Ch 35: Optical Instruments

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 35: Optical Instruments

Problem 37a

Chapter 35, Problem 37a

Marooned on a desert island and with a lot of time on your hands, you decide to disassemble your glasses to make a crude telescope with which you can scan the horizon for rescuers. Luckily you’re farsighted, and, like most people, your two eyes have different lens prescriptions. Your left eye uses a lens of power +4.5 D, and your right eye’s lens is +3.0 D. Which lens should you use for the objective and which for the eyepiece? Explain.

Verified step by step guidance

Step 1: Understand the purpose of the telescope. A telescope consists of two lenses: the objective lens, which gathers light from distant objects and forms an image, and the eyepiece lens, which magnifies that image for viewing.

Step 2: Recall the properties of lens power. Lens power (measured in diopters, D) is the reciprocal of the focal length (in meters). A lens with a higher power has a shorter focal length, while a lens with a lower power has a longer focal length. The focal length is given by \( f = \frac{1}{P} \), where \( P \) is the lens power.

Step 3: Determine the roles of the lenses in the telescope. The objective lens should have a longer focal length to gather light from distant objects, while the eyepiece lens should have a shorter focal length to magnify the image.

Step 4: Compare the powers of the two lenses. The left eye's lens has a power of +4.5 D, corresponding to a shorter focal length, while the right eye's lens has a power of +3.0 D, corresponding to a longer focal length.

Step 5: Assign the lenses to their roles. Use the right eye's lens (+3.0 D) as the objective lens because it has a longer focal length, and use the left eye's lens (+4.5 D) as the eyepiece because it has a shorter focal length, which is better for magnification.

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Key Concepts

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

Lens Power and Focal Length

The power of a lens, measured in diopters (D), is the reciprocal of its focal length in meters. A positive power indicates a converging lens, which brings light rays together. In this scenario, the left eye's lens has a higher power (+4.5 D) than the right eye's lens (+3.0 D), meaning it has a shorter focal length and is more suitable for focusing light from distant objects.

Telescope Configuration

A telescope typically consists of two lenses: the objective lens, which gathers light and forms an image, and the eyepiece lens, which magnifies that image for viewing. For optimal performance, the objective lens should have a longer focal length than the eyepiece lens. In this case, the lens with +4.5 D should be used as the objective, while the +3.0 D lens serves as the eyepiece.

Magnification in Telescopes

The magnification of a telescope is determined by the ratio of the focal lengths of the objective lens to the eyepiece lens. This relationship allows the telescope to enlarge the image formed by the objective lens. By using the +4.5 D lens as the objective and the +3.0 D lens as the eyepiece, the system will provide a suitable magnification for viewing distant objects, enhancing the chances of spotting rescuers.

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Related Practice

Textbook Question

A microscope with a tube length of 180 mm achieves a total magnification of 800x with a 40x objective and a 20x eyepiece. The microscope is focused for viewing with a relaxed eye. Approximately how far is the sample from the objective lens?

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Textbook Question

A simple and relatively inexpensive microscope eyepiece is the Ramsden eyepiece shown in FIGURE P35.40. Two plano-convex lenses have their curved surfaces facing each other, which a more advanced analysis shows is the orientation that minimizes spherical aberration. That same analysis finds that chromatic aberration is minimized with lens spacing L = 1/2 (f₁ + f₂). Your task is to design a 10x Ramsden eyepiece in which the first lens has a focal length of 30 mm. What are (a) the focal length and (b) the spacing of the second lens?

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Textbook Question

Mars (6800 km diameter) is viewed through a telescope on a night when it is 1.1 x 10⁸ km from the earth. Its angular size as seen through the eyepiece is 0.50°, the same size as the full moon seen by the naked eye. If the eyepiece focal length is 25 mm, how long is the telescope?

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Textbook Question

The cornea, a boundary between the air and the aqueous humor, has a 3.0 cm focal length when acting alone. What is its radius of curvature?

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Textbook Question

A 15-cm-focal-length converging lens is 20 cm to the right of a 7.0-cm-focal-length converging lens. A 1.0-cm-tall object is distance L to the left of the 7.0-cm-focal-length lens. What are the height and orientation of the final image?

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Textbook Question

Modern microscopes are more likely to use a camera than human viewing. This is accomplished by replacing the eyepiece in Figure 35.14 with a photo-ocular that focuses the of the objective to a real on the sensor of a digital camera. Suppose the sensor is 22.5 mm wide, a typical value, with 4.0 μm x 4.0 μm pixels. The photo of a cell is 120 pixels in diameter. What is the cell’s actual diameter, in μm?

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