Problem 66 - Physics for Scientists and Engineers with Modern Physics

Question

(II) What is the magnifying power of an astronomical telescope using a reflecting mirror whose radius of curvature is \(6.4 \mathrm{~m}\) and an eyepiece whose focal length is \(2.8 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

The magnifying power of the telescope is approximately 114.29.

Step 1: Understanding the Problem

We are tasked with finding the magnifying power of an astronomical telescope. The problem provides the radius of curvature of the reflecting mirror and the focal length of the eyepiece. We know that using these parameters, we can compute the magnifying power of the telescope.

Step 2: Calculate the Focal Length of the Mirror

The focal length of a mirror is given by half the radius of curvature. Thus, we can calculate the focal length (\(f_o\)) using the formula:\[ f_o = \frac{R}{2} \]where \(R = 6.4 \mathrm{~m} = 640 \mathrm{~cm}\).So,\[ f_o = \frac{640}{2} = 320 \mathrm{~cm} \]

Step 3: Identify the Focal Length of the Eyepiece

The focal length of the eyepiece (\( f_e \)) is provided directly as 2.8 cm.

Step 4: Apply the Magnifying Power Formula

The magnifying power (\(M\)) of an astronomical telescope is given by the formula:\[ M = \frac{f_o}{f_e} \]Substituting the given values:\[ M = \frac{320}{2.8} \]

Step 5: Calculate the Magnifying Power

Perform the division to find the magnifying power:\[ M \approx 114.29 \]

Key Concepts

Astronomical Telescope

An astronomical telescope is a device that helps us observe distant celestial objects. Its main purpose is to make these objects appear larger and more distinct to our eyes. Telescopes work by collecting and focusing light to create an enlarged image of the observed object.
These telescopes typically consist of two main components: the objective lens or mirror and the eyepiece.

• The objective lens or mirror gathers light from the object and focuses it to form an image.
• The eyepiece magnifies the image formed by the objective, allowing us to see a larger image.

The magnifying power of an astronomical telescope is determined by the combination of these two components. Understanding how they work together is crucial for anyone interested in astronomy or telescope design.

Reflecting Mirror

In the context of astronomical telescopes, a reflecting mirror acts as the primary means of gathering and focusing light. Unlike lenses, mirrors reflect light and can be made much larger. This enables them to capture more light from faint celestial bodies.
Reflecting telescopes rely on one or more mirrors to form an image. The key advantages of using mirrors include reduced chromatic aberration and the ability to support larger apertures.
Reflecting mirrors usually come in the form of a large concave mirror. The light hits the mirror's surface and converges to a point called the focal point. The shape and size of the mirror determine how effectively it can focus light.

Focal Length

The focal length is a critical property of optical components, such as lenses and mirrors, in an astronomical telescope. It is the distance between the lens or mirror and the point where it focuses light to form a clear image.

• Objective Focal Length: For a mirror, the focal length is half the radius of curvature. It dictates the distance over which light is focused, impacting image size and clarity.
• Eyepiece Focal Length: This is the distance from the eyepiece where the eye perceives a sharp image. Shorter focal lengths result in higher magnification.

Longer focal lengths often mean higher magnification but can also lead to a narrower field of view. Properly combining the focal lengths of both components is vital for achieving the desired magnifying power of a telescope.

Radius of Curvature

The radius of curvature of a reflecting mirror is a measure of its shape. It is defined as the radius of the sphere from which a section of the mirror is shaped. This geometric property is directly linked to how a mirror focuses light.
The focal length of a mirror can be derived from its radius of curvature, as they are inversely related. The focal length is precisely half the radius of curvature.

• This relationship lets us compute the focal point, which is essential for forming clear images.
• A larger radius of curvature implies a longer focal length and a wider range of focusing. This means the images can appear larger at a given distance.

Understanding the radius of curvature is crucial because it affects a telescope's ability to gather and focus light, thereby impacting its performance in viewing distant objects.