An elderly sailor is shipwrecked on a desert island, but...

An elderly sailor is shipwrecked on a desert island, but manages to save his eyeglasses. The lens for one eye has a power of $+1.20$ diopters, and the other lens has a power of $+9.00$ diopters. (a) What is the magnifying power of the telescope he can construct with these lenses? (b) How far apart are the lenses when the telescope is adjusted for minimum eyestrain?

Key Concepts

Diopter to Focal Length Relationship

The diopter is a unit that indicates the optical power of a lens and is defined as the reciprocal of its focal length (in meters). This relationship is fundamental in optics because it allows one to easily calculate the focal length of a lens when given its power, and vice versa, which is essential in the design and analysis of optical instruments.

Telescope Magnification

Telescope magnification is determined by the ratio of the focal lengths of the objective and the eyepiece lenses. In an astronomical telescope, the absolute value of the magnification is given by dividing the focal length of the objective lens by that of the eyepiece. This concept is critical to understanding how telescopes enlarge distant objects and to achieving the desired level of magnification.

Afocal Optical Systems

Telescopes are designed as afocal systems, meaning they take incoming parallel rays (from distant objects) and output parallel rays through the eyepiece, which minimizes additional focusing requirements for the human eye. This setup is important for ensuring that the final image appears at infinity, reducing eye strain and enabling comfortable long-term observation.

Eye Relief and Minimizing Eyestrain

Adjusting the distance between the lenses in a telescope such that the final image is formed at infinity is crucial for minimizing eyestrain. This typically means setting the separation equal to the sum of the individual focal lengths of the objective and eyepiece. Understanding this concept ensures that optical instruments are user-friendly and comfortable for extended viewing periods.

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