#### Question

Draw a labelled ray diagram of an astronomical telescope to show the image formation of a distant object. Write the main considerations required in selecting the objective and eyepiece lenses in order to have large magnifying power and high resolution of the telescope.

#### Solution

**Astronomical telescope**

When the final image is formed at the least distance of distinct vision:

Magnifying power, `M =β/α`

Since *α* and *β* are small, we have:

∴ `M= tanβ/tanα ...... (1)`

In `ΔA'B'C_2, tanβ = (A'B')/(C_2B') `

In `ΔA'B'C_1, tanβ = (A'B')/(C_2B') `

From equation (i), we get:

`M = (A'B')/(C_2B') xx (C_1B')/(A'B')`

\[\Rightarrow\] `M = (C_1B')/(C_2B')`

Here, `C_1B' = +f_0`

\[\Rightarrow\] `C_2B' = -u_e`

\[\Rightarrow\] `M = f_0/ -u_e .......... (2)`

Using the lens equation `(1/v-1/u=1/f)`for the eyepieces `(1/-D-1/-u_e=1/f_e,)`we get:

`(-1/D+1/u_e=1/f_e)`

\[\Rightarrow\] `(1/u_e=1/(f_e)+1/D)`

\[\Rightarrow\] `(f_0)/u_e =(f_0)/(f_e )(1+f_e/D)`

\[\Rightarrow\] `(-f_0)/u_e =(-f_0)/(f_e )(1+f_e/D) or M = -f_0/(f_e) (1+f_e/D) `

In order to have a large magnifying power and high resolution of the telescope, its objective lens should have a large focal length and the eyepiece lens should have a short focal length.