Problem 89 - Physics

Question

Mars subtends an angle of \(8.0 \times 10^{-5} \mathrm{rad}\) at the unaided eye. An astronomical telescope has an eyepiece with a focal length of \(0.032 \mathrm{~m}\). When Mars is viewed using this telescope, it subtends an angle of \(2.8 \times 10^{-3} \mathrm{rad}\). Find the focal length of the telescope's objective lens.

Step-by-Step Solution

Answer

The focal length of the telescope's objective lens is 1.12 meters.

Step 1: Understand the Problem Setup

We need to find the focal length of the telescope's objective lens based on the given angle subtended by Mars at the unaided eye and when viewed through the telescope.

Step 2: Establish the Telescope Magnification Formula

The angular magnification (or magnifying power) of a telescope is given by the formula \( M = \frac{\theta'}{\theta} = \frac{f_o}{f_e} \), where \( \theta' \) is the angle subtended through the telescope, \( \theta \) is the angle subtended to the unaided eye, \( f_o \) is the focal length of the objective lens, and \( f_e \) is the focal length of the eyepiece.

Step 3: Substitute Known Values into the Formula

We are given: \( \theta = 8.0 \times 10^{-5} \, \text{rad} \), \( \theta' = 2.8 \times 10^{-3} \, \text{rad} \), and \( f_e = 0.032 \, \text{m} \). Substituting these into the formula yields: \[ M = \frac{2.8 \times 10^{-3}}{8.0 \times 10^{-5}} = \frac{f_o}{0.032} \]

Step 4: Calculate the Magnification

Calculate the magnifying power \( M \):\[ M = \frac{2.8 \times 10^{-3}}{8.0 \times 10^{-5}} = 35 \]

Step 5: Solve for the Objective Lens Focal Length

With the calculated magnification, substitute back into the equation \( M = \frac{f_o}{f_e} \) to solve for \( f_o \):\[ 35 = \frac{f_o}{0.032} \]Solve for \( f_o \):\[ f_o = 35 \times 0.032 = 1.12 \, \text{m} \]

Step 6: Conclusion

The focal length of the telescope's objective lens is 1.12 meters. This means the telescope effectively enhances the angular size of Mars from an unaided view to what is seen as enhanced through the telescope by this objective focal length.

Key Concepts

Focal Length Calculation

Calculating the focal length of a telescope's objective lens is essential to understanding how telescopes work. The focal length determines how strongly the lens bends or focuses light.
In our exercise, we need to find the focal length of the objective lens using given angles and the focal length of the eyepiece.

• The sky object, like Mars, subtends an angle of \(8.0 \times 10^{-5}\) radians to the unaided eye.
• Using the telescope, the angle becomes \(2.8 \times 10^{-3}\) radians.
• The eyepiece focal length is \(0.032\) meters.

We use the formula for angular magnification \( M = \frac{f_o}{f_e} \) to find \( f_o \), the focal length of the objective lens. By substituting the magnification and the eyepiece focal length, we solve to find \( f_o = 1.12 \) meters. This calculation shows how the telescope collects and focuses light to magnify distant objects.

Angular Magnification

Angular magnification, or magnifying power, is a measure of how much larger a telescope can make objects appear. It's crucial in choosing or designing a telescope that fits specific observational needs.The formula for angular magnification is:\[ M = \frac{\theta'}{\theta} = \frac{f_o}{f_e} \]

• \( \theta' \) is the angle seen through the telescope.
• \( \theta \) is the angle to the unaided eye.
• \( f_o \) is the focal length of the objective lens.
• \( f_e \) is the focal length of the eyepiece.

In practice, this means that by changing the eyepiece or the objective lens, you can adjust the magnification level. For the given values, the telescope magnified Mars 35 times. Understanding this formula helps users know how different telescope parts contribute to magnification and how it affects one's ability to observe distant celestial objects.

Astronomical Observation

Astronomical observations allow us to explore and study distant celestial bodies like Mars and other planets. Telescopes play a vital role in this exploration by magnifying these distant objects, making them visible or more detailed to the human eye. The focal lengths of the lenses and the angular magnification power are key factors in effective astronomical observations. By using a telescope:

• The light from celestial objects can be collected and focused efficiently.
• Details not visible to the unaided eye, such as surface features of planets, can be examined.
• Changes or movements in these celestial bodies can be monitored.

An understanding of how focal length and angular magnification work together helps hobbyists and astronomers alike choose the right telescope for their specific needs, enhancing the beauty and understanding of the universe.