Question

The relationship between the radius of curvature R and focal length f of a spherical mirror of small aperture could be represented as:

• A. R = 1/f
• B. R = 2f
• C. R = 3f
• D. R = f

Hint:

The focus of a spherical mirror is always halfway to its center of curvature:
\(f = \frac{R}{2}\) or \(R = 2f\).
Keep this simple relation in mind for numerical problems.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the radius of curvature (\(R\)) and the focal length (\(f\)) of a spherical mirror with a small aperture.

Step 2: Key Formula or Approach:
For spherical mirrors with a small aperture (where paraxial rays are considered), the principal focus lies exactly halfway between the pole and the center of curvature.

Step 3: Detailed Explanation:

• Let \(C\) be the center of curvature and \(P\) be the pole of the mirror. The distance \(PC\) is equal to the radius of curvature \(R\).

• Let \(F\) be the principal focus. The distance \(PF\) is the focal length \(f\).

• Geometry shows that for a spherical mirror of small aperture, the principal focus \(F\) is the midpoint of the line segment \(PC\).

• Therefore:
\[ PF = \frac{PC}{2} \implies f = \frac{R}{2} \]

• Rearranging this equation to solve for \(R\), we get:
\[ R = 2f \]

• This relation is valid for both concave and convex mirrors, provided their aperture is small compared to their radius of curvature.


Step 4: Final Answer:
The correct relationship is \(R = 2f\).