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Chapter 34: Problem 4

A concave mirror has a radius of curvature of \(34.0 \mathrm{~cm}\). (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
The focal length of the concave mirror is \(17.0 \mathrm{~cm}\) in air and \(12.8 \mathrm{~cm}\) when it is immersed in water.

Step by step solution

01

Find the initial focal length

For a mirror, whether it is concave or convex, the radius of curvature \(R\) is twice the focal length \(f\). Hence, we can find the focal length by dividing the radius of curvature by 2. In this case, the radius of curvature \(R\) is given as \(34.0 \mathrm{~cm}\). Therefore, we can calculate the focal length \(f\) as \(f = R/2 = 34.0 \mathrm{~cm} / 2 = 17.0 \mathrm{~cm}\).
02

Calculate the new focal length in water

When a mirror is immersed in a medium other than air, its focal length changes. We can calculate the new focal length \(f^{'}\) using the formula \(f^{'} = f/n\), where \(n\) is the refractive index of the medium. In this case, the refractive index of water is given as \(n = 1.33\). Therefore, the new focal length in water \(f^{'}\) is \(f^{'} = 17.0 \mathrm{~cm} / 1.33 = 12.8 \mathrm{~cm}\).

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Key Concepts

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

Radius of Curvature
In the realm of optics, the radius of curvature is a fundamental concept that signifies the radius of an imaginary circle that has the same curvature as a given optical surface at a specific point. For mirrors, this is particularly important as it defines how parallel rays will converge after reflection.

For concave mirrors, such as in the exercise under consideration, the radius of curvature is essentially twice the distance from the mirror's surface to its focal point鈥攖hat is, the point where light rays converge after reflection. The relationship is mathematically expressed as \( R = 2f \) , where \( R \) is the radius of curvature, and \( f \) is the focal length. Understanding this relationship is crucial for answering part (a) of the exercise, where we directly use it to deduce the focal length from the given radius of curvature.
Optics in Different Media
The interaction of light with different media is a key aspect of optics. When light passes through different media or reflects off surfaces in different media, its behavior changes. Optical phenomena such as reflection, refraction, and dispersion are influenced by the medium in which they occur.

In the given exercise, we see an example of this with a concave mirror being immersed in water. The presence of water, a medium with a different density than air, modifies how the light is reflected by the mirror. This immersion doesn't change the physical curvature of the mirror but changes the effective focal length of the system鈥攖his is because the speed of light and its interaction with the mirror鈥檚 surface are altered due to the optical density of the new medium.
Refractive Index
The refractive index, denoted as \( n \) , is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium: \( n = c / v \) , where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium. The refractive index determines how much the path of light is bent, or refracted, when entering a material.

In part (b) of our exercise, this concept is used to calculate the new focal length of the concave mirror when immersed in water. Water's refractive index (1.33) indicates that light travels slower in water than in air. By applying the refractive index to the mirror's focal length in air, we can find the focal length in the new medium, highlighting the refractive index's influence on optical systems.

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Most popular questions from this chapter

The radii of curvature of the surfaces of a thin converging meniscus lens are \(R_{1}=+12.0 \mathrm{~cm}\) and \(R_{2}=+28.0 \mathrm{~cm} .\) The index of refraction is \(1.60 .\) (a) Compute the position and size of the image of an object in the form of an arrow \(5.00 \mathrm{~mm}\) tall, perpendicular to the lens axis, \(45.0 \mathrm{~cm}\) to the left of the lens. (b) A second converging lens with the same focal length is placed \(3.15 \mathrm{~m}\) to the right of the first. Find the position and size of the final image. Is the final image erect or inverted with respect to the original object? (c) Repeat part (b) except with the second lens \(45.0 \mathrm{~cm}\) to the right of the first.

A small tropical fish is at the center of a water-filled, spherical fish bowl \(28.0 \mathrm{~cm}\) in diameter. (a) Find the apparent position and magnification of the fish to an observer outside the bowl. The effect of the thin walls of the bowl may be ignored. (b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl?

A converging lens with a focal length of \(9.00 \mathrm{~cm}\) forms an image of a \(4.00-\mathrm{mm}\) -tall real object that is to the left of the lens. The image is \(1.30 \mathrm{~cm}\) tall and erect. Where are the object and image located? Is the image real or virtual?

The focal length of a simple magnifier is \(8.00 \mathrm{~cm}\). Assume the magnifier is a thin lens placed very close to the eye. (a) How far in front of the magnifier should an object be placed if the image is formed at the observer's near point, \(25.0 \mathrm{~cm}\) in front of her eye? (b) If the object is \(1.00 \mathrm{~mm}\) high, what is the height of its image formed by the magnifier?

A converging lens with a focal length of \(12.0 \mathrm{~cm}\) forms a virtual image \(8.00 \mathrm{~mm}\) tall, \(17.0 \mathrm{~cm}\) to the right of the lens. Determine the position and size of the object. Is the image erect or inverted? Are the object and image on the same side or opposite sides of the lens? Draw a principal-ray diagram for this situation.
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