A rear view mirror of a vehicle is cylindrical having radius...

A rear view mirror of a vehicle is cylindrical having radius of curvature $5 \mathrm{~cm}$ and length of arc of curved surface is $10 \mathrm{~cm}$. The field of view in radian, if it is assumed that the eye of the driver is at a large distance from the mirror, is: (a) $0.5$ (b) 1 (c) 2 (d) 4

Key Concepts

Cylindrical Mirror

A cylindrical mirror is a type of curved mirror where the reflecting surface is a segment of a cylinder. Unlike spherical or parabolic mirrors, cylindrical mirrors have curvature in one direction only, which affects how they form images and project views. Their unique geometry is useful in applications where a panoramic or extended view in one plane is desired, such as in vehicular rear view mirrors.

Arc Length and Central Angle Relationship

The fundamental relationship between the arc length of a circular segment and its central angle is given by the formula: arc length = radius × central angle (in radians). This formula is crucial in geometrically connecting the physical size of a curved surface to the angular extent that it subtends at the center of the circle. It allows one to convert a measured length along the circumference to an angle, which can be used in further calculations involving fields of view or coverage.

Field of View Calculation

Field of view in optical devices refers to the angular extent that can be seen through the device, determined by the geometry of the reflective or refractive surfaces. When a viewer is positioned at a significant distance, the actual field of view can be approximated by the angle subtended by the mirror’s reflective surface at the viewer's eye. Using the arc length and central angle relationship for the mirror design provides a way to compute this viewing angle, revealing the range of vision offered by the mirror.

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