In geometrical optics, what is the focal length of a concave mirror that has a radius of curvature of 20.0 cm?

Introduction / Context:
Concave mirrors are widely used in devices such as shaving mirrors, head lights, reflecting telescopes, and many optical instruments. A key parameter that describes any spherical mirror is its focal length, which tells us how it converges or diverges light rays coming from infinity. In this question from basic geometrical optics, we are given the radius of curvature of a concave mirror and are asked to determine its focal length. Understanding the relationship between radius of curvature and focal length is very important for quick numerical calculations in physics examinations and practical applications in optical design and laboratory work.

Given Data / Assumptions:
• The mirror is a concave spherical mirror. • Radius of curvature R = 20.0 cm. • We assume the standard sign convention for spherical mirrors is used in theory, but for this simple magnitude question only the absolute value is needed. • We are asked for the focal length f in centimetres.

Concept / Approach:
In geometrical optics, a spherical mirror has a simple relationship between its focal length f and its radius of curvature R. The focus is located at the midpoint between the pole of the mirror and its center of curvature. For both concave and convex spherical mirrors, the magnitude of the focal length is given by f = R / 2. For a concave mirror, the focal length is treated as negative in sign convention, but when a question asks for focal length without focusing on sign, usually only the magnitude is expected in the answer. We therefore focus on the simple proportional relationship and compute half of the given radius of curvature.

Step-by-Step Solution:
Step 1: Recall the mirror formula relating focal length and radius of curvature: f = R / 2. Step 2: Substitute the given radius of curvature R = 20.0 cm into the formula. Step 3: Calculate f = 20.0 cm / 2 = 10.0 cm. Step 4: Report the focal length as 10 cm in magnitude, which is the required answer for this conceptual numerical question.

Verification / Alternative check:
One way to remember this relationship is geometric. For a spherical mirror, light rays coming from infinity and parallel to the principal axis reflect through a point that is midway between the mirror surface and the center of curvature. That midpoint is, by definition, the focal point. Since the center of curvature is at a distance R from the pole, the midpoint is at R / 2. This geometric interpretation is independent of the sign convention and simply confirms that the focal length in magnitude is half of the radius of curvature. You can also recall that textbook ray diagrams for spherical mirrors always show the focus halfway to the center of curvature, which is another confirmation.

Why Other Options Are Wrong:
Option a (5 cm): This would correspond to f = R / 4, which has no standard basis for spherical mirrors. Option c (15 cm): This value is three quarters of the radius and does not match the required half relationship. Option d (20 cm): This equals the full radius of curvature and would only be correct if focal length equalled radius, which is not true. Option e (25 cm): This is larger than the radius and has no physical meaning in the context of the standard mirror formula.

Common Pitfalls:
Students sometimes confuse focal length with radius of curvature and choose R instead of R / 2. Another frequent mistake is mixing sign convention with magnitude and worrying about negative signs when the question clearly asks only for the numerical value. Some learners also confuse formulas for lenses and mirrors, but the relationship f = R / 2 for spherical mirrors is a very direct one and should be memorised as a basic result. Always read the question carefully to see whether it is asking for magnitude or for a sign sensitive value in a coordinate system.

Final Answer:
The focal length of the concave mirror is 10 cm.