What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person's face is \(12.0 \mathrm{cm}\) away? Explicitly show how you follow the steps in the Problem-Solving Strategy: Spherical Mirrors.

Calculating focal length plays a crucial role in understanding how spherical mirrors affect light. In the context of a makeup mirror and our given problem, it's the distance from the mirror to the point where parallel light rays converge or appear to diverge from. When a person's face is placed at a distance, the mirror shifts the reflected rays to create a magnified image.

To calculate the focal length, we use the image distance, which can be derived from the magnification formula. With the object distance and magnification known, focal length emerges from a simple algebraic manipulation of the mirror formula, \( 1/f = 1/do + 1/di \). This equation balances the reciprocal of the focal length with the sum of the reciprocals of the object and image distances, ultimately guiding us to solve for the unknown focal length.

The mirror formula is a foundational equation in the study of spherical mirrors physics. It provides a relation among focal length (f), object distance (do), and image distance (di) and is given by \( 1/f = 1/do + 1/di \).

This equation helps determine the location of the image formed by the mirror. It is applicable to both concave and convex mirrors, and you must consider the sign convention to obtain the correct value. In a step-by-step approach like in our example, first clear out what information is given and what is unknown, and then rearrange the mirror formula accordingly to find the missing variable.

The magnification formula is significant when dealing with images produced by mirrors. It is the ratio of the height of the image to the height of the object, and in terms of distances, it is given by \( m = -di/do \).

Magnification (m) gives us a sense of how much larger or smaller the image is compared to the object. In our exercise, a magnification of 1.50 indicates that the image is 1.5 times larger than the object. The negative sign in the formula signifies the nature and orientation of the image; a negative magnification indicates a real, inverted image, while a positive value would indicate a virtual, upright image.

A concave mirror is a spherical mirror that curves inward, resembling a portion of the interior of a sphere. These mirrors can produce both real and virtual images based on the distance between the object and the mirror.

In scenarios like our exercise, where a real image is formed, the light rays actually converge at a point. Concave mirrors are used in a variety of applications, including makeup mirrors, telescopes, and headlight reflectors, due to their ability to focus light and produce magnified images. These mirrors are fundamental in optical physics and practical instruments, emphasizing their importance in both theoretical and real-world contexts.

In the realm of physics, a real image is an image that is formed where rays of light actually intersect. In contrast with virtual images, a real image can be projected onto a screen or surface. In our mentioned problem, the negative value of image distance (di) signifies that the image formed by the concave mirror is real.

Real images are inverted with respect to the object, which is an essential aspect for understanding the behavior of concave mirrors. Whether an image is real or virtual has significant implications on how we apply mirrors in practical situations, from the design of optical devices to everyday use mirrors.