One of the largest refracting telescopes made to date has an...

One of the largest refracting telescopes made to date has an objective lens with a focal length of approximately $20 \mathrm{m}$. When this telescope is used to look at the full Moon, what is the angle subtended by the edges of the Moon in the image formed by the objective? If the eyepiece has a focal length of $5.0 \mathrm{cm},$ what is the total magnification?

Key Concepts

Focal Length

Focal length is the distance from a lens (or mirror) at which parallel rays of light converge to a point to form a sharp image. In optical systems, it governs the size of the resulting image; for distant objects, the linear size of the image is approximately equal to the focal length multiplied by the object's angular extent when the small angle approximation is valid.

Small Angle Approximation

The small angle approximation is a mathematical simplification used when an object’s angular size is very small (typically measured in radians). It assumes that for small angles, the sine or tangent of the angle is approximately equal to the angle itself, allowing one to relate the physical size of an image to the product of the optical system's focal length and the object's angular size.

Angular Magnification in Telescopes

Angular magnification in telescopes describes how much larger an object appears when viewed through the instrument compared to the naked eye. It is typically defined as the ratio of the focal length of the objective to that of the eyepiece, meaning that a longer objective or a shorter eyepiece leads to higher magnification, which allows finer details of distant objects to be observed.

Image Formation in Optical Instruments

Image formation in optical instruments involves concentrating light from a distant object to form a real image using a lens or mirror. This process capitalizes on geometrical optics principles, wherein rays from the object are refracted or reflected to meet at a focal plane, creating an image whose dimensions are dictated by the focal length and the object's angular size.

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