Question

In an astronomical telescope, the distance between the objective and the eyepiece is $d = 36cm$ and the final image is formed at infinity. The focal length ${f_o}$​ of the objective and the focal length ${f_e}$​ of the eyepiece are
(A) ${f_o} = 45cm,{f_e} = - 9cm$
(B) ${f_o} = 50cm,{f_e} = 20cm$
(C) ${f_o} = 7.2cm,{f_e} = 5cm$
(D) ${f_o} = 30cm,{f_e} = 6cm$

Answer 1

We use the formula for distance between objective lens and eyepiece. The formula for the distance between the two lenses is simply the sum of the focal lengths of both the lenses. The focal lengths of both the lenses are given in the options. We see which of the following focal lengths given in the options satisfies the formula for distance between the two lenses.
Formula used: Distance between the objective and eyepiece in an astronomical telescope is $d = {f_o} + {f_e}$
Where,
${f_e}$ is the focal length of eyepiece
${f_o}$ is the focal length of objective lens
$d$ is the distance between the two lenses

Complete step by step solution: From the question the distance between the eyepiece and the objective lens is
$d = 36cm$
In an astronomical telescope the distance between the two lenses is the sum of focal lengths of the lenses
$d = {f_o} + {f_e}$
From the given options the sum of focal lengths that gives us $d = 36cm$ is only option (A) and (D)
$d = 45 - 9 = 36cm$
$d = 30 + 6 = 36cm$

Hence option (A) ${f_o} = 45cm,{f_e} = - 9cm$ and (D) ${f_o} = 30cm,{f_e} = 6cm$ are the correct answers.

Additional information: An astronomical telescope having an objective with a long focal length and an eyepiece with a short focal length, usually used for observing celestial bodies like the moon and the planets in the solar system.

Note: In an astronomical telescope both the lenses are convex lenses. The light from the object at infinity gets refracted and forms an image at focal length of the objective lens. This image becomes the object for eye piece. The image of the eyepiece is magnified. This is why the distance between the two lenses is the sum of their focal lengths.