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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

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \( 17.0 \text{ cm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A transparent liquid fills a cylindrical tank to a depth of 3.60 m. There is air above the liquid. You look at normal incidence at a small pebble at the bottom of the tank. The apparent depth of the pebble below the liquid's surface is 2.45 m. What is the refractive index of this liquid?

A convex spherical mirror with a focal length of magnitude 24.0 cm is placed 20.0 cm to the left of a plane mirror. An object 0.250 cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closest to the spherical mirror, and how tall is each image?

The focal length of the eyepiece of a certain microscope is 18.0 mm. The focal length of the objective is 8.00 mm. The distance between objective and eyepiece is 19.7 cm. The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

A lensmaker wants to make a magnifying glass from glass that has an index of refraction \(n\) = 1.55 and a focal length of 20.0 cm. If the two surfaces of the lens are to have equal radii, what should that radius be?

An object is 16.0 cm to the left of a lens. The lens forms an image 36.0 cm to the right of the lens. (a) What is the focal length of the lens? Is the lens converging or diverging? (b) If the object is 8.00 mm tall, how tall is the image? Is it erect or inverted? (c) Draw a principal-ray diagram.

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