The thin glass shell shown in Fig. E34.15 has a spherical shape with a radius of curvature of 12.0 cm, and both of its surfaces can act as mirrors. A seed 3.30 mm high is placed 15.0 cm from the center of the mirror along the optic axis, as shown in the figure. Calculate the location and height of the of this seed.

Verified step by step guidance

1. Identify the type of mirror: The thin glass shell acts as a spherical mirror with a radius of curvature of 12.0 cm. Since the seed is placed outside the mirror, it is a convex mirror.
2. Use the mirror equation to find the image location: 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. For a convex mirror, the focal length \( f \) is negative and equal to half the radius of curvature, \( f = -\frac{R}{2} = -6.0 \text{ cm} \). The object distance \( d_o \) is 15.0 cm.
3. Calculate the image distance \( d_i \): Rearrange the mirror equation to solve for \( d_i \): \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). Substitute the values for \( f \) and \( d_o \) to find \( d_i \).
4. Determine the magnification \( m \): The magnification is given by \( m = -\frac{d_i}{d_o} \). Use the calculated \( d_i \) and given \( d_o \) to find \( m \).
5. Calculate the image height: The image height \( h_i \) is related to the object height \( h_o \) and magnification \( m \) by \( h_i = m \times h_o \). Given \( h_o = 3.30 \text{ mm} \), use the magnification to find \( h_i \).

Key Concepts

Mirror Equation

The mirror equation relates the object distance (p), image distance (q), and focal length (f) of a spherical mirror: 1/f = 1/p + 1/q. This equation is essential for determining the position of the image formed by the mirror. In this problem, the radius of curvature is given, allowing us to find the focal length using f = R/2, where R is the radius of curvature.

Magnification

Magnification (M) in mirrors is defined as the ratio of the image height (h') to the object height (h), given by M = h'/h = -q/p. This concept helps in calculating the size of the image formed by the mirror. The negative sign indicates that the image is inverted relative to the object. Understanding magnification is crucial for determining the height of the image in this scenario.

Spherical Mirrors

Spherical mirrors, such as the one in the problem, can be concave or convex, affecting how they reflect light and form images. The curvature of the mirror influences the focal point and the nature of the image (real or virtual, inverted or upright). In this case, the thin glass shell acts as a spherical mirror, and its properties are used to analyze the image formation along the optic axis.