Problem 61 - Physics for Scientists and Engineers with Modern Physics

Question

(I) The overall magnification of an astronomical telescope is desired to be \(35 \times\). If an objective of \(88 \mathrm{~cm}\) focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?(I) The overall magnification of an astronomical telescope is desired to be \(35 \times\). If an objective of \(88 \mathrm{~cm}\) focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?

Step-by-Step Solution

Answer

The eyepiece focal length is approximately 2.51 cm, and the telescope length is about 90.51 cm.

Step 1: Understand the Magnification Formula

For an astronomical telescope, the magnification \( M \) is given by the formula: \( M = \frac{f_o}{f_e} \), where \( f_o \) is the focal length of the objective lens and \( f_e \) is the focal length of the eyepiece.

Step 2: Substitute Known Values into the Formula

We are given that the desired magnification \( M \) is \( 35 \) and the focal length of the objective lens \( f_o \) is \( 88 \) cm. Substitute these values into the magnification formula: \( 35 = \frac{88}{f_e} \).

Step 3: Solve for the Focal Length of the Eyepiece

To find \( f_e \), rearrange the equation to \( f_e = \frac{88}{35} \). Calculating this gives \( f_e \approx 2.51 \) cm.

Step 4: Calculate the Overall Length of the Telescope

For a telescope adjusted for a relaxed eye, the overall length \( L \) is given by \( L = f_o + f_e \). Substitute the known values: \( L = 88 + 2.51 \), which simplifies to \( L \approx 90.51 \) cm.

Key Concepts

Telescope Magnification

Understanding telescope magnification is crucial when working with astronomical telescopes. Magnification tells us how much larger an object appears when viewed through the telescope compared to the naked eye. This is particularly useful for observing distant celestial bodies.

In astronomical terms, magnification (M) is calculated with a basic formula: \(M = \frac{f_o}{f_e}\). Here, \(f_o\) is the focal length of the objective lens and \(f_e\) is the focal length of the eyepiece lens. This formula helps determine how much the telescope enlarges the view of distant objects.

• High magnification allows for close-up views of distant stars and planets.
• Understanding this concept helps astronomers choose the right lenses for specific viewing needs.

Focal Length

The focal length is a fundamental concept in optics, particularly for telescopes. It refers to the distance between the lens and the point where light rays converge to form a clear image. In the context of telescopes, it helps determine the magnifying power and overall efficiency of the system.

The longer the focal length of an objective lens, the greater the potential for magnification. With our example, a focal length of 88 cm for the objective lens provides strong magnification potential. In contrast, the focal length of the eyepiece affects the overall zoom level of the telescope's image.

By manipulating these focal lengths, astronomers can tailor telescopes to specific observational needs, enhancing specific details in the view of distant objects.

• The objective lens usually has a longer focal length for clear far-distance viewing.
• Different combinations of focal lengths can produce a range of zoom levels and image sizes.

Objective Lens

An essential component of astronomical telescopes, the objective lens gathers light from a distant object to create a clear image. Larger and longer focal lengths in objective lenses capture more light, enhancing the clarity and detail of celestial objects.

This lens is key to determining the telescope's capability. In our example, with a focal length of 88 cm , the lens significantly contributes to achieving the desired magnification of 35 times . By using a powerful objective lens, more distant objects can be viewed in greater detail, making it a critical part of any telescope setup.

Some key considerations for objective lenses include:

• They are essential for collecting light and defining image quality.
• Higher quality lenses can enhance color and contrast, providing a better viewing experience.

Eyepiece Lens

The eyepiece lens is an important part of the telescope that works in tandem with the objective lens to enhance observation. It further magnifies the image formed by the objective lens and determines the final magnification power of the telescope.

In this context, the focal length of the eyepiece lens (\(f_e\)) is calculated based on the desired magnification and the known focal length of the objective lens using the formula: \(M = \frac{f_o}{f_e}\). In our example, the eyepiece lens with a focal length of approximately 2.51 cm allows the astronomical telescope to reach the specified 35 times magnification.

This adjustment is crucial because:

• Itfine-tunes the magnifying power for clear, detailed images.
• A variety of eyepiece lenses can be swapped out to provide different levels of magnification depending on the user's needs.