Mars subtends an angle of 8.0 ×10^-5 rad at the unaided eye....

Mars subtends an angle of $8.0 \times 10^{-5} \mathrm{rad}$ at the unaided eye. An astronomical telescope has an eyepiece with a focal length of $0.032 \mathrm{~m} .$ When Mars is viewed using this telescope, it subtends an angle of $2.8 \times 10^{-3}$ rad. Find the focal length of the telescope's objective lens.

Key Concepts

Focal Length Relationship in Optical Instruments

In optical systems like telescopes, the focal lengths of the objective and the eyepiece are critical in determining the overall performance of the instrument. The focal length of the objective lens is directly related to its ability to gather light and resolve distant objects, while the eyepiece enables the eye to view the magnified image. The magnification formula, which states that magnification equals the focal length of the objective divided by the focal length of the eyepiece, underscores the significance of these focal lengths in the design and functionality of telescopes.

Small Angle Approximation

The small angle approximation is a mathematical simplification used in optics when dealing with very small angles measured in radians. It allows the sine, tangent, and the angle itself to be considered nearly equivalent, facilitating easier calculations in problems involving the angular size of distant objects. This approximation is particularly useful in astronomical contexts where objects subtend extremely small angles at the observer's eye.

Angular Magnification

Angular magnification is the factor by which an optical instrument, such as a telescope, increases the apparent angular size of an object. It is given by the ratio of the angle subtended by the image to that subtended by the object when observed with the unaided eye. In telescopic systems, this magnification is typically expressed as the ratio of the focal lengths of the objective and the eyepiece, which allows astronomers to see distant objects, like planets, in greater detail.

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