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Chapter 24: Problem 57

You have two lenses of focal length \(25.0 \mathrm{cm}\) (lens 1 ) and $5.0 \mathrm{cm}$ (lens 2 ). (a) To build an astronomical telescope that gives an angular magnification of \(5.0,\) how should you use the lenses (which for objective and which for eyepiece)? Explain. (b) How far apart should they be?

Short Answer

Expert verified
Short Answer: To achieve the desired angular magnification of 5, lens 1 (focal length: 25 cm) should be used as the objective lens and lens 2 (focal length: 5 cm) should be used as the eyepiece lens. The lenses should be placed 30 cm apart.

Step by step solution

01

Identify the formula for angular magnification

For a telescope, the angular magnification (M) is equal to the ratio of the focal length of the objective lens (F_obj) to the focal length of the eyepiece lens (F_eye): M = F_obj / F_eye We are given the angular magnification (M) and the focal lengths of the two lenses (lens 1 and lens 2). Our task is to arrange these lenses to act as the objective and eyepiece to achieve the desired magnification.
02

Calculate the possible combinations

First, let's consider the two possible ways to arrange the lenses and see which combination gives the desired magnification. 1. Lens 1 as objective and Lens 2 as eyepiece: M = F_obj / F_eye = 25 cm / 5 cm = 5 2. Lens 2 as the objective and Lens 1 as the eyepiece: M = F_obj / F_eye = 5 cm / 25 cm = 1/5 Since the desired magnification is 5, the first combination (Lens 1 as objective and Lens 2 as eyepiece) is the correct one.
03

Answer part (a)

To build an astronomical telescope with an angular magnification of 5, lens 1 (focal length: 25 cm) should be the objective lens, and lens 2 (focal length: 5 cm) should be the eyepiece lens.
04

Calculate lens separation

To find the distance between the lenses, we will use the lensmaker's equation for a system of two lenses. The equivalent focal length (F) of the two-lens system is given by the formula 1/F = 1/F_obj + 1/F_eye. The distance (d) between the lenses is the sum of their respective focal lengths. First, let's find the equivalent focal length (F): 1/F = 1/25 cm + 1/5 cm = 6/25 cm F = 25 cm / 6 = 4.17 cm Now that we have the equivalent focal length, we can calculate the distance between the lenses (d) by summing their respective focal lengths: d = F_obj + F_eye = 25 cm + 5 cm = 30 cm
05

Answer part (b)

To build the astronomical telescope with the given lenses and desired angular magnification of 5, the lenses should be placed 30 cm apart.

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Most popular questions from this chapter

A camera with a 50.0 -mm lens can focus on objects located from $1.5 \mathrm{m}$ to an infinite distance away by adjusting the distance between the lens and the film. When the focus is changed from that for a distant mountain range to that for a flower bed at \(1.5 \mathrm{m},\) how far does the lens move with respect to the film?
A statue is \(6.6 \mathrm{m}\) from the opening of a pinhole camera, and the screen is \(2.8 \mathrm{m}\) from the pinhole. (a) Is the image erect or inverted? (b) What is the magnification of the image? (c) To get a brighter image, we enlarge the pinhole to let more light through, but then the image looks blurry. Why? (d) To admit more light and still have a sharp image, we replace the pinhole with a lens. Should it be a converging or diverging lens? Why? (e) What should the focal length of the lens be?
An object is placed \(12.0 \mathrm{cm}\) in front of a lens of focal length $5.0 \mathrm{cm} .\( Another lens of focal length \)4.0 \mathrm{cm}\( is placed \)2.0 \mathrm{cm}$ past the first lens. (a) Where is the final image? Is it real or virtual? (b) What is the overall magnification? (interactive: virtual optics lab).
Callum is examining a square stamp of side \(3.00 \mathrm{cm}\) with a magnifying glass of refractive power \(+40.0 \mathrm{D}\). The magnifier forms an image of the stamp at a distance of \(25.0 \mathrm{cm} .\) Assume that Callum's eye is close to the magnifying glass. (a) What is the distance between the stamp and the magnifier? (b) What is the angular magnification? (c) How large is the image formed by the magnifier?
The Ortiz family is viewing slides from their summer vacation trip to the Grand Canyon. Their slide projector has a projection lens of \(10.0-\mathrm{cm}\) focal length and the screen is located \(2.5 \mathrm{m}\) from the projector. (a) What is the distance between the slide and the projection lens? (b) What is the magnification of the image? (c) How wide is the image of a slide of width \(36 \mathrm{mm}\) on the screen? (See the figure with Problem 16 .)
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