Problem 41 - College Physics

Question

\(\bullet\) Galileo's telescopes, I. While Galileo did not invent the telescope, he was the first known person to use it astronomically, beginning around \(1609 .\) Five of his original lenses have survived (although he did work with others). Two of these have focal lengths of 1710 \(\mathrm{mm}\) and 980 \(\mathrm{mm}\) . (a) For greatest magnification, which of these two lenses should be the eye- piece and which the objective? How long would this telescope be between the two lenses? (b) What is the greatest angular magnification that Galileo could have obtained with these lenses? (Note: Galileo actually obtained magnifications up to about \(30 \times\) but by using a diverging lens as the eye- piece.) (c) The Moon subtends an angle of \(\frac{10}{2}\) when viewed with the naked eye. What angle would it subtend when viewed through this telescope (assuming that all of it could be seen)?

Step-by-Step Solution

Answer

Objective: 1710 mm, Eyepiece: 980 mm; Length: 2690 mm; Max magnification: 1.75; Moon angle: 2.86.

Step 1: Identifying Telescope Components

In a simple refracting telescope, the lens with the greater focal length should be used as the objective lens, whereas the lens with the shorter focal length should be the eyepiece. This setup maximizes magnification.

Step 2: Assigning Focal Lengths

Given focal lengths are 1710 mm and 980 mm. Therefore, the lens with a focal length of 1710 mm should be the objective lens, and the lens with 980 mm focal length should be the eyepiece.

Step 3: Calculating Telescope Length

The length of the telescope is the sum of the focal lengths of the objective lens and the eyepiece lens. Thus, the length of the telescope is 1710 mm + 980 mm = 2690 mm.

Step 4: Calculating Angular Magnification

The angular magnification (M) of a telescope is given by the ratio of the focal length of the objective lens (\( f_o \)) to the focal length of the eyepiece lens (\( f_e \)). Thus, \( M = \frac{f_o}{f_e} = \frac{1710}{980} \approx 1.75.\) This is the greatest magnification using the given lenses.

Step 5: Calculating Subtended Angle of the Moon

The angle subtended by an object through a telescope is the angle subtended by the object with the naked eye divided by the magnification.The Moon's angle with the naked eye is \( \frac{10}{2} = 5.\)Therefore, the angle subtended through the telescope is \( \frac{5}{1.75} \approx 2.86.\)

Key Concepts

Understanding the Refracting Telescope

A refracting telescope is a device used to observe distant objects. It uses lenses to bend and focus light. The word 'refracting' comes from refraction, which is the bending of light as it passes through different materials. There are two main lenses in a refracting telescope:

• The objective lens, which gathers light from a distant object and focuses it into an image.
• The eyepiece lens, which magnifies the image formed by the objective lens so that it can be viewed clearly by the human eye.

These telescopes have been used for centuries to explore the heavens, famously by Galileo. They can provide clear images of celestial bodies like the moon and planets, making them important tools in astronomy.
The basic principle behind their operation is straightforward, and it involves optical physics.

Exploring Focal Length

The focal length of a lens is the distance from the center of the lens to its focus point, where light converges to form a clear image. In a refracting telescope, each lens has a specific focal length.
Focal length is crucial for determining both the magnification and the size of the telescope:

• A longer focal length in the objective lens allows more light to be captured, creating a brighter and more detailed image.
• A shorter focal length in the eyepiece magnifies this image, enhancing the details further for observation.

This combination helps in optimizing the telescope's power and clarity. For instance, in the exercise about Galileo’s experiments, choosing the right focal lengths impacts both the telescope’s length and the magnification it provides.

Diving into Angular Magnification

Angular magnification is how much larger an object appears when viewed through a telescope compared to the naked eye. It is a critical measurement for telescope performance:

• The formula used is \( M = \frac{f_o}{f_e} \), where \( f_o \) is the focal length of the objective lens and \( f_e \) is that of the eyepiece lens.
• A higher magnification indicates that the object appears larger through the telescope.

In Galileo's telescope, the choice of lenses with specific focal lengths allowed his telescope to have a notable angular magnification, even though technological limitations were present at that time. Understanding how to calculate this magnification helps astronomers to choose the right lenses for their specific needs.

The Role of the Objective Lens

The objective lens is perhaps the most pivotal part of the refracting telescope. Its main job is to collect light and focus it to create a clear image.
Here’s how it works:

• It is the larger of the two lenses, ensuring it can gather more light from the object being observed.
• With a longer focal length, it brings distant images into sharper relief, which is ideal for astronomical observations.

In historical telescopes like Galileo's, the objective lens was carefully chosen to optimize performance, highlighting its significance in the overall design and efficiency in viewing celestial bodies.

Eyepiece Lens Explained

The eyepiece lens is an essential component that works in concert with the objective lens to allow for effective observation. Its primary duty is magnifying the focused image provided by the objective lens.

• The eyepiece lens is smaller and positioned close to the viewer’s eye, making it convenient for detailed viewing.
• A shorter focal length in the eyepiece equates to higher magnification, enabling the viewer to see finer details of the object being observed.

The eyepiece lens was vital in Galileo's telescopes as it allowed him to study celestial phenomena with greater detail and precision. Choosing the right eyepiece is key in achieving the desired level of magnification in any optical telescope.