Problem 25 - University Physics with Modern Physics

Question

The radius of curvature of a convex mirror is \(-25 \mathrm{~cm} .\) What is its focal length?

Step-by-Step Solution

Answer

Answer: The focal length of the convex mirror is -12.5 cm.

Step 1: Write down the mirror equation for focal length

The mirror equation for the focal length (f) of a mirror in terms of its radius of curvature (R) is given by: f = R / 2

Step 2: Identify the radius of curvature

The problem tells us that the radius of curvature of the convex mirror is -25 cm. So, we have: R = -25 cm

Step 3: Calculate the focal length

Substituting the given radius of curvature into the mirror equation: f = (-25 cm) / 2 f = -12.5 cm

Step 4: State the answer

The focal length of the convex mirror is -12.5 cm.

Key Concepts

Mirror Equation

The mirror equation is a fundamental concept in optics that relates the focal length of a mirror to its radius of curvature. This equation can be expressed as:

• \( f = \frac{R}{2} \)

In this formula, \( f \) represents the focal length and \( R \) is the radius of curvature. The radius of curvature is the distance from the mirror's surface to the center of curvature, which is the point on the optical axis opposite to the focus. Each mirror, whether concave or convex, adheres to this formula.
Convex mirrors, like the one in the exercise, have a negative radius of curvature and focal length due to their shape. This results in diverging reflected light rays, creating virtual images. By using the mirror equation, we can easily determine the focal length if we know the radius of curvature. This is crucial for understanding how mirrors form images.

Radius of Curvature

The radius of curvature is an essential parameter in determining a mirror's optical properties. It is defined as the distance from the mirror's surface to its center of curvature, where all reflected light beams seemingly originate or converge. For convex mirrors, this radius is noted as negative because such mirrors diverge light.
In our exercise, the convex mirror has a radius of curvature of \(-25 \text{ cm}\). When dealing with optics problems, the sign of the radius is critical since it affects the calculation of the focal length and the nature of the image formed. For concave mirrors, this sign would be positive. Understanding the radius of curvature is central to mastering mirror equations and predicting how light will behave around reflective surfaces.

Optics

Optics is the area of physics that studies the behavior and properties of light. It covers various phenomena, including reflection, refraction, and diffraction. Convex mirrors, central to this field, are strong examples of optical devices designed to manipulate light paths. They reflect light outwards, creating a virtual image that appears smaller and further away than actual size.
Key principles in optics aiding in understanding convex mirrors include:

• Reflection: Light reflecting off the surface of a convex mirror follows the law of reflection, whereby the angle of incidence equals the angle of reflection.
• Virtual Images: Convex mirrors produce virtual images, as the reflected rays appear to originate from a focal point behind the mirror.

Understanding these concepts in optics helps explain why convex mirrors are used in various applications, such as vehicles' side mirrors and security in stores, where wide-angle visibility is essential.