Prove that the angular magnification of a telescope having...

Prove that the angular magnification of a telescope having an objective focal length of $f_{\mathrm{obj}}$ and an eyepiece focal length of $f_{\text {eye is given by Eq. }}(9)$ when the objective and the eyepiece are separated by the sum of their focal lengths, $f_{\mathrm{obj}}+f_{\mathrm{eye}}$ $$m=\frac{f_{\mathrm{obj}}}{f_{\mathrm{eye}}}$$

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Key Concepts

Angular Magnification

Angular magnification is the ratio of the angular size of the image seen through an optical instrument to the angular size of the object seen with the naked eye. In telescopes, it quantifies how much larger distant objects appear, and it is derived from the geometry of the light paths through the system.

Focal Length

The focal length of a lens is the distance over which parallel rays of light converge to a point. In optical instruments, both the objective and eyepiece have focal lengths that determine how light is brought to focus and subsequently magnified.

Telescope Optical System

A telescope is an optical system that uses an objective lens to collect light and form a real image, which is then magnified by an eyepiece. The design relies on careful alignment and spacing of the lenses to ensure that the image is properly formed and enlarged.

Lens Separation in Telescopes

In telescopes, when the separation between the objective and eyepiece equals the sum of their focal lengths, the system is configured so that the light exiting the eyepiece is parallel. This collimated output is essential for comfortable viewing and leads directly to the angular magnification relationship m = f_obj / f_eye.

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