Problem 62 - College Physics

Question

\(\cdot\) A 3.80 -mm-tall object is 24.0 \(\mathrm{cm}\) from the center of a sil- vered spherical glass Christmas tree ornament 6.00 \(\mathrm{cm}\) in diameter, What are the position and height of its image?

Step-by-Step Solution

Answer

Image position: -1.41 cm, Image height: 0.0223 cm

Step 1: Identify the Known Values

The object height is given as 3.80 mm or 0.38 cm (converted to cm for consistency). The object distance from the center of the ornament is 24.0 cm. The radius of the spherical ornament is half of 6.00 cm, so 3.00 cm.

Step 2: Determine the Mirror Type

Since the ornament is spherical and silvered, it acts as a convex mirror. The focal length is half the radius but negative in sign, so \( f = -\frac{r}{2} = -1.5 \text{ cm} \).

Step 3: Use the Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substitute the known values: \( \frac{1}{-1.5} = \frac{1}{24} + \frac{1}{d_i} \).

Step 4: Solve for Image Distance

Solving the equation: \( \frac{1}{d_i} = \frac{1}{-1.5} - \frac{1}{24} \). Calculating gives \( \frac{1}{d_i} = -0.6667 - 0.0417 = -0.7083 \). Therefore, \( d_i = \frac{1}{-0.7083} \approx -1.41 \text{ cm} \).

Step 5: Use the Magnification Equation

The magnification equation \( m = -\frac{d_i}{d_o} = \frac{h_i}{h_o} \). Substitute \( m = -\frac{-1.41}{24} \approx 0.05875 \).

Step 6: Solve for Image Height

Using \( m = \frac{h_i}{0.38} \), substitute the values, \( h_i = 0.58 \times 0.38 \approx 0.0223 \text{ cm} \).

Step 7: Conclusion

The image distance is -1.41 cm, indicating that the image is virtual and located 1.41 cm behind the mirror. The image height is 0.0223 cm.

Key Concepts

Mirror Equation

The mirror equation is a fundamental tool in geometrical optics, particularly useful for analyzing spherical mirrors. It relates the object distance, image distance, and focal length of the mirror through the equation:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]This equation allows us to calculate the position of an image formed by a mirror when the object distance and the focal length are known.

• \(f\) is the focal length of the mirror.
• \(d_o\) is the object distance from the mirror.
• \(d_i\) is the image distance from the mirror.

For convex mirrors, the focal length is always negative, reflecting the fact that the mirror diverges light rays.

Image Distance

The image distance \(d_i\) is essential to understanding where an image will appear in relation to a mirror. It is the distance between the image and the mirror's surface. To find it, rearrange the mirror equation once the focal length and object distance are known.
In the case of convex mirrors, since they diverge light, the image distance is always depicted as a negative value. This denotes that the image itself is virtual, forming on the side of the mirror opposite the object.
In our problem, by substituting known values into the mirror equation, we calculated the image distance as approximately \(-1.41\, \text{cm}\), which confirms the virtual nature of the image, located behind the mirror.

Magnification

Magnification describes how much larger or smaller the image is compared to the object. It's determined by the ratio of the image height to the object height, but can also be found using the object and image distances.
The magnification formula is given by:
\[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]

• \(m\) is the magnification.
• \(h_i\) and \(h_o\) are the image and object heights, respectively.

A positive magnification implies an upright image, while a negative one suggests an inverted image. For this exercise, the magnification was calculated as approximately 0.05875, indicating that the image is significantly smaller and also upright.

Convex Mirror

Convex mirrors, also known as diverging mirrors, are characterized by their outwardly curved reflective surfaces. They cause incoming parallel rays of light to spread apart, giving the effect of a virtual image that appears smaller and farther away.
Convex mirrors are commonly used for safety and security purposes because they enable a wider field of view. Any image formed by a convex mirror is virtual, diminished in size, and located behind the mirror. This feature stems from the fact that the reflected rays seem to converge at a point behind the mirror.
In the exercise, the spherical glass Christmas tree ornament acts as a convex mirror. Despite being spherical, its mirrored surface creates these typical convex mirror effects, leading to an upright, diminished, and virtual image.