A dentist is examining a dental filling in a patient’s...

A dentist is examining a dental filling in a patient’s tooth. The diameter of the filling is 2.4 mm, and the dentist’s near point is 17.0 cm. To get a better look at the filling, the dentist dons safety goggles fitted with magnifying glasses $(f=6.0 \mathrm{cm}) .$ Find the greatest possible angular size (in radians) of the patient's filling when viewed by the dentist, both (a) without and $\quad$ (b) with the magnifying glasses.

Key Concepts

Angular Size

Angular size is a measure of how large an object appears to an observer. It is defined as the angle subtended at the observer’s eye by the object, and for small angles, it can be approximated by the ratio of the object’s physical size to its distance from the observer. This concept is crucial in optics as it relates to the resolving capability of the eye and imaging systems.

Near Point

The near point is the closest distance at which the human eye can focus on an object clearly. It is an important parameter in vision as objects closer than the near point appear blurred. In optical instruments like magnifiers, adjusting the apparent position of the object to the near point can maximize the angular size and perceived clarity of the image.

Magnifying Glass

A magnifying glass is a simple convex lens used to produce an enlarged virtual image of an object. It works by bending light rays so that the object appears closer to the observer than it actually is, effectively increasing its angular size. The effectiveness of a magnifying glass is often discussed in terms of its focal length, with shorter focal lengths typically offering higher magnification when the image is adjusted to the near point.

Lens Formula

The lens formula relates the object distance, image distance, and focal length of a lens, and is given by the equation 1/f = 1/d_o + 1/d_i. This formula is fundamental in optical calculations as it helps in determining where an image will be formed. In devising optical aids like magnifiers, the lens formula is used to set up conditions that maximize the angular size of the viewed object by positioning the image, typically at the near point.

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