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Ch 34: Ray Optics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 34: Ray OpticsProblem 64
Chapter 34, Problem 64

BIO A keratometer is an optical device used to measure the radius of curvature of the eye's cornea—its entrance surface. This measurement is especially important when fitting contact lenses, which must match the cornea's curvature. Most light incident on the eye is transmitted into the eye, but some light reflects from the cornea, which, due to its curvature, acts like a convex mirror. The keratometer places a small, illuminated ring of known diameter 7.5 cm in front of the eye. The optometrist, using an eyepiece, looks through the center of this ring and sees a small virtual image of the ring that appears to be behind the cornea. The optometrist uses a scale inside the eyepiece to measure the diameter of the image and calculate its magnification. Suppose the optometrist finds that the magnification for one patient is 0.049. What is the absolute value of the radius of curvature of her cornea?

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Understand the problem: The cornea acts as a convex mirror, and the keratometer measures the magnification of the reflected image of a ring. The goal is to calculate the radius of curvature of the cornea using the given magnification and the diameter of the ring.
Recall the magnification formula for a mirror: \( M = \frac{-q}{p} \), where \( M \) is the magnification, \( q \) is the image distance (virtual image distance for a convex mirror), and \( p \) is the object distance (distance from the ring to the cornea). Rearrange this formula to find \( q \): \( q = -M \cdot p \).
Use the mirror equation: \( \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \), where \( f \) is the focal length of the mirror. Substitute \( q = -M \cdot p \) into the equation to express \( f \) in terms of \( p \) and \( M \).
Recall the relationship between the focal length \( f \) and the radius of curvature \( R \) for a mirror: \( R = 2f \). Use this relationship to calculate \( R \) once \( f \) is determined.
Substitute the given values: \( M = 0.049 \) and \( p = 7.5 \, \text{cm} \) into the equations derived in the previous steps to calculate the absolute value of \( R \). Ensure all units are consistent during the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radius of Curvature

The radius of curvature is a measure of how sharply a surface curves. In the context of the eye's cornea, it refers to the radius of the spherical shape that approximates the cornea's surface. A smaller radius indicates a steeper curve, while a larger radius indicates a flatter surface. This measurement is crucial for applications like contact lens fitting, as it ensures that the lens conforms properly to the cornea.
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Magnification

Magnification is the process of enlarging the appearance of an object through optical means. In this scenario, it refers to the ratio of the size of the virtual image seen through the keratometer to the actual size of the illuminated ring. The magnification value helps the optometrist determine the cornea's curvature by relating the observed image size to the known dimensions of the ring, allowing for precise calculations.
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Optical Devices and Virtual Images

Optical devices, like the keratometer, manipulate light to create images that can be observed. A virtual image is formed when light rays appear to diverge from a point behind the optical device, making it seem as if the image is located at that point. Understanding how virtual images are formed is essential for interpreting measurements taken with optical instruments, as it directly influences the calculations needed to determine physical properties like the radius of curvature.
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Example 1
Related Practice
Textbook Question

A 25-cm-long rod lies along the optical axis of a converging lens, perpendicular to the lens plane. The lens has a 30 cm focal length. The rod's real , along the optical axis on the other side of the lens, is also 25 cm long. What is the distance from the lens to the nearest end of the rod?

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Textbook Question

BIO A dentist uses a curved mirror to view the back side of teeth in the upper jaw. Suppose she wants an upright image with a magnification of 1.5 when the mirror is 1.2 cm from a tooth. Should she use a convex or a concave mirror? What focal length should it have?

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Textbook Question

An old-fashioned slide projector needs to create a 98-cm-high of a 2.0-cm-tall slide. The screen is 300 cm from the slide. What focal length does the lens need? Assume that it is a thin lens.

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Textbook Question

You're visiting the shark tank at the aquarium when you see a 2.5-m-long shark that appears to be swimming straight toward you at 2.0 m/s. What is the shark's actual speed through the water? You can ignore the glass wall of the tank.

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Textbook Question

A 2.0-cm-tall candle flame is 2.0 m from a wall. You happen to have a lens with a focal length of 32 cm. How many places can you put the lens to form a well-focused image of the candle flame on the wall? For each location, what are the height and orientation of the image?

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Textbook Question

Paraxial light rays approach a transparent sphere parallel to an optical axis passing through the center of the sphere. The rays come to a focus on the far surface of the sphere. What is the sphere's index of refraction?

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