91Ó°ÊÓ

In the world of telescopes, understanding the focal length is essential. Focal length, denoted as \(f\), is the distance between the lens and the point where parallel rays of light converge to a focus. This measurement is crucial because it influences the magnification and clarity of the image your telescope will produce.
The focal length of a lens determines how much it will magnify objects. In this scenario, we have two lenses with focal lengths of 95.0 cm and 15.0 cm. The 95.0 cm lens is used as the objective lens, which is the larger lens at the front of the telescope. This lens collects light and focuses it to form an image.
The eyepiece, with a focal length of 15.0 cm, is the lens you look through. It magnifies the image formed by the objective lens. This combination of lenses helps in creating a detailed view of distant objects like stars and planets.
Understanding focal lengths also helps in calculating the angular magnification of the telescope, which is determined by the ratio of the focal lengths of the objective and eyepiece lenses.

The formation of an image in a telescope involves capturing light rays from distant objects and focusing them to create a visual representation. This process begins with the objective lens. In our case, it has a focal length of 95.0 cm. This lens gathers light from the object, like a building, and brings it to a focus, forming a real, inverted image.
In telescopes, when we say that both the object being viewed and the final image are at infinity, it means the light rays entering the telescope are almost parallel. This setup is optimal for viewing celestial objects, which are far away. The real image formed by the objective lens for a building of 60.0 meters, located 3.00 km away, can be calculated. Converting these measurements to the same units, the objective lens forms an image 1.9 cm tall. This small yet detailed image is then magnified by the eyepiece.

Angular size is a crucial concept when talking about telescopes and how we perceive distant objects. It refers to the perceived size of an object as seen from a particular point and is usually measured in degrees or radians. Angular size depends on both the actual size of the object and its distance from the observer.
In our example, after the objective lens forms an image of the building, we determine the angular size of this image when viewed through the telescope's eyepiece. Using the height of the image, 1.9 cm, and the focal length of the eyepiece, 15.0 cm, we can calculate the angular size. The formula for angular size \(θ\) is \(θ = h / f_e\), where \(h\) is the image height. Substituting in our values gives an angular size of 0.127 radians.
This angular size can also be converted to degrees using the conversion \(1 \text{ radian} = 57.3 \text{ degrees}\), resulting in approximately 7.27°. This measurement helps astronomers understand the scale of what they’re observing through the telescope.