Problem 5 - College Physics

Question

A concave spherical mirror has a radius of curvature of 10.0 \(\mathrm{cm} .\) Calculate the location and size of the image formed of an 8.00 -mm-tall object whose distance from the mirror is (a) \(15.0 \mathrm{cm},(\mathrm{b}) 10.0 \mathrm{cm},(\mathrm{c}) 2.50 \mathrm{cm},\) and (d) 10.0 \(\mathrm{m}\)

Step-by-Step Solution

Answer

(a) Image at 7.5 cm, -4.0 mm; (b) Image at 10 cm, -8.0 mm; (c) Image at -2.5 cm, 8.0 mm; (d) Image at 5 cm, -0.04 mm.

Step 1: Determine Focal Length

The focal length \( f \) of a concave mirror is half of the radius of curvature. Given the radius of curvature \( R = 10.0 \text{ cm} \), the focal length is \( f = \frac{R}{2} = 5.0 \text{ cm} \).

Step 2: Use Mirror Equation for Location of Image

The mirror equation is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) where \( d_o \) is the object distance and \( d_i \) is the image distance.

Step 3: Apply Mirror Equation for (a)

For part (a), \( d_o = 15.0 \text{ cm} \). Plug into the mirror equation: \( \frac{1}{5.0} = \frac{1}{15.0} + \frac{1}{d_i} \). Solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{15.0} = \frac{2}{15} \), so \( d_i = 7.5 \text{ cm} \).

Step 4: Calculate Magnification for (a)

The magnification \( m \) is given by \( m = -\frac{d_i}{d_o} \). For \( d_i = 7.5 \text{ cm} \) and \( d_o = 15.0 \text{ cm} \), \( m = -\frac{7.5}{15.0} = -0.5 \). Image height is \( m \times 8.0 \text{ mm} = -4.0 \text{ mm} \).

Step 5: Apply Mirror Equation for (b)

For part (b), \( d_o = 10.0 \text{ cm} \). Plug into the mirror equation: \( \frac{1}{5.0} = \frac{1}{10.0} + \frac{1}{d_i} \). Solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{10.0} = \frac{1}{10} \), so \( d_i = 10.0 \text{ cm} \).

Step 6: Calculate Magnification for (b)

The magnification is \( m = -\frac{d_i}{d_o} = -\frac{10.0}{10.0} = -1 \). Image height is \( -1 \times 8.0 \text{ mm} = -8.0 \text{ mm} \).

Step 7: Apply Mirror Equation for (c)

For part (c), \( d_o = 2.50 \text{ cm} \). Plug into the mirror equation: \( \frac{1}{5.0} = \frac{1}{2.50} + \frac{1}{d_i} \). Solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{2.50} = -\frac{2}{5} \), so \( d_i = -2.5 \text{ cm} \).

Step 8: Calculate Magnification for (c)

The magnification is \( m = -\frac{d_i}{d_o} = -\frac{-2.5}{2.5} = 1 \). Image height is \( 1 \times 8.0 \text{ mm} = 8.0 \text{ mm} \).

Step 9: Apply Mirror Equation for (d)

For part (d), \( d_o = 1000 \text{ cm} \). Plug into the mirror equation: \( \frac{1}{5.0} = \frac{1}{1000} + \frac{1}{d_i} \). Solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{5.0} - \frac{1}{1000} \approx \frac{1}{5.0} \), so \( d_i \approx 5.0 \text{ cm} \).

Step 10: Calculate Magnification for (d)

The magnification is \( m = -\frac{d_i}{d_o} \approx -\frac{5.0}{1000} = -0.005 \). Image height is \( -0.005 \times 8.0 \text{ mm} = -0.04 \text{ mm} \).

Key Concepts

Focal Length Determination

The focal length of a concave mirror is an essential concept to grasp when working with optical systems. It is the distance from the mirror to the focal point, where parallel rays of light converge after reflection. To determine the focal length ( f ) of a concave mirror, use the formula:

• \( f = \frac{R}{2} \)
• where \( R \) is the radius of curvature.

In this exercise, we are given a radius of curvature of \( R = 10.0 \) cm. By applying the formula, we find that the focal length is 5.0 cm. Understanding and finding the focal length is the first critical step when dealing with mirrors as it influences how you calculate other properties like image formation and magnification.

Mirror Equation

The mirror equation is a powerful tool used to relate the object distance ( d_o ), image distance ( d_i ), and focal length ( f ) of a mirror. This equation is formulated as:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
• where \( f \) is the focal length, \( d_o \) is the distance from the object to the mirror, and \( d_i \) is the distance from the image to the mirror.

By rearranging this equation, you can solve for any one of these variables if the other two are known. In practice, the mirror equation enables you to predict where the image will be formed based on the object's position relative to the mirror. It's crucial for understanding how images are constructed in concave mirror systems.

Image Magnification

Image magnification is the ratio of the height of the image to the height of the object. This concept is essential in understanding the scale and orientation of the image produced by a mirror. The magnification ( m ) can be calculated using the formula:

• \( m = -\frac{d_i}{d_o} \)
• where \( d_i \) is the image distance and \( d_o \) is the object distance.

The negative sign indicates that the image is inverted relative to the object. If magnification is less than one, the image is smaller than the object. If more than one, the image is larger. This calculation allows you to not only determine the size but also the orientation of the image produced by a concave mirror.

Radius of Curvature

The radius of curvature ( R ) is the radius of the sphere from which the mirror is a segment. It's a crucial characteristic that influences the mirror's overall properties, especially in defining its focal length. The radius of curvature is directly related to the focal length by the formula:

• \( R = 2f \)
• where \( f \) is the focal length.

In practical terms, the radius of curvature helps to identify how severely the mirror will curve. A smaller radius means a more sharply curved mirror, leading to different image formation characteristics compared to a mirror with a larger radius. Understanding the radius of curvature is fundamental for analyzing and predicting the behavior of a concave mirror in optical systems.