{1}{0.133} + \frac{1}{8.00} \). Calculate \( f \).

Step 5: Determine the Radius of Curvature

Recall that the focal length \( f \) is related to the radius of curvature \( R \) by \( f = \frac{R}{2} \). Rearrange to find \( R = 2f \). Use the calculated \( f \) from Step 4 to find \( R \).

Key Concepts

Mirror Equation

The mirror equation is fundamental when studying concave mirrors, as it relates the focal length, object distance, and image distance. The equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. This equation helps in determining the position where an object should be placed in order to form a clear image at a particular location.
In our given exercise, the image distance \( d_i \) is provided as 8.00 meters, which means the image forms 8.00 meters away from the mirror. This equation is critical for finding either \( f \) or \( d_o \) if one of them is unknown, as was done in step four of the original solution to calculate the focal length.

Magnification

Magnification is another key concept when working with mirrors. It describes how much larger or smaller the image is compared to the object. The formula for magnification \( m \) is given by the equation \( m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \). Here, \( h_i \) is the height of the image, \( h_o \) is the height of the object, and \( d_i \) and \( d_o \) are the image and object distances, respectively.

In the scenario depicted in the exercise, we start by determining the magnification using the heights of the image and object:

• Image height \( h_i = 36.0 \text{ cm} \)
• Object height \( h_o = 6.00 \text{ mm} = 0.600 \text{ cm} \)

By calculating \( m = \frac{36.0}{0.600} = 60 \), it tells us that the image is 60 times taller than the object. Such calculations inform us about the relationship between object and image distances, facilitating the computation of \( d_o \) when needed.

Focal Length

In concave mirrors, the focal length \( f \) is a measure of how strongly the mirror converges or diverges light. The equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) is rearranged to solve for \( f \) when \( d_o \) and \( d_i \) are known.
For the given problem, you substitute the values obtained earlier:

• Object distance \( d_o = 0.133 \text{ m} \)
• Image distance \( d_i = 8.00 \text{ m} \)

Substituting these into the mirror equation gives us the formula to calculate the focal length. Knowing the focal length is essential for designing optical systems because it influences how images are projected, their clarity, and position. It’s a critical factor in determining where objects must be positioned to form images effectively.

Radius of Curvature

The radius of curvature \( R \) of a mirror is twice the focal length, expressed by the equation \( R = 2f \). It reflects the size of the mirror's spherical part. A larger radius indicates a flatter mirror curve, while a smaller radius means the mirror is more curved.
In the original task, once the focal length \( f \) was calculated, the radius of curvature \( R \) was determined using this relationship. This step is crucial for manufacturing and grinding the mirror to obtain a specific focus depth. Understanding \( R \) allows engineers to tailor the mirror's curvature to meet particular design requirements, ensuring that reflection and image formation are achieved as needed.