Understanding Concave Mirror Focal Length and Radius of Curvature

The question asks for the focal length of a concave mirror given its radius of curvature. This involves a fundamental concept in optics related to spherical mirrors.

Relationship Between Focal Length and Radius of Curvature

For any spherical mirror, whether concave or convex, the focal length (\(f\)) is directly related to its radius of curvature (\(R\)). The focal point is located exactly halfway between the mirror's pole (center of the mirror surface) and the center of curvature.

This means the magnitude of the focal length is half the magnitude of the radius of curvature. The relationship is expressed by the formula:

\[ f = \frac{R}{2} \]

For a concave mirror, both the focal length and the radius of curvature are typically considered negative according to the Cartesian sign convention when light comes from the left and reflects off the concave surface towards the right. However, the question provides the magnitude as a positive value (20.0 cm), and the options are also positive values representing magnitudes. We will work with the magnitudes for this calculation.

Calculating the Focal Length

Given:

• Radius of curvature, \(R = 20.0\) cm

Using the formula \(f = \frac{R}{2}\), we can calculate the focal length:

\[ f = \frac{20.0 \text{ cm}}{2} \]

\[ f = 10.0 \text{ cm} \]

Comparing with Options

Let's look at the provided options:

Our calculated focal length is 10.0 cm, which matches Option 4.

Conclusion on Concave Mirror Calculation

The focal length of a concave mirror with a radius of curvature of 20.0 cm is 10.0 cm, as the focal length is half the radius of curvature for a spherical mirror.

Revision Table: Concave Mirror Properties

Additional Information: Optics Concepts

Understanding spherical mirrors, like the concave mirror in this question, is key in optics. Here are some related concepts:

• Center of Curvature (C): The center of the sphere from which the mirror is a part. For a concave mirror, it's in front of the mirror.
• Pole (P): The geometrical center of the spherical mirror surface.
• Principal Axis: A straight line passing through the pole and the center of curvature.
• Principal Focus (F): The point on the principal axis where rays of light parallel to the principal axis converge after reflection from a concave mirror. For a convex mirror, parallel rays appear to diverge from this point (virtual focus).
• Sign Convention: A set of rules (like the Cartesian sign convention) used to assign signs to distances and lengths in optical systems to ensure consistent results when using formulas like the mirror equation (\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)).

The relationship \(f = R/2\) is a good approximation and holds true for paraxial rays (rays close to and parallel to the principal axis) incident on a spherical mirror. For rays far from the principal axis, spherical aberration occurs, where the rays do not converge at a single focal point.