Question

A concave mirror has a radius of curvature of 34.0 cm. (a) What is its focal length? (b) If the mirror is immersed in water (refractive index 1.33), what is its focal length?

Short answer

(a) The focal length is 17.0 cm. (b) In water, it remains 17.0 cm.

Step-by-step solution

Understand the Relation Between Radius of Curvature and Focal Length

The focal length \( f \) of a mirror is half of its radius of curvature \( R \). This is given by the formula \( f = \frac{R}{2} \). In this case, we need to calculate the focal length of a concave mirror with a radius of curvature of \( 34.0 \) cm.

Calculate the Focal Length in Air

Using the formula from Step 1, we calculate the focal length: \( f = \frac{34.0 \text{ cm}}{2} = 17.0 \text{ cm} \). This is the focal length of the mirror in air.

Consider Effect of Water on Focal Length

The focal length of a mirror does not change with the medium in which it is placed. This is because the focal length of a mirror is determined by its shape, not by the medium. Therefore, even when the mirror is immersed in water, its focal length remains unchanged.

Confirm the Focal Length in Water

Since the focal length of a mirror is independent of the medium, the focal length of the concave mirror when immersed in water remains \( 17.0 \text{ cm} \).

Key concepts

Focal Length

The focal length is a key concept in optics, representing the distance between the mirror's surface and its focal point - the point where parallel rays of light converge after reflection. For a concave mirror, which curves inward, understanding how to calculate focal length is essential.
The formula to determine the focal length (\( f \)) of a concave mirror is simple: it is half the radius of curvature (\( R \)). This can be expressed as \( f = \frac{R}{2} \).
In our exercise, with a radius of curvature of 34.0 cm, the calculation becomes straightforward: \( f = \frac{34.0 \text{ cm}}{2} = 17.0 \text{ cm} \).
This derived focal length of 17.0 cm is vital for understanding how light behaves when it hits the mirror. It tells us where light converges, helping us predict image formation accurately.

Radius of Curvature

The radius of curvature is the radius of the imaginary sphere from which a mirror segment is "cut". It is fundamental in defining the mirror's shape and impacts how it directs light.
For concave mirrors, this involves understanding that the larger the radius, the more "spread out" the curved surface is, affecting the focal length. In our problem, we had a radius of 34.0 cm. This determines how strongly the mirror focuses light.
To relate this to focal length, remember that the radius of curvature is twice the focal length for spherical mirrors. Thus, it directly influences optical behavior. With more curvature (smaller radius), light is focused more sharply, leading to a smaller focal length.
This relationship is essential for students of physics education to understand how mirrors work, set the foundation for exploring more complex optical systems.

Refractive Index

The refractive index is crucial in optics, measuring how much the speed of light is reduced inside a medium compared to a vacuum. It affects how light refracts but has interesting interactions with mirrors.
When dealing with lenses, the refractive index significantly impacts focal length. However, with mirrors, the medium's refractive index doesn't alter the focal length.
In this case, even when the concave mirror is immersed in water (refractive index of 1.33), its focal length remains 17.0 cm. This is because mirrors reflect light rather than refract it, making their behavior independent of the surrounding medium. Understanding this concept emphasizes the unique properties of mirrors in comparison to lenses in physics education.

Physics Education

Physics education is fundamental for making sense of the vast world of optics. It bridges theoretical concepts and practical applications, making understanding tools like concave mirrors approachable.
Through the study of focal length, radius of curvature, and refractive index, students gain insight into both simple and complex optical systems.

• Understanding concave mirrors lays the groundwork for grasping more advanced concepts in optical engineering.
• Real-world applications, from telescopes to solar concentrators, rely on the foundational physics taught in school.

By connecting these fundamental physics concepts to real-life examples, educators can inspire curiosity and innovation amongst students. This holistic approach equips students with the knowledge to explore and innovate within the wider field of physics.