The correct option is R = 2f.
Explanation
In the optics of spherical mirrors, the geometric relationship between the curvature of the surface and its focusing capability is derived using the laws of reflection. The principal parameters involved are the radius of curvature ($R$) and the focal length ($f$).
Analysis:
- The radius of curvature ($R$) is the distance between the pole of the mirror and the center of the sphere of which the mirror is a part.
- The focal length ($f$) is the distance between the pole and the principal focus, where parallel rays converge (concave mirror) or appear to diverge from (convex mirror).
- For a spherical mirror with a small aperture, rays falling close to the principal axis (paraxial rays) focus at a single point. Under this approximation, the principal focus lies exactly midway between the pole and the center of curvature.
- Consequently, the radius of curvature is twice the focal length:
$R = 2f$
Key Takeaway: For spherical mirrors with small apertures, the radius of curvature is strictly double the focal length ($R = 2f$). This relationship assumes paraxial approximation to minimize spherical aberration.