### Correct Option: A. Concave mirror of focal length 5cm #### Step-by-Step Explanation: 1. **Understanding the Geometry of the Sphere**: - The problem states that we have a hollow metal sphere with a diameter of 20 cm. This means the radius (r) of the sphere is half of the diameter: \[ r = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \] 2. **Identifying the Type of Mirror**: - The inside portion of the hollow sphere is polished. When the inner surface of a sphere is polished, it behaves like a concave mirror. This is because the reflective surface curves inward, which is characteristic of concave mirrors. 3. **Calculating the Focal Length**: - The focal length (f) of a spherical mirror is related to its radius of curvature (R) by the formula: \[ f = \frac{R}{2} \] - For our hollow sphere, the radius of curvature (R) is equal to the radius of the sphere, which we calculated as 10 cm. Therefore, we can substitute this value into the formula: \[ f = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \] 4. **Conclusion**: - Since the inner surface of the hollow sphere acts as a concave mirror and we have calculated the focal length to be 5 cm, the correct answer is option A: **A concave mirror of focal length 5 cm**. #### Why Other Options Are Incorrect: - **Option B: Concave mirror of focal length 10 cm**: - This option incorrectly states the focal length. The focal length is derived from the radius of curvature, which is 10 cm, but the focal length is half of that, which is 5 cm. Therefore, this option is incorrect. - **Option C: Convex mirror of focal length 5 cm**: - This option is incorrect because the inner surface of the hollow sphere is concave, not convex. A convex mirror would have a reflective surface that bulges outward, which is not the case here. - **Option D: Convex mirror of focal length 20 cm**: - Similar to option C, this option is incorrect for the same reason. The inner surface is not convex, and even if it were, the focal length would not be 20 cm. The focal length for a convex mirror is also positive and would be calculated differently. #### Common Pitfalls: - Confusing the terms "concave" and "convex" mirrors. Remember that concave mirrors curve inward and are used to focus light, while convex mirrors curve outward and diverge light. - Miscalculating the focal length by not halving the radius of curvature. Always remember the relationship \( f = \frac{R}{2} \). ### Revision Summary: - A hollow metal sphere with a polished inner surface acts as a concave mirror. - The radius of the sphere is 10 cm, leading to a focal length of 5 cm. - The correct answer is option A: concave mirror of focal length 5 cm. - Remember the formula \( f = \frac{R}{2} \) for calculating the focal length of spherical mirrors.