Final answer:
Using the law of reflection and the small-angle approximation, it is shown that the focal length of a spherical mirror is equal to half the radius of curvature, or f = R/2, for mirrors whose diameter is significantly smaller than their radius of curvature.
Step-by-step explanation:
To demonstrate that the focal length of a mirror (f) is half its radius of curvature (R), we can refer to the physical properties of spherical mirrors under the law of reflection. According to the law of reflection, the angle of incidence is equal to the angle of reflection. Now, consider a ray of light that is incident parallel to the principal axis of a spherical mirror and reflects through the mirror's focal point.
The focal point (F) is defined as the point through which all parallel rays after reflection pass for a spherical mirror. In the case of a spherical mirror whose diameter is much smaller compared to its radius of curvature, this setup simplifies to a right-angled isosceles triangle, given the small-angle approximation where the sine of a small angle is approximately equal to the angle itself (sin θ ≈ θ).
In the isosceles triangle formed by the axis of the mirror, the incident ray, and the line segment joining the center of curvature (C) with the point of incidence (X), the lengths of the sides from the center of curvature to the point of incidence (CX) and from the point of incidence to the focal point on the axis (XF) will be equal (CX = XF), given the small angle and the reflection properties.
The radius of curvature (R) is then twice the distance from the focal point to the center of curvature (R = CF + FP), and since CF = FP, we can deduce that R = 2FP = 2f. Hence, it is shown that the focal length is half of the radius of curvature under the condition mentioned (f = R/2).