Two telescopes of different lengths produce the same angular...

Two telescopes of different lengths produce the same angular
magnification. Is the focal length of the long telescope's eyepiece
greater than or less than the focal length of the short telescope's
eyepiece? Explain.

Key Concepts

Angular Magnification

Angular magnification is the measure of how much larger an object appears through an optical instrument compared to the naked eye. In telescopes, it is determined by the ratio of the focal length of the objective to the focal length of the eyepiece. Maintaining the same angular magnification requires that this ratio remains constant, even when other parameters such as the tube length change.

Focal Length of Optical Components

The focal length of an optical component is the distance from the lens at which parallel incoming rays converge to a point. In telescopes, the objective's focal length largely determines the overall tube length, while the eyepiece's focal length works in conjunction with it to set the angular magnification. If an instrument has a longer overall focal length due to a longer tube, the eyepiece focal length must also be proportionally larger to maintain the same magnification ratio.

Telescope Design Relations

In telescope design, the overall tube length is approximately the sum of the objective and eyepiece focal lengths when set up for relaxed eye viewing of distant objects. For two telescopes with equal angular magnification, a longer tube typically implies a longer objective focal length, which means that to preserve the same f_obj/f_eye ratio, the eyepiece in the longer telescope must also have a greater focal length than that in the shorter telescope.

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