Problem 132 - Fundamentals of Physics

Question

A millipede sits \(1.0 \mathrm{~m}\) in front of the nearest part of the surface of a shiny sphere of diameter \(0.70 \mathrm{~m}\). (a) How far from the surface does the millipede's image appear? (b) If the millipede's height is \(2.0\) \(\mathrm{mm}\), what is the image height? (c) Is the image inverted?

Step-by-Step Solution

Answer

(a) Image appears 0.138 m in front of the surface inside the sphere. (b) Image height is 0.424 mm. (c) The image is inverted.

Step 1: Identify the problem type

This problem involves reflection in a concave spherical mirror. We need to find image distance, image height, and determine if the image is inverted.

Step 2: Calculate the radius of curvature

The diameter of the sphere is given as 0.70 m, so the radius of curvature \( R \) is half of the diameter: \( R = \frac{0.70}{2} = 0.35 \) m.

Step 3: Use the mirror formula

The mirror formula is \( \frac{1}{f} = \frac{1}{R} \), which gives the focal length \( f \). With \( R = 0.35 \) m, the focal length is \( f = \frac{0.35}{2} = 0.175 \) m.

Step 4: Apply the mirror equation

Using the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( d_o \) is the object distance (1.0 m) and \( d_i \) is the image distance, we solve for \( d_i \). \[ \frac{1}{0.175} = \frac{1}{1.0} + \frac{1}{d_i} \] \[ \frac{1}{d_i} = \frac{1}{0.175} - \frac{1}{1.0} = 5.71 - 1 = 4.71 \] \( d_i = \frac{1}{4.71} \approx 0.212 \) m. Since the sign of \( d_i \) is positive, the image appears 0.212 m away from the mirror.

Step 5: Calculate the image distance from the surface

The image distance from the mirror is \( d_i \), which is from the center. To find the distance from the surface: distance from surface = \( d_i - R \). So, \( 0.212 - 0.35 = -0.138 \) m, meaning the image is 0.138 m in front of the surface inside the sphere.

Step 6: Calculate the magnification

The magnification \( m \) is given by \( m = \frac{h_i}{h_o} = \frac{d_i}{d_o} \), where \( h_i \) is the image height and \( h_o \) is the object height (2.0 mm or 0.002 m). \[ m = \frac{0.212}{1.0} = 0.212 \] Thus, \( h_i = m \times h_o = 0.212 \times 0.002 = 0.000424 \) m (0.424 mm).

Step 7: Determine if the image is inverted

In concave mirrors, a real image for objects beyond the focal point is inverted. Since the image formed is real and in front of the mirror surface, it is inverted.

Key Concepts

Understanding Concave Mirrors

Concave mirrors are spherical mirrors with a reflecting surface curving inward, resembling a portion of the interior of a sphere. These mirrors gather incoming light and converge it at a particular point, known as the focal point. Concave mirrors are widely used in applications that require focusing light, such as in telescopes and headlights.

Key features of concave mirrors include:

• The principal axis: a straight line passing through the center of curvature and the midpoint of the mirror.
• The focal point: where reflected light converges.
• The radius of curvature (R): the radius of the sphere from which the mirror segment is derived.

Understanding these basic components helps in determining how an image is formed and where it appears.

The Mirror Formula Explained

The mirror formula is a crucial equation used to relate the object distance, image distance, and the focal length of a spherical mirror. It 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 distance from the object to the mirror.
• \(d_i\) is the distance from the image to the mirror.

This equation helps us find unknown values when two parameters are known. To solve for the image distance, rearrange the formula and plug in the known values.

Using the mirror formula in exercises helps solve for image characteristics like position and size, making it invaluable for studying optics.

Calculating Image Distance

To find out how far an image appears from the surface of the concave mirror, we use the mirror formula in combination with other known measurements. For instance, a common problem may provide you with the object distance (\(d_o\)) and the focal length (\(f\)) and ask for the image distance (\(d_i\)).

The process involves these steps:

• Calculate the focal length using the given radius of curvature: \(f = \frac{R}{2}\).
• Rearrange the mirror formula to solve for \(d_i\): \(\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}\).
• Solve the equation for \(d_i\) to find the distance from the mirror.
• Subtract \(R\) from the result to find the distance from the surface of the mirror.

This method helps determine where the image "forms" relative to the mirror, which is crucial for understanding reflection and image properties.

Image Height Calculation

The height of an image formed by a concave mirror can be determined by calculating the magnification. Magnification relates the image height to the object height and the image distance to the object distance. The formula for magnification \(m\) is: \[m = \frac{h_i}{h_o} = \frac{d_i}{d_o}\]Where:

• \(h_i\) is the height of the image.
• \(h_o\) is the height of the object.
• \(d_i\) and \(d_o\) are the image and object distances, respectively.

To find the image height \(h_i\), rearrange the formula to: \(h_i = m \times h_o\). By knowing the object's height and the distances involved, this calculation becomes straightforward.

This concept is vital for predicting how large or small the reflected image will appear compared to the actual object, and whether the image is upright or inverted.