Question

A Galilean telescope measures 9 cm from the objective to the eye-piece. The focal length of the objective is 15 cm. Its magnifying power is
A. $2.5$
B. ${\displaystyle \frac{2}{5}}$
C. ${\displaystyle \frac{5}{3}}$
D. ${\displaystyle \frac{3}{5}}$

Answer 1

Hint: In a Galilean telescope, the focal length of the objective lens is equal to the difference between the length between the two lenses and the focal length of the eye-piece, $f_o=L+f_e$, where $f_o,f_e$ are the focal lengths of objective lens and eye-piece, L is the distance between them. Substitute the values of L and focal length of the objective lens to find the value of focal length of the eye-piece. Magnification is the negative ratio of focal length of the objective lens and focal length of the eye-piece.

Complete step by step answer:

We are given that a Galilean telescope measures 9 cm from the objective to the eye-piece and the focal length of the objective is 15 cm.
We have to find its magnifying power.
As we can see in the above figure, the objective is a convex lens and the eye-piece is a concave lens.
So the focal length is negative for the concave (eye-piece) lens.
As we can see, the focal length of the objective lens is the sum of the focal length of the eye-piece and the distance between them.
$f_o=L+f_e$, where $f_o,f_e$ are the focal lengths of objective lens and eye-piece, L is the distance between them and $f_e$ should be taken negatively.
$$\begin{gathered} f_o=L+\left(-f_e\right) \\ ⟹f_o=L-f_e \\ ⟹f_e=L-f_o \end{gathered}$$
Substitute the values of the L and $f_o$ as they are given in the question.
$$\begin{gathered} f_o=15cm,L=9cm \\ ⟹f_e=L-f_o \\ =9-15 \\ \therefore f_e=-6cm \end{gathered}$$
Magnification is the negative ratio of focal length of the objective lens and focal length of the eye-piece.
$$\begin{gathered} m={\displaystyle \frac{-f_o}{f_e}} \\ {f}_o=15cm,f_e=-6cm \\ m={\displaystyle \frac{-15}{-6}} \\ ⟹m={\displaystyle \frac{5}{2}} \\ \therefore m=+2.5 \end{gathered}$$
The magnifying power of the given Galilean telescope is 2.5.
So, the correct answer is “Option A”.

Note:
The objective lens is concave, which is a spherical lens and focuses the light rays passed through it to a specific point. Eye-piece is a convex lens, which is a spherical lens and diverges the light rays refracted from it. The lenses used in objective and eye-piece are not the same in a Galilean telescope. Be careful in considering the lenses.