A 20x telescope has a 12-cm-diameter objective lens. What minimum diameter must the eyepiece lens have to collect all the light rays from an on-axis distant source?

Verified step by step guidance

1. Understand the problem: A telescope magnifies distant objects, and the eyepiece lens must collect all the light rays from the objective lens. The magnification of the telescope is given as 20x, and the diameter of the objective lens is 12 cm. We need to determine the minimum diameter of the eyepiece lens.
2. Recall the relationship between magnification and the diameters of the objective and eyepiece lenses. The magnification (M) is given by the ratio of the focal lengths of the objective lens (fₒ) and the eyepiece lens (fₑ): M = fₒ / fₑ. Additionally, the diameters of the lenses are proportional to their focal lengths.
3. Set up the proportionality equation for the diameters of the lenses: \( \frac{Dₒ}{Dₑ} = \frac{fₒ}{fₑ} \), where \( Dₒ \) is the diameter of the objective lens, \( Dₑ \) is the diameter of the eyepiece lens, \( fₒ \) is the focal length of the objective lens, and \( fₑ \) is the focal length of the eyepiece lens. Since magnification \( M \) is 20x, we know \( \frac{fₒ}{fₑ} = 20 \).
4. Substitute the known values into the proportionality equation: \( \frac{Dₒ}{Dₑ} = 20 \). Rearrange to solve for \( Dₑ \): \( Dₑ = \frac{Dₒ}{20} \). Here, \( Dₒ \) is the diameter of the objective lens, which is 12 cm.
5. Perform the division to find the minimum diameter of the eyepiece lens: \( Dₑ = \frac{12}{20} \). This gives the minimum diameter of the eyepiece lens required to collect all the light rays from the objective lens.

Key Concepts

Telescope Magnification

Telescope magnification is the ratio of the focal lengths of the telescope's objective lens to its eyepiece lens. In this case, a 20x magnification means that the image seen through the eyepiece appears 20 times larger than the object. Understanding this concept is crucial for determining the relationship between the objective and eyepiece lenses in a telescope.

Lens Diameter and Light Gathering

The diameter of a lens, known as its aperture, directly affects its ability to gather light. A larger diameter allows the lens to collect more light rays, which is essential for forming a clear image, especially in low-light conditions. For the eyepiece lens to effectively collect all light rays from a distant source, its diameter must be sufficient to match the light-gathering capability of the objective lens.

Ray Optics and On-Axis Light

Ray optics, or geometrical optics, describes how light travels in straight lines and interacts with lenses. When considering an on-axis distant source, the light rays are parallel as they enter the telescope. The eyepiece must be designed to capture these parallel rays effectively, ensuring that all light is focused properly to create a clear image, which is essential for determining the required diameter of the eyepiece lens.