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Chapter 34: Problem 60

A Cassegrain telescope is a refiecting telescope that uses two mirrors, the secondary mirror focusing the image through a hole in the primary mirror (similar to that shown in Fig. 34.55 ). You wish to focus the image of a distant galaxy onto the detector shown in the figure. If the primary mirror has a focal length of \(2.5 \mathrm{m},\) the secondary mirror has a focal length of \(-1.5 \mathrm{m}\) and the distance from the vertex of the primary mirror to the detector is 15 \(\mathrm{cm}\) . What should be the distance between the vertices of the two mirrors?

Short Answer

Expert verified
The distance between the vertices of the two mirrors is 2.35 m.

Step by step solution

01

Identify the given information

We are given the focal lengths of the primary mirror \(f_1 = 2.5 \text{ m}\) and the secondary mirror \(f_2 = -1.5 \text{ m}\). The distance from the primary mirror vertex to the detector is \(d = 0.15 \text{ m}\). We are to find the distance between the vertices of the two mirrors \(d_{12}\).
02

Determine effective focal length equation

The effective focal length \(F\) of the Cassegrain telescope is given by the formula:\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d_{12}}{f_1 f_2} \] where \(f_1\) and \(f_2\) are the focal lengths of the primary and secondary mirrors.
03

Evaluate effective focal length

Since the image is focused on the detector at the focus of the system, the effective focal length \(F\) equals \(f_1 = 2.5\; \text{m}\). Thus:\[ \frac{1}{F} = \frac{1}{2.5} = 0.4 \]
04

Substitute and solve for mirror distance

Substitute the known values into the effective focal length equation to find \(d_{12}\). Plug in the values:\[ \frac{1}{2.5} = \frac{1}{2.5} + \frac{1}{-1.5} - \frac{d_{12}}{2.5 \times (-1.5)} \]This simplifies to:\[ 0 = -0.6667 + \frac{d_{12}}{3.75} \]\[ 0.6667 \cdot 3.75 = d_{12} \]Solving gives \(d_{12} = 2.5 \text{ m}\).
05

Calculate final distance to mirror

The actual distance from the primary mirror vertex to the secondary mirror should account for the position of the detector at \(15\; \text{cm}\). Thus:\[ d_{12} = 2.5 - 0.15 = 2.35 \text{ m} \]

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Key Concepts

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

Reflecting Telescopes
Reflecting telescopes are powerful tools in astronomy, using mirrors to gather and focus light. They differ from refracting telescopes, which use lenses to bend light. Reflecting telescopes offer several advantages:
  • Less optical distortion: Mirrored surfaces avoid chromatic aberration, which occurs in lenses.
  • Larger diameter: Mirrors can be made larger than lenses, allowing for more light collection.
  • More compact design: Reflecting telescopes can be shorter for the same aperture size compared to refractors.
This makes reflecting telescopes ideal for observing faint and distant astronomical objects. The Cassegrain telescope is a type of reflecting telescope that uses two curved mirrors to focus light through a small hole in the primary mirror, making it a popular choice for both professional and amateur astronomers.
Focal Length
Focal length is a crucial aspect of optical systems, representing the distance between the mirror or lens and its focus point. It's a measure of how strongly the optical device converges or diverges light. For mirrors in a telescope:
  • The primary mirror has a positive focal length, gathering light and starting the initial focusing process.
  • The secondary mirror typically has a negative focal length, further refining the focus and directing light to a focal point.
In the Cassegrain telescope design, the focal lengths of both the primary and secondary mirrors work in tandem to achieve a precise focal point on the detector. Understanding these focal lengths helps astronomers determine the magnification and resolution capabilities of their telescopes.
Optical Systems
Optical systems in telescopes are designed to manipulate light to produce a clear and detailed image. These systems consist of various components like mirrors and lenses that work together:
  • Primary mirror: Gathers light from distant objects.
  • Secondary mirror: Redirects light to the eyepiece or detector, allowing for a compact and efficient design.
  • Detector or eyepiece: Captures or magnifies the focused image for observation or recording.
The effectiveness of an optical system is determined by how well it balances the different focal lengths and mirror placements to reduce optical aberrations. Cassegrain telescopes exemplify sophisticated optical systems, providing high-quality images suitable for a range of astronomical applications.
Astronomy
Astronomy is the scientific study of celestial objects, such as stars, planets, and galaxies. Telescopes have revolutionized this field by providing detailed observations beyond the capabilities of the naked eye. In astronomy, telescopes help to:
  • Observe distant galaxies and stars.
  • Study planetary systems and celestial phenomena.
  • Discover new astronomical objects and phenomena.
Instruments like the Cassegrain telescope have enabled astronomers to explore the universe with greater clarity and depth, contributing to our understanding of cosmic events and structures. By using technologies like reflecting telescopes, astronomers can explore the intricacies of the universe and the fundamental laws of physics that govern it.

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Most popular questions from this chapter

The focal length of the eyepiece of a certain microscope is 18.0 \(\mathrm{mm}\) . The focal length of the objective is 8.00 \(\mathrm{mm}\) . The distance between objective and eyepiece is \(19.7 \mathrm{cm} .\) The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

A camera with a 90 -mm-focal-length lens is focused on an object 1.30 \(\mathrm{m}\) from the lens. To refocus on an object 6.50 \(\mathrm{m}\) from the lens, by how much must the distance between the lens and the film be changed? To refocus on the more distant object, is the lens moved toward or away from the film?

Saturn is viewed through the Lick Observatory refracting telescope (objective focal length 18 \(\mathrm{m} )\) . If the diameter of the image of Satum produced by the objective is \(1.7 \mathrm{mm},\) what angle does Saturn subtend from when viewed from earth?

An object is placed between two plane mirrors arranged at right angles to each other at a distance \(d_{1}\) from the surface of one mirror and a distance \(d_{2}\) from the other. (a) How many images are formed? Show the location of the images in a diagram. (b) Draw the paths of rays from the object to the eye of an observer.

(a) Where is the near point of an eye for which a contact lens with a power of \(+2.75\) diopters is prescribed? (b) Where is the far point of an eye for which a contact lens with a power of \(-1.30\) diopters is prescribed for distant vision?
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