Question

A refracting telescope is \(45.0 \mathrm{cm}\) long and the caption states that the telescope magnifies images by a factor of \(30.0 .\) Assuming these numbers are for viewing an object an infinite distance away with minimum eyestrain, what is the focal length of each of the two lenses?

Short answer

The focal length of the objective is 43.5 cm, and the focal length of the eyepiece is 1.45 cm.

Step-by-step solution

Understanding the Telescope Length

The length of the telescope is the sum of the focal lengths of the objective lens \( (f_o) \) and the eyepiece lens \( (f_e) \). Given that the total length is \(45.0\, \text{cm}\), we can write the equation:\[f_o + f_e = 45.0\, \text{cm}\]

Applying the Magnification Formula

The magnification \( (M) \) of the telescope is given by the ratio of the focal length of the objective to the focal length of the eyepiece:\[M = \frac{f_o}{f_e}\]We know the magnification is \(30.0\), so:\[\frac{f_o}{f_e} = 30.0\]

Solving the System of Equations

We have two equations:1. \(f_o + f_e = 45.0\)2. \(\frac{f_o}{f_e} = 30.0\) (which can be rewritten as \(f_o = 30.0 \times f_e\))Substitute \(f_o = 30.0 \times f_e\) into the first equation:\[30.0f_e + f_e = 45.0\]\[31.0f_e = 45.0\]Solve for \(f_e\):\[f_e = \frac{45.0}{31.0} \approx 1.45\, \text{cm}\]

Finding the Focal Length of the Objective Lens

Using \(f_o = 30.0 \times f_e\), substitute \(f_e = 1.45\, \text{cm}\) to find \(f_o\):\[f_o = 30.0 \times 1.45 = 43.5\, \text{cm}\]

Key concepts

Focal Length

The focal length of a lens is a crucial concept when discussing telescopes and lenses. It is the distance over which parallel rays of light are brought to a focus. In a refracting telescope, understanding focal length is essential because it directly impacts how the telescope forms an image.
A refracting telescope consists of at least two lenses, each with its own focal length. The total length of the telescope is the sum of the focal lengths of its lenses: the objective lens and the eyepiece lens.
In our exercise, the focal lengths of the objective and the eyepiece add up to a total telescope length of 45 cm. This simple addition becomes more complex when calculating each lens's focal length individually, as it requires knowledge of the telescope’s magnification as well.

Magnification

Magnification is a measure of how much larger or closer an object appears when viewed through a telescope. It is an essential factor in determining the power of a telescope to enlarge small or distant objects.
Magnification is calculated by dividing the focal length of the objective lens by the focal length of the eyepiece lens. Thus, the magnification can be expressed through:

• \( M = \frac{f_o}{f_e} \)
• Where \( M \) is the magnification, \( f_o \) is the focal length of the objective lens, and \( f_e \) is the focal length of the eyepiece lens.

In our exercise scenario, the telescope’s magnification is given as 30. This means that the image seen through the telescope is 30 times larger than it appears to the naked eye. Understanding how magnification interacts with focal lengths helps ensure accurate imaging with minimal eyestrain.

Objective Lens

The objective lens is one of the key components of a refracting telescope, often positioned at the front. Its role is to gather light from a distant object and bring it into focus. When light passes through the objective lens, it bends or refracts. This light converges at a point known as the focal point, which is crucial for creating clear images.
For the refracting telescope in our exercise, the objective lens has a focal length determined in relation to both the eyepiece lens and the telescope’s overall length and magnification. After performing calculations, the focal length of the objective lens is found to be 43.5 cm. This value highlights the objective’s capacity to gather and focus light effectively, impacting the quality of the observed image.

Eyepiece Lens

The eyepiece lens is located at the opposite end of the telescope from the objective lens. Its primary purpose is to magnify the focused image produced by the objective lens. This component allows your eye to view the enlarged image effectively.
In the context of our exercise, the eyepiece lens has a focal length that complements the length of the telescope and the magnification. Calculated through our given equations, the eyepiece has a focal length of approximately 1.45 cm. This smaller focal length, combined with the longer focal length of the objective lens, is what gives the telescope its magnifying power.
Understanding how the eyepiece works alongside the objective lens provides insight into how telescopes can be tailored for viewing specific objects, whether they are planets in our solar system or distant galaxies far from our reach.