Problem 69 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An astronomical telescope longer than about \(50 \mathrm{~cm}\) is not easy to hold by hand. Based on this fact, estimate the maximum angular magnification achievable for a telescope designed to be handheld. Assume its eyepiece lens, if used as a magnifying glass, provides a magnification of \(5 \times\) for a relaxed eye with near point \(N=25 \mathrm{~cm}\).

Step-by-Step Solution

Answer

Maximum angular magnification is approximately 9x.

Step 1: Understanding Telescope Length and Effectiveness

A handheld telescope should be easy to hold, limiting its focal length to at most 50 cm. The total length of a telescope is the sum of the focal lengths of the objective (large) lens and the eyepiece (small) lens.

Step 2: Defining Angular Magnification

Angular magnification \(M\) of a telescope is given by \( M = \frac{f_o}{f_e} \), where \(f_o\) is the focal length of the objective lens and \(f_e\) is the focal length of the eyepiece lens. For maximum magnification, \(f_o\) should be near the maximum length constraint.

Step 3: Calculating Eyepiece Focal Length

Given that the eyepiece provides a magnification of \(5\times\) as a magnifying glass for relaxed eye, \( M_e = \frac{N}{f_e} = 5\). Thus, \( f_e = \frac{N}{5} = \frac{25 \, cm}{5} = 5 \, cm \).

Step 4: Estimating Maximum Angular Magnification

To achieve maximum angular magnification with \(f_o + f_e \leq 50 \, cm\), and knowing \(f_e = 5 \, cm\), the maximum \(f_o = 50 - 5 = 45 \, cm\). Thus, \( M = \frac{f_o}{f_e} = \frac{45 \, cm}{5 \, cm} = 9 \).

Key Concepts

Telescope Focal Length

The focal length of a telescope is the combined distance of how light is focused through its lenses. It is often the sum of the focal lengths of the objective lens and the eyepiece lens.

For a handheld telescope, this distance is crucial because it must not be too long to comfortably hold and use. Typically, a handheld telescope reaches the maximum usable focal length of about 50 cm.

This limitation means the combined focal lengths of both crucial lenses must fit within this boundary. Since the focal length impacts both how much a telescope can zoom and its overall length, balancing these is key.

By choosing focal lengths wisely, a user maximizes magnification while keeping the telescope compact.

• A shorter focal length in the eyepiece means more magnification.
• A longer focal length in the objective lens gathers more light, providing clearer images.

Objective Lens

The objective lens is the larger lens at the front of a telescope. Its main job is to gather light from distant objects and bring it to focus. This lens is essential for a clear and bright image, as more light gathering means better visual details.

The focal length of the objective lens directly impacts the angular magnification of the telescope.
For maximum magnification, extending the focal length of the objective lens is beneficial—hence the original target of around 45 cm when keeping a total length limit of 50 cm.

Remember:

• The larger the focal length of the objective lens, the larger the telescope, but more light is collected.
• A balance is crucial in handheld models to ensure usability and comfort.

Eyepiece Lens

The eyepiece lens is the smaller lens located at the back end of a telescope, through which you look. It serves to magnify the image formed by the objective lens.

In this exercise, the eyepiece's focal length was found using its known magnification as a standalone magnifying glass. Given it provides a magnification of 5x, and the near point of a relaxed eye is 25 cm:

• The formula used is \( M_e = \frac{N}{f_e} \).
• Thus, the focal length \(f_e\) equals 5 cm, derived from \( \frac{25 \, \text{cm}}{5} \).

The eyepiece is crucial for achieving the final magnified view, and its focal length determines its power.

Key points:

• Shorter eyepiece focal lengths lead to higher magnification power.
• Balancing this with the objective lens is crucial to staying within practical design constraints, such as keeping the telescope easy to handle and operate.