Question

The eyepiece of a refracting telescope (see Fig. 34.53) has a focal length of 9.00 cm. The distance between objective and eyepiece is 1.20 m, and the final image is at infinity. What is the angular magnification of the telescope?

Short answer

The angular magnification of the telescope is 13.33.

Step-by-step solution

Identify the Known Values

We know the focal length of the eyepiece, \( f_e = 9.00 \) cm, and the distance between the objective and the eyepiece, \( L = 1.20 \) m. The final image being at infinity implies that the focal length of the objective, \( f_o \), is equal to the distance \( L \) (because when the image is at infinity, the objective lens focuses light at its focal length). Therefore, \( f_o = 1.20 \) m = 120 cm.

Use the Angular Magnification Formula

The angular magnification \( M \) of a telescope, when the final image is at infinity, 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.

Calculate the Angular Magnification

Substitute the known values into the angular magnification formula: \[ M = \frac{120\, \text{cm}}{9\, \text{cm}} = \frac{120}{9} = 13.33 \] Thus, the angular magnification of the telescope is 13.33.

Key concepts

Refracting Telescope

A refracting telescope is a type of optical telescope that uses lenses to gather and focus light. It consists of two main lenses: the **objective lens** and the **eyepiece lens**. The objective lens is responsible for collecting light and creating an image, while the eyepiece lens magnifies this image for the viewer.

These telescopes are popular for their simple design and ease of use, especially for amateur astronomers. Refracting telescopes are often used for celestial observations because they provide clear, high-quality images.

However, larger refracting telescopes can be costly due to the large lenses required, and they are susceptible to chromatic aberration, where different colors are focused at slightly different positions, causing fringes around the edges of images. Despite these drawbacks, their straightforward mechanism makes them a favorite among astronomy enthusiasts.

Focal Length

The focal length of a lens is a fundamental concept when dealing with telescopes. It is defined as the distance from the lens to the point where it converges or diverges light to form an image. In the context of a refracting telescope:

• The objective lens has its focal length where it focuses incoming light to form an image.
• The eyepiece lens has a focal length that determines how much the image gets magnified.

The focal lengths of these lenses decide the power and capability of the telescope. A longer focal length in the objective lens allows for a stronger magnification potential. However, in practical telescope applications, the combined focal lengths of both lenses are crucial to achieving the desired balance between magnification and image clarity.

Telescope Formula

The telescope formula is pivotal in understanding how a refracting telescope functions. The formula for the angular magnification, which describes how much larger an object appears through the telescope than to the naked eye, is: \[ M = \frac{f_o}{f_e} \]This simple equation tells us that magnification (M) is the ratio of the focal length of the objective lens ( f_o) to the focal length of the eyepiece lens ( f_e).

When using this formula, understanding the focal lengths' measurement units is vital since they must be consistent. This relationship directly affects how the telescope's magnification changes if either lens is swapped for one with a different focal length, offering flexibility in designing telescopes to specific observational needs.

Objective Lens

The objective lens, as a core component of a refracting telescope, is the first lens that incoming light encounters. It plays a significant role by gathering light from distant objects and focusing it to form an image. This lens is usually much larger than the eyepiece lens to maximize its light-gathering ability.

The quality and size of the objective lens determine the telescope's capacity to magnify distant objects. Seasoned astronomers often consider the diameter of the objective lens as an indicator of a telescope’s prowess, termed as its "aperture." A larger aperture allows more light to be captured, resulting in brighter and clearer images.

In our example, the objective lens has a focal length equal to the distance between the lenses, which perfectly focuses the light to an image at infinity, crucial for achieving maximum angular magnification.

Eyepiece Lens

The eyepiece lens is the part of the refracting telescope where the observer places their eye. Its job is to magnify the image created by the objective lens, allowing for a closer examination of the distant object. The eyepiece's focal length is a key player in determining the overall magnification of the telescope.

Different eyepiece lenses can be swapped out to provide various levels of magnification, making telescopes highly versatile tools for exploring the night sky.

The quality of the eyepiece also affects the viewing experience. Higher-quality eyepieces can enhance the clarity and detail seen, making them a worthy investment for serious astronomers. In our specific case, the eyepiece focal length is 9 cm, which, along with the objective lens, contributes to calculating the telescope's angular magnification as 13.33.