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 Noting that the expression $M \approx-f_{\mathrm{o}} / f_{\mathrm{e}}$ is no longer applicable because the object is so close, use the thin-lens and magnification equations to find the angular magnification of this telescope.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 $F_{\mathrm{o}}$ and $F_{\mathrm{e}}$ are so close together that the distance between them may be ignored.)

Key Concepts

Thin-Lens Equation

The thin-lens equation, given by 1/f = 1/d_o + 1/d_i, relates the focal length (f) of a lens to the object distance (d_o) and the image distance (d_i). This fundamental equation is crucial for understanding how lenses form images and is applicable under the assumption that the lens has negligible thickness, simplifying the analysis in optical systems such as telescopes.

Linear (Lateral) Magnification

Linear, or lateral, magnification describes the ratio of the image size to the object size and is defined as m = -d_i/d_o. While this concept is primarily concerned with the physical dimensions of the image, it lays the groundwork for understanding how optical devices modify the appearance of objects, a necessary step before relating physical magnification to angular magnification.

Angular Magnification

Angular magnification refers to the ratio of the angular size of the image as seen through an optical instrument to the angular size of the object when viewed with the naked eye. In telescopes, this concept is central because the human eye perceives enlargements in terms of angles subtended at the eye. This becomes particularly important when the standard approximations (such as M ? -f_o/f_e) fail due to the finite distance of the object.

Focal Length in Optical Systems

Focal length is a key parameter in optical systems that determines the converging or diverging power of a lens. In a telescope, the objective and eyepiece have specific focal lengths that play a vital role in image formation. Understanding the role of these focal lengths is essential not only for designing telescopes but also for analyzing image properties, such as magnification, especially when object distances are non-ideal.

Limitations of Approximations in Optical Instruments

Many optical calculations rely on approximations that assume objects are at infinity, making formulas like M ? -f_o/f_e applicable. However, when objects are relatively close, these approximations break down. Recognizing the limitations of such simplifications is important for correctly applying the thin-lens and magnification equations to obtain accurate measures of angular magnification in practical scenarios.

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