An astronomical telescope is being used to examine a...

An astronomical telescope is being used to examine a relatively close object that is only $114.00 \mathrm{~m}$ away from the objective of the telescope. The objective and eyepiece have focal lengths of $1.500$ and $0.070 \mathrm{~m}$, respectively. Noting that the expression $M \approx-f_{\mathrm{o}} / f_{\mathrm{c}}$ is no longer applicable because the object is so close, use the thin-lens and magnification equations to find the angular magnification of this telescope. (Hint: See Figure $26.41$ and note that the focal points $F_{0}$ and $F_{\mathrm{c}}$ are so close together that the distance between them may be ignored.)

Key Concepts

Astronomical Telescope

An astronomical telescope is an optical instrument designed to view distant objects by collecting light through a large objective lens and then magnifying it with an eyepiece. Traditionally, these telescopes are calibrated for objects at infinity, simplifying many calculations. However, when objects are closer, as in the problem scenario, modifications to the standard approximations (like using precise forms of the thin-lens and magnification equations) become necessary.

Angular Magnification

Angular magnification refers to the enhancement of the apparent angle under which an object is seen through an optical system in comparison to the naked eye. It is particularly important in telescopes where the objective is to make distant objects or objects that have a small angular size appear larger. Unlike linear magnification, angular magnification considers the observer's perspective, making it a more relevant measure for instruments like telescopes.

Focal Length

Focal length, a key characteristic of lenses, indicates the distance over which parallel rays of light converge to a point (the focal point). In optical instruments such as telescopes, different focal lengths for the objective and eyepiece determine the overall magnification and resolution. The relationship between these focal lengths is foundational when deriving magnification formulas and understanding the behavior of optical systems.

Thin-Lens Equation

The thin-lens equation, 1/f = 1/d? + 1/d?, is a fundamental relation in geometric optics that connects the focal length of a lens (f) with the object distance (d?) and the image distance (d?). This equation is essential for analyzing how lenses form images, enabling one to calculate the positions where images are formed when light passes through thin lenses.

Linear Magnification

Linear magnification is defined as the ratio of the image height to the object height, and equivalently as the ratio of the image distance to the object distance (m = d?/d?) in the paraxial regime. This concept is crucial for understanding how much larger or smaller an image will be compared to the object, which directly stems from the lens equations and directly relates to image formation in optical systems.

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