Question

With help of labelled diagram, write principle of astronomical telescope and derive expression for magnifying power of an astronomical telescope, when final image is formed at least distance of distinct vision.

Hint:

The magnifying power of an astronomical telescope increases with the focal length of the objective lens and decreases with the focal length of the eyepiece.

Answer 1

Step 1: Principle of Astronomical Telescope.
An astronomical telescope is an optical instrument used to observe distant objects, such as stars and planets. It consists of two lenses: - **Objective lens**: A large lens with a short focal length that collects light from a distant object and forms a real, inverted, and diminished image. - **Eyepiece**: A smaller lens with a long focal length that magnifies the image formed by the objective lens.
The principle of the astronomical telescope is based on the combination of these two lenses. The objective lens forms an image of the distant object at a position where the eyepiece can view it.
Step 2: Magnifying Power of the Telescope.
The magnifying power \( M \) of an astronomical telescope is the ratio of the angular size of the image to the angular size of the object when viewed with the naked eye. The formula for magnifying power is: \[ M = \dfrac{\theta_{\text{image}}}{\theta_{\text{object}}} \] where \( \theta_{\text{image}} \) is the angular size of the image formed by the objective lens, and \( \theta_{\text{object}} \) is the angular size of the object.
For an astronomical telescope, when the final image is formed at the least distance of distinct vision \( D \), the magnifying power is given by: \[ M = \dfrac{f_o}{f_e} \] where: - \( f_o \) is the focal length of the objective lens, - \( f_e \) is the focal length of the eyepiece.
Thus, the magnifying power of an astronomical telescope is the ratio of the focal lengths of the objective and eyepiece lenses. Step 3: Diagram.
The diagram for an astronomical telescope with final image at the least distance of distinct vision is as follows: \[ \begin{array}{c} \text{Object at Infinity} \quad \longrightarrow \quad \text{Objective lens} \quad \longrightarrow \quad \text{Real Image} \quad \longrightarrow \quad \text{Eyepiece} \quad \longrightarrow \quad \text{Virtual Image at D} \end{array} \] Step 4: Conclusion.
The magnifying power of an astronomical telescope is given by the ratio of the focal length of the objective lens to the focal length of the eyepiece: \[ M = \dfrac{f_o}{f_e} \]