Draw a labelled ray diagram of an astronomical telescope in the near point adjustment position. A giant refracting telescope at an observatory has an objective lens of focal length 15 m and an eyepiece of focal length 1.0 cm. If this telescope is used to view the Moon, find the diameter of the image of the Moon formed by the objective lens. The diameter of the Moon is 3.48 x 10⁶ m and the radius of lunar orbit is 3.8 x 10⁸ m.
Astronomical telescope in near points adjustment:

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Detailed Solution

$Here, f_{0} = 15 m, fe = 1 cm = 0 .1 m$
$Diameter of moon, d = 3 .48 \times 10^{6} m$
$Radius of lunar orbit u_{0} = 3 .8 \times 10^{8} m$
$size of the image of moon, I = ?$
The angle subtended by the moon at the objective lens,
$α = \frac{d}{|u_{0}|} . . . (i)$
And the angle subtended by the image of the moon at the objective lens,
$β = \frac{I}{f_{o}} . . . (ii)$
Both these angles in equations (i) and (ii) will be equal.
$\begin{array}{l} ∴ \frac{d}{|u_{0}|} = \frac{I}{f_{0}} \\ \Rightarrow I = \frac{d \times f_{0}}{|u_{0}|} \\ = \frac{3 .48 \times 10^{6} \times 15}{3 .8 \times 10^{8}} m \\ = 13 .73 \times 10^{- 2} m \\ = 13 .73 cm \end{array}$