Problem 91 - Physics

Question

An astronomical telescope has an angular magnification of -184 and uses an objective with a focal length of \(48.0 \mathrm{~cm}\). What is the focal length of the eyepiece?

Step-by-Step Solution

Answer

The focal length of the eyepiece is approximately 0.261 cm.

Step 1: Understand the Formula for Angular Magnification

The angular magnification \(M\) of a telescope is given by the formula \( M = - \frac{f_o}{f_e} \), where \(f_o\) is the focal length of the objective lens, and \(f_e\) is the focal length of the eyepiece lens. The negative sign indicates the image is inverted.

Step 2: Substitute Given Values

We are given that the angular magnification \(M\) is \(-184\) and the focal length of the objective \(f_o\) is \(48.0 \text{ cm} \). Substitute these values into the formula: \( -184 = - \frac{48.0 \text{ cm}}{f_e} \).

Step 3: Solve for the Focal Length of the Eyepiece

To find \(f_e\), rearrange the equation from the previous step to get \(f_e = \frac{48.0 \text{ cm}}{184} \).

Step 4: Calculate the Focal Length of the Eyepiece

Calculate \(f_e\) by performing the division: \(f_e = \frac{48.0}{184} \approx 0.261 \text{ cm} \).

Key Concepts

Angular Magnification

Angular magnification is a measure of how much larger or smaller an object appears when viewed through a telescope compared to the naked eye. It is a critical concept in astronomy and optics as it affects the way we perceive distant objects.

• The formula for angular magnification is given by: \( M = - \frac{f_o}{f_e} \), where:
• \( M \) is the angular magnification
• \( f_o \) is the focal length of the objective lens
• \( f_e \) is the focal length of the eyepiece lens
• The negative sign indicates that telescopes typically produce an inverted image.

In our example, the angular magnification was calculated to be -184, which means the image is magnified 184 times and inverted.By understanding this formula, you can determine how powerful a telescope is relative to the size of its lenses.

Objective Lens

The objective lens of a telescope is one of the fundamental components that plays a crucial role in gathering light.It is the main lens through which light enters the telescope and is responsible for forming the primary image.

• The focal length of the objective lens, denoted as \( f_o \), is the distance from the lens to the point where it converges light to form an image.
• In an astronomical telescope, longer focal lengths allow you to see further into space and with greater detail.

In the problem, the objective lens has a focal length of 48.0 cm. This value is essential for calculating the angular magnification and understanding the telescope's capability to magnify distant objects.

Eyepiece Lens

The eyepiece lens of a telescope is another essential component that magnifies the image formed by the objective lens.After the light has been gathered and converged by the objective lens, the eyepiece takes over to provide a further enlarged view to the observer.

• The focal length of the eyepiece, \( f_e \), helps to determine the telescope's total magnification along with the objective lens.
• A shorter focal length in the eyepiece lens results in greater magnification.

In our calculation, the eyepiece lens had a calculated focal length of approximately 0.261 cm.This short focal length relative to the objective's focal length leads to the high angular magnification of -184.It highlights how the eyepiece contributes to making distant objects appear closer.

Focal Length

Focal length is a core concept in the study of lenses and optics.It refers to the distance between a lens and its focused image.Whether for the objective or eyepiece lens, understanding focal length is crucial for manipulating and optimizing a telescope.

• A longer focal length in a telescope's lens usually means the ability to see clearer and more detailed images.
• It is used as a primary factor in determining the angular magnification when combined with the focal length of another lens in the telescope.

In our solution, we utilized the definition of focal length to determine the unknown variable—\( f_e \)—which stands for the focal length of the eyepiece.This calculation allowed us to find the precise magnification for the telescope setup, enhancing our understanding of how telescopes function.