(II) A 4.2-cm-tall object is placed 26 cm in front of a spherical mirror. It is desired to produce a virtual image that is upright and 3.2 cm tall.
(a) What type of mirror should be used?
(b) Where is the image located?
(c) What is the focal length of the mirror?
(d) What is the radius of curvature of the mirror?

Verified step by step guidance

1. To determine the type of mirror, recall that a virtual, upright image is produced by a **concave mirror** when the object is placed within the focal length. Convex mirrors also produce virtual, upright images, but they always reduce the size of the image. Since the image is taller than the object, a concave mirror must be used.
2. To find the image location, use the magnification formula: , where $m$ is the magnification, $d\_i$ is the image distance, and $d\_o$ is the object distance. The magnification is also given by the ratio of the image height to the object height: . Substitute $h\_i\; =\; 3.2\; ext\{\; cm\}$, $h\_o\; =\; 4.2\; ext\{\; cm\}$, and $d\_o\; =\; 26\; ext\{\; cm\}$ to solve for $d\_i$.
3. To find the focal length of the mirror, use the mirror equation: , where $f$ is the focal length, $d\_o$ is the object distance, and $d\_i$ is the image distance. Use the value of $d\_i$ found in the previous step to calculate $f$.
4. To find the radius of curvature of the mirror, use the relationship between the focal length and the radius of curvature: $R\; =\; 2f$, where $R$ is the radius of curvature and $f$ is the focal length. Substitute the value of $f$ from the previous step to find $R$.
5. Verify the results by checking the consistency of the calculations. Ensure that the image distance, focal length, and radius of curvature satisfy the mirror equation and the magnification condition. This step ensures the solution is physically meaningful and accurate.

Key Concepts

Types of Mirrors

There are two main types of spherical mirrors: concave and convex. Concave mirrors can produce real or virtual images depending on the object's position relative to the focal point, while convex mirrors always produce virtual images that are upright and smaller than the object. Understanding the type of mirror is crucial for determining the characteristics of the image formed.

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror: 1/f = 1/v + 1/u. This formula is essential for calculating the position of the image and the focal length of the mirror. It helps in understanding how the distances are interrelated and how they affect image formation.

Magnification

Magnification (m) is the ratio of the height of the image (h') to the height of the object (h), given by m = h'/h = -v/u. It indicates how much larger or smaller the image is compared to the object. For virtual images, magnification is positive, and understanding this concept is key to determining the size and orientation of the image produced by the mirror.