Question

A telescope consisting of an objective of focal length \(60 \mathrm{~cm}\) and a single-lens eyepiece of focal length \(5 \mathrm{~cm}\) is focussed at a distant object in such a way that parallel rays emerge from the eye piece. If the object subtends an angle of 2 at the objective, then find the angular width of the image. (1) 6 (2) 12 (3) 24 (4) \(1 / 6\)

Short answer

24 degrees

Step-by-step solution

- Determine the Angular Magnification

The angular magnification of a telescope can be calculated using 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. Plug in the given values: \( f_o = 60 \, \text{cm} \) and \( f_e = 5 \, \text{cm} \). Thus, \[ M = \frac{60}{5} = 12 \].

- Relate Object Angle to Image Angle

The angular magnification equation tells us that the angular width of the image is the product of the angular magnification and the angular width of the object. This is given by: \[ \theta_i = M \times \theta_o \] where \( \theta_i \) is the angular width of the image and \( \theta_o \) is the angular width of the object. The object subtends an angle of 2 degrees at the objective.

- Calculate the Angular Width of the Image

Using the magnification and object's angle obtained from the previous steps: \[ \theta_i = 12 \times 2 = 24 \text{ degrees} \]

Key concepts

Focal Length

The focal length is a critical aspect when working with optical instruments like telescopes. In simple terms, the focal length is the distance between the lens and the point where parallel rays of light converge to a single point. It dictates how strongly the lens converges or diverges light.

Using a longer focal length for the objective lens allows the telescope to gather more light and provide a clearer image of distant objects. In the exercise, the objective lens has a focal length of 60 cm, while the eyepiece lens has a focal length of 5 cm.

Why do these differences matter? Well, the focal length influences the magnification power of the telescope, which brings us to the next core concept.

Optical Instruments

Optical instruments like telescopes and microscopes are designed to enhance our visual capabilities. Telescopes, for instance, allow us to see distant celestial bodies by magnifying the light they emit.

The basic components of a simple telescope include:

• An objective lens, which gathers light from the distant object and focuses it to form an image.
• An eyepiece lens, which magnifies this image.

For the telescope in the problem, the given focal lengths for both lenses help determine how much the image is magnified.

Optical instruments work on the principle of refraction and reflection to manipulate light. The arrangement and properties of the lenses or mirrors significantly influence the instrument's performance.

Angular Magnification

Angular magnification describes how much larger an angular object appears when viewed through an optical instrument compared to the naked eye. In telescopes, this is calculated using the formula: