Final answer:
The focal length of a spherical concave mirror is half its radius of curvature, so if the radius of curvature is R, the focal length is R/2. The smaller the radius, the shorter the focal length and the more powerful the mirror. The answer is E. R/2.
Step-by-step explanation:
If a spherical concave mirror has a radius of curvature R, its focal length is R/2.
This is because, under the small-angle approximation, the relationship between the focal length (f) of a concave spherical mirror and its radius of curvature (R) is defined by the equation R = 2f. Therefore, the focal length is half the radius of curvature.
The correct answer to the given question is E. R/2.
The smaller the radius of curvature, the smaller the focal length, and thus, the more powerful the mirror.
This indicates that a concave mirror with a smaller radius of curvature will have a shorter focal length and therefore a greater power.
Using the law of reflection and simple trigonometry, the relationship confirms that for a concave mirror, the focal length is indeed half the radius of curvature.
For the practice problem provided, the focal length of a concave mirror with a radius of curvature of 0.8 m would be 0.4 m, displaying the direct application of the relationship where the focal length is half the radius of curvature.