Problem 89 - Physics for Scientists and Engineers with Modern Physics

Question

An astronomical telescope has a magnification of \(8.0 \times .\) If the two lenses are \(28 \mathrm{~cm}\) apart, determine the focal length of each lens.

Step-by-Step Solution

Answer

The focal length of the objective lens is about 24.89 cm, and the eyepiece is about 3.11 cm.

Step 1: Understand Telescope Magnification Formula

The magnification (8.0) of a telescope is the ratio of the focal length of the objective lens ( f_o ) to the focal length of the eyepiece ( f_e ). The formula is given by \( M = \frac{f_o}{f_e} \). We use this to establish a relationship between the objective's and the eyepiece's focal lengths.

Step 2: Establish the Lens Equation

Given the distance between the lenses is 28 cm, we have the equation for the total length of the telescope as \( f_o + f_e = 28 \). This equation uses the fact that in normal adjustment, the intermediate image is at the focal point of the eyepiece.

Step 3: Solve the System of Equations

With \( M = 8 \) and \( f_o + f_e = 28 \), substitute \( f_o = 8f_e \) which is derived from \( M = \frac{f_o}{f_e} \). Substitute into the total length equation: \( 8f_e + f_e = 28 \), which simplifies to \( 9f_e = 28 \).

Step 4: Calculate the Eyepiece Focal Length

Solve \( 9f_e = 28 \) for \( f_e \), giving \( f_e = \frac{28}{9} \approx 3.11 \) cm. This is the focal length of the eyepiece.

Step 5: Calculate the Objective Focal Length

Using \( f_o = 8f_e \), compute \( f_o = 8 \times 3.11 \approx 24.89 \) cm. This is the focal length of the objective.

Key Concepts

Telescope Magnification

Telescope magnification is a fundamental concept that helps us understand how much larger an object appears when viewed through a telescope compared to the naked eye. It gives us an idea of the telescope's power. For a basic astronomical telescope, the magnification is calculated using the focal lengths of the two main lenses: the objective lens and the eyepiece lens. The formula for telescope magnification is: \[ M = \frac{f_o}{f_e} \] where \( M \) is the magnification factor, \( f_o \) is the focal length of the objective lens, and \( f_e \) is the focal length of the eyepiece. This formula shows us that the magnification is directly proportional to the focal length of the objective lens and inversely proportional to the focal length of the eyepiece. A higher magnification means objects will appear larger, but it's important to balance this with the clarity and quality of the image.

Objective Lens Focal Length

The objective lens in an astronomical telescope is crucial because it captures light from distant objects and focuses it to form a clear image. The focal length of the objective lens, \( f_o \), affects how much light is collected and how detailed the image is. A larger focal length:

• collects more light, providing better image brightness and detail.
• results in higher magnification when paired with a fixed eyepiece focal length.
• is typically used in telescopes intended for observing distant objects like stars and galaxies.

For example, in our exercise, the objective lens has a focal length calculated as \( f_o = 8 \times 3.11 \approx 24.89 \) cm. This relatively long focal length is suitable for astronomical observations, allowing us to see stars more clearly.

Eyepiece Focal Length

The eyepiece lens of a telescope is the lens through which we actually view the image; it magnifies the image formed by the objective lens. The focal length of the eyepiece, \( f_e \), is a critical factor in determining the telescope's total magnification.Key points about eyepiece focal length:

• A shorter focal length eyepiece increases magnification.
• It can result in a smaller field of view, sometimes limiting the amount of sky visible at once.
• Must be chosen to provide a balance between magnification and image quality.

In the problem provided, the eyepiece focal length is found to be \( f_e = \frac{28}{9} \approx 3.11 \) cm. This results in a significant magnification for a detailed view without compromising image clarity._

Lens Distance Calculation

The distance between the objective and the eyepiece lenses in a telescope, also known as the optical tube length, is a sum of their respective focal lengths. This distance needs to be carefully managed to ensure the image is focused correctly at the eyepiece. Calculation process:

• In typical use, the total length of the telescope, \( L \), is given by: \[ L = f_o + f_e \]
• This indicates that when the telescope is focused for infinity, the intermediate image falls at the focal point of the eyepiece.
• Ensures that both lenses work in harmony to present a clear, magnified image.

In the sample exercise, the total distance is specified as 28 cm, facilitating the calculations for each lens's focal length as shown in the step-by-step solution. Proper lens distance calculation is vital for effective telescope use, ensuring that both magnification and image quality are optimized.