Problem 20 - College Physics

Question

A virtual image of magnification +2.0 is produced when an object is placed \(7.0 \mathrm{~cm}\) in front of a spherical mirror. (a) The mirror is (1) convex, (2) concave, (3) flat. Explain. (b) Find the radius of curvature of the mirror.

Step-by-Step Solution

Answer

(a) The mirror is concave. (b) The radius of curvature is 28.0 cm.

Step 1: Understand Magnification of a Mirror

Given a magnification of +2.0, this signifies that the image is virtual and upright. For mirrors, a positive magnification indicates that the mirror is a concave type because concave mirrors can form virtual images when the object is placed between the focal point and the mirror.

Step 2: Identify the Type of Mirror

Since the magnification is +2.0 and considering the property of mirrors in forming virtual images, the mirror must be concave. Convex mirrors always produce images that are smaller and upright, and flat mirrors have a magnification of +1.

Step 3: Analyze the Mirror Formula

The mirror formula is given by: \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. Magnification \(m\) is also given by: \(m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\). With \(m = +2.0\) and \(d_o = 7.0\, \text{cm}\), the image distance \(d_i\) is calculated as \(d_i = -2 \times 7.0 \text{ cm} = -14.0 \text{ cm}\).

Step 4: Calculate the Focal Length

Substitute the values into the mirror formula: \(\frac{1}{f} = \frac{1}{7.0} + \frac{1}{-14.0}\). \(\frac{1}{f} = \frac{2}{14} - \frac{1}{14} = \frac{1}{14}\). Therefore, \(f = 14.0 \text{ cm}\).

Step 5: Determine the Radius of Curvature

The radius of curvature \(R\) is related to the focal length by \(R = 2f\). So, \(R = 2 \times 14.0 \text{ cm} = 28.0 \text{ cm}\).

Key Concepts

Magnification

Magnification is an important concept in optics that helps us understand how the size of an image compares to the size of the object. In the context of spherical mirrors, it tells us about the nature of the image produced. Magnification (\(m\)) is defined as the ratio of the height of the image (\(h_i\)) to the height of the object (\(h_o\)). Mathematically, it is expressed as:

• \[m = \frac{h_i}{h_o}\]
• It can also be related to distances by the formula:\[m = -\frac{d_i}{d_o}\].

A positive magnification means the image is upright relative to the object and typically happens with concave mirrors when the object is positioned between the focal point and the mirror. A concave mirror with a magnification of +2.0, like in our example, indicates that the image is not only virtual and upright but also twice the size of the object. Understanding magnification helps to determine the type of mirror and the positioning of the object and image.

Virtual Image

A virtual image is one where the rays of light appear to diverge from a point behind the mirror, creating an image that cannot be projected onto a screen. These images are usually formed when light rays reflect from a surface such as a concave mirror under certain conditions. In the exercise example, the virtual image is formed with a magnification of +2.0. This implies that:

• The image is indeed virtual and is formed behind the mirror.
• It appears upright and is larger than the actual object.
• Virtual images in concave mirrors occur when the object lies between the focal point and the mirror.

In contrast, convex mirrors generally form virtual images that are smaller and upright, while plane mirrors create images that are of equal size to the object and also virtual. Understanding the nature of virtual images in different mirror types helps in predicting the behavior and application of these mirrors in real-world scenarios.

Radius of Curvature

The radius of curvature (\(R\)) of a mirror is a key property that relates to its bending and focal properties. It corresponds to the radius of the sphere from which the mirror segment is taken. This concept is central to understanding how much a mirror can focus and magnify images.The radius of curvature is directly connected to the focal length (\(f\)) through the formula:

• \[R = 2f\].

In our exercise example, the focal length was determined to be 14 cm, making the radius of curvature 28 cm.

• A larger radius indicates a less curved mirror, leading to a longer focal length.
• Conversely, a smaller radius of curvature means a more sharply curved mirror with a shorter focal length.

Understanding the radius of curvature helps predict how light will focus upon reflecting off the mirror, affecting the type and characteristics of the images it produces.