Question

A giant refracting telescope at an observatory has an objective lens of focal length $15$ m. If this telescope is used to view the moon, what is the diameter of the image of the moon formed by the objective lens? The diameter of the moon is $3.48 \times 10^{6}$, and the radius of lunar orbit is $3.8 \times 10^{8}$.

Answer 1

Focal length of the objective lens, diameter of the moon and the radius of lunar orbit is given. The ratio of diameter of the moon to the radius of lunar orbit is equal to the ratio of the diameter of the moon to the focal length of the objective lens because angle subtended by the diameter of the moon is equal to the angle subtended by the image.

Complete step-by-step solution:
Given focal length of the objective lens, $f_{o}=15 m$
Diameter of the moon, $d = 3.48 \times 10^{6}$
Radius of the lunar orbit, $r_{o} = 3.8 \times 10^{8}$
Let $d_{m}$ be the diameter of the image of the moon formed by the objective lens.
The angle subtended by the diameter of the moon is equal to the angle subtended by the image. So, $\dfrac{d}{r_{o}}= \dfrac{d_{m}}{f_{o}}$
Put the values of $f_{o}$, $r_{o}$ and d in the above formula. We get,
$\dfrac{3.48 \times 10^{6}}{3.8 \times 10^{8}}= \dfrac{d_{m}}{15 }$
$\implies d_{m} = 13.73 \times 10^{-2} m = 13.73 cm$
Hence, the diameter of the moon’s image formed by the objective lens is $13.73$ cm

Additional Information: - When light is traveling corner ways from one medium to a different medium, then the direction of spread of light shifts in the second medium; the phenomenon is refraction of light. A refracting telescope or reflector utilizes lenses to focus the beam. Reflectors provide a fantastic view of the minor details in the night sky that encloses us.

Note: A refracting telescope uses lenses, i.e., the objective and the eyepiece, to provide an image of a distant object. The objective lens produces a picture of a distant object at its focus, and the eyepiece lens makes it more significant. Refraction of light is used to produce the image in this case.