Animated Solution for Physics - Optics: A planet is observed by an astronomical refracting telescope having an objective of focal length $16\text{ m}$ and an eyepiece of focal length $2\text{ cm}$

Imagine you are looking at a distant planet through an astronomical refracting telescope. The telescope consists of two lenses: a large objective lens that gathers light from the distant object, and a smaller eyepiece lens that magnifies the image for your eye.

In normal adjustment, the final image is formed at infinity. This means the image formed by the objective lens falls exactly at the focal point of the eyepiece. Therefore, the focal points of both lenses coincide.

The distance between the objective and the eyepiece, often called the tube length $L$, is simply the sum of their focal lengths.

We are given the focal length of the objective $f_o=16\text{ m}$ and the eyepiece $f_e=2\text{ cm}$. First, let's convert the eyepiece focal length to meters to keep our units consistent: $f_e=0.02\text{ m}$.

Substituting these values into our equation:

This confirms that the distance between the objective and the eyepiece is indeed $16.02\text{ m}$.

Next, let's determine the angular magnification $M$ of the telescope. For an astronomical telescope in normal adjustment, the magnification is given by the ratio of the focal length of the objective to the focal length of the eyepiece. The negative sign indicates that the final image is inverted relative to the original object.

Let's plug in our values:

The angular magnification is $-800$. This confirms two things: the magnitude of magnification is $800$, and the negative sign means the image is inverted.

Finally, by the very design of an astronomical telescope, the objective lens must have a large aperture and a large focal length to gather as much light as possible from faint, distant stars and planets. The eyepiece, on the other hand, has a small focal length to provide high magnification. Therefore, the objective is always larger than the eyepiece.

All four statements provided in the question are absolutely correct!