Use the law of reflection to prove that the focal length of a mirror is half its radius of curvature. That is, prove that \(f=R / 2 .\) Note this is true for a spherical mirror only if its diameter is small compared with its radius of curvature.

When learning about the behavior of light as it interacts with surfaces, the law of reflection is fundamental. According to this law, when a light ray strikes a reflective surface, the angle at which the light hits the surface, known as the angle of incidence, is equal to the angle at which it reflects away, called the angle of reflection. These angles are measured from the normal, an imaginary line perpendicular to the surface at the point of incidence.

The law is concisely expressed as: \( \angle_i = \angle_r \), where \( \angle_i \) is the angle of incidence and \( \angle_r \) is the angle of reflection. This principle applies perfectly to flat (planar) mirrors, but when dealing with spherical mirrors, the angles and geometry can be a bit more complicated due to the curved surface. However, even for spherical mirrors, the law of reflection holds true for each point on the mirror, helping us understand how images are formed.

In the context of spherical mirrors, the radius of curvature (denoted as R) is the radius of the sphere from which the mirror is a section. It plays a pivotal role in determining the properties of the mirror, such as its focal length. The center of curvature (C), which is not physically part of the mirror but a conceptually important point, lies on the surface of this hypothetical sphere at a distance R from any point on the mirror.

The relationship between the radius of curvature and the focal length (f) of mirrors is quite intuitive—focal length is essentially determined by the curvature of the mirror. For a spherical mirror, the focal length is the distance from the mirror's surface to the focus, the point where parallel light rays converge after reflection (for a concave mirror) or appear to diverge from (for a convex mirror). The assertion that the focal length is half the radius of curvature, \( f = \frac{R}{2} \), assumes that the mirror is a small section of a larger sphere, which ensures the approximation holds and the geometry remains consistent.

The geometry of spherical mirrors is based on segments of a sphere, which affects how light is reflected. There are two main types of spherical mirrors: concave and convex. A concave mirror, often used in telescopes, is a spherical segment that curves inward, like a bowl. A convex mirror, found in security mirrors and car side mirrors, bulges outward. For concave mirrors, light rays that are parallel to the principal axis (an imaginary line that passes through the center of curvature and the midpoint of the mirror) will converge at a common point called the focus after reflecting off the mirror.

In the proof provided in the exercise solution, geometry plays a crucial role in establishing the relationship between focal length and the radius of curvature. The solution mentions the approximation made when the diameter of the mirror is significantly smaller than the radius of curvature; this assumption allows us to treat angles as small and hence assume a linear relationship in a segment of the mirror's curve. As a result, the simplification \( f = \frac{R}{2} \) holds true under these specific conditions, illustrating the interplay between physical laws and geometric constraints in optical systems.