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Chapter 26: Problem 123

The angular magnification of a telescope is 32800 times as large when you look through the correct end of the telescope as when you look through the wrong end. What is the angular magnification of the telescope?

Short Answer

Expert verified
The angular magnification of the telescope is approximately 181.

Step by step solution

01

Understanding Angular Magnification

Angular magnification (M) of a telescope is defined as the ratio of the angle subtended by the image seen through the telescope to the angle subtended by the object when observed with the naked eye. It is represented as:\[M = \frac{f_{obj}}{f_{eye}}\]where \(f_{obj}\) and \(f_{eye}\) are the focal lengths of the objective lens and the eyepiece, respectively.
02

Identifying Magnification Ratios

When looking through the wrong end, the telescope acts inversely, and the magnification is the reciprocal of looking through the correct end. Let \(M_c\) be the magnification through the correct end and \(M_w\) through the wrong end, then:\[M_c = 32800 \times M_w\]And, \(M_w = \frac{1}{M_c}\), thus:\[M_c = 32800 \times \frac{1}{M_c}\]
03

Setting Up the Equation

Replace \(M_w\) with \(\frac{1}{M_c}\) in the equation obtained from step 2:\[M_c^2 = 32800\]
04

Solving for Angular Magnification

To find \(M_c\), take the square root of both sides:\[M_c = \sqrt{32800}\]Using a calculator,\[M_c 鈮� 181\]
05

Conclusion

The angular magnification of the telescope when looking through the correct end is approximately 181 times.

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Key Concepts

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

Focal Length
Understanding the focal length is crucial when studying telescopes. The focal length of a lens is the distance between the lens and the point where light rays parallel to the optical axis converge, known as the focus.
In the context of telescopes, two focal lengths are critical: the focal length of the objective lens and the focal length of the eyepiece. Each plays a unique role in determining the overall behavior of the telescope.
  • Focal Length of the Objective Lens ( f_{obj}) - This is usually much longer and is responsible for gathering light from a distant object, providing initial magnification.
  • Focal Length of the Eyepiece ( f_{eye}) - This lens magnifies the image formed by the objective lens. It typically has a shorter focal length.

By adjusting the focal lengths, astronomers can enhance the magnification and resolution capabilities of the telescope. Longer focal lengths generally provide greater magnification by reducing the field of view.
Objective Lens
The objective lens is one of the most vital components of a telescope. This lens (or mirror, in the case of a reflecting telescope) is responsible for collecting light from distant objects and forming an image. Its main characteristics and functions include:
  • Collecting Light - Allows distant celestial bodies to be seen clearly by gathering as much light as possible.
  • Image Formation - Forms a real image of a distant object at its focal point.
  • Large Size - The objective lens is usually larger to ensure that more light can be captured, improving the clarity and brightness of the image.

The larger the diameter of the objective lens, the more light it can gather, which becomes especially important when observing faint objects. The design and quality of an objective lens affect both the resolution and the sharpness of the images that the telescope can produce.
Eyepiece
The eyepiece is another critical part of a telescope. It plays a pivotal role in magnifying the image produced by the objective lens, making it easier to see with the human eye. Here are some of its key roles and features:
  • Magnification - By using the eyepiece, the small real image created by the objective lens can be magnified to fit the eye's perspective.
  • Focal Length - The focal length of the eyepiece is selected to provide the desired level of magnification in combination with the objective lens.
  • Field of View - A smaller eyepiece focal length reduces the field of view but increases magnification, making it suitable for detailed observations.

Various eyepieces can be interchanged to alter the magnification levels, offering flexibility depending on observational needs.
Correct and Wrong End Observation
Observing through the correct and wrong end of a telescope can dramatically change the experience. When you observe objects through the correct end, the designed magnification shows a detailed and enhanced view of distant objects.
If you mistakenly observe through the wrong end, the telescope acts inversely, significantly minimizing the image rather than magnifying it. Here's how it impacts the observation:
  • Correct End - Provides actual magnification, leveraging the objective lens to focus and the eyepiece to enlarge the image. For example, a telescope with a magnification of 181 would make an object appear 181 times larger than when seen with the naked eye.
  • Wrong End - Acts like a reversed telescope, leading to a very tiny image due to the reciprocal magnification effect. Thus, it shows objects much smaller, as evidenced by an extreme difference in magnification ratios (like the 32800 times reversal mentioned in the exercise).

Understanding how each component works together is essential for proper observatory practices and striving for higher image accuracy and clarity.

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Most popular questions from this chapter

A converging lens has a focal length of \(88.00 \mathrm{cm} .\) An object \(13.0 \mathrm{cm}\) tall is located \(155.0 \mathrm{cm}\) in front of this lens. (a) What is the image distance? (b) Is the image real or virtual? (c) What is the image height? Be sure to include the proper algebraic sign.

The moon's diameter is \(3.48 \times 10^{6} \mathrm{m},\) and its mean distance from the earth is \(3.85 \times 10^{8} \mathrm{m}\). The moon is being photographed by a camera whose lens has a focal length of \(50.0 \mathrm{mm}\). (a) Find the diameter of the moon's image on the slide film. (b) When the slide is projected onto a screen that is \(15.0 \mathrm{m}\) from the lens of the projector \((f=110.0 \mathrm{mm})\) what is the diameter of the moon's image on the screen?

A dentist is examining a dental filling in a patient's tooth. The diameter of the filling is \(2.4 \mathrm{mm},\) and the dentist's near point is \(17.0 \mathrm{cm}\) To get a better look at the filling, the dentist dons safety goggles fitted with magnifying glasses \((f=6.0 \mathrm{cm}) .\) Find the greatest possible angular size (in radians) of the patient's filling when viewed by the dentist, both (a) without and (b) with the magnifying glasses.

In an ultra-low-temperature experiment, a collection of sodium atoms enter a special state called a Bose-Einstein condensate in which the index of refraction is \(1.57 \times 10^{7} .\) What is the speed of light in this condensate?

A plate glass window \((n=1.5)\) has a thickness of \(4.0 \times 10^{-3} \mathrm{m}\). How long does it take light to pass perpendicularly through the plate?
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