Relation Between Focal Length and Radius of Curvature Class 10

I. Definition

The relationship between the focal length (f) and the radius of curvature (R) of a spherical mirror is a fundamental concept in geometrical optics. This relationship is given by the equation ( R = 2f ).

II. Explanation

The radius of curvature (R) of a spherical mirror is the radius of the sphere of which the mirror is a part. The focal length (f) is the distance between the pole (P) of the mirror and its focal point (F).

In a spherical mirror, the focal point is the point where parallel rays of light either converge (concave mirror) or appear to diverge (convex mirror) after reflecting from the mirror. The relationship ( R = 2f ) means that the focal length is half the radius of curvature.

III. Derivation of the Relationship

1. Concave Mirror:

• Consider a concave mirror with the center of curvature (C) and the focal point (F). When a parallel ray of light incident on the mirror passes through the focal point after reflection, the angle of incidence (i) and angle of reflection (r) are equal.
• Using simple geometry and the property of similar triangles, it can be shown that:
  [R = 2f]

1. Convex Mirror:

• In a convex mirror, the focal point (F) is virtual and lies behind the mirror. Parallel rays of light appear to diverge from this point after reflection.
• The same geometrical principles apply, leading to the relationship:
  [R = 2f]

IV. Applications

Understanding the relationship between the focal length and the radius of curvature is crucial in designing optical instruments such as telescopes, microscopes, and cameras. It is also fundamental in the study of image formation by mirrors.

V. Diagrams and Illustrations

• Diagram for Concave Mirror:
• A ray diagram showing a concave mirror, the center of curvature (C), the focal point (F), and the principal axis.
• Diagram for Convex Mirror:
• A ray diagram showing a convex mirror, the center of curvature (C), the focal point (F), and the principal axis.

Short Answer Type Questions

1. Question: What is the relationship between the focal length (f) and the radius of curvature (R) of a spherical mirror?
   Answer: The relationship is ( R = 2f ), where R is the radius of curvature and f is the focal length.
2. Question: How is the focal length (f) related to the radius of curvature (R) in a concave mirror?
   Answer: The focal length (f) is half of the radius of curvature (R), given by ( f =R/2 ).
3. Question: How is the focal length (f) related to the radius of curvature (R) in a convex mirror?
   Answer: The focal length (f) is half of the radius of curvature (R), given by ( f =R/2 ).
4. Question: If the radius of curvature (R) of a concave mirror is 20 cm, what is its focal length (f)?
   Answer: The focal length (f) is 10 cm, calculated using ( f =R/2 ).
5. Question: If the focal length (f) of a convex mirror is 15 cm, what is its radius of curvature (R)?
   Answer: The radius of curvature (R) is 30 cm, calculated using ( R = 2f ).
6. Question: Why is the relationship ( R = 2f ) important in optical design?
   Answer: This relationship is crucial for accurately designing and constructing optical instruments, ensuring proper focusing and image formation.
7. Question: How does the focal length (f) affect the image formation in a concave mirror?
   Answer: The focal length (f) determines where the parallel rays of light converge to form an image, affecting the image’s size and position.
8. Question: How does the radius of curvature (R) affect the curvature of a mirror?
   Answer: The radius of curvature (R) determines the mirror’s curvature; a smaller R means a more curved mirror and a larger R means a less curved mirror.
9. Question: Can the relationship ( R = 2f ) be applied to both concave and convex mirrors?
   Answer: Yes, the relationship ( R = 2f ) applies to both concave and convex mirrors.
10. Question: How is the focal point (F) defined in the context of spherical mirrors?
    Answer: The focal point (F) is the point where parallel rays of light either converge (concave mirror) or appear to diverge from (convex mirror) after reflecting from the mirror.
11. Question: In a convex mirror, why is the focal length (f) considered positive?
    Answer: The focal length (f) is positive because the focal point is virtual and located behind the mirror.
12. Question: In a concave mirror, why is the focal length (f) considered negative?
    Answer: The focal length (f) is negative because the focal point is real and located in front of the mirror.
13. Question: What is the significance of the center of curvature (C) in spherical mirrors?
    Answer: The center of curvature (C) is the center of the sphere of which the mirror is a part, and it helps determine the mirror’s focal length.
14. Question: How do you calculate the focal length if you know the radius of curvature?
    Answer: The focal length (f) can be calculated using the formula ( f =R/2 ).
15. Question: How do you calculate the radius of curvature if you know the focal length?
    Answer: The radius of curvature (R) can be calculated using the formula ( R = 2f ).
16. Question: Explain why the focal length is half the radius of curvature in spherical mirrors.
    Answer: This is a geometrical property of spherical mirrors where the focal point is midway between the pole and the center of curvature.
17. Question: What would happen to the focal length if the radius of curvature is doubled?
    Answer: If the radius of curvature is doubled, the focal length will also be doubled.
18. Question: What would happen to the focal length if the radius of curvature is halved?
    Answer: If the radius of curvature is halved, the focal length will also be halved.
19. Question: How does the focal length affect the field of view in optical instruments?
    Answer: A shorter focal length provides a wider field of view, while a longer focal length provides a narrower field of view.
20. Question: Why is the focal length important in determining the magnification of images?
    Answer: The focal length affects the size and position of the image, which in turn determines the magnification.
21. Question: How is the focal point related to the principal axis?
    Answer: The focal point lies on the principal axis, midway between the pole and the center of curvature.
22. Question: What is the principal axis in the context of spherical mirrors?
    Answer: The principal axis is the straight line that passes through the pole and the center of curvature of the mirror.
23. Question: Can the relationship ( R = 2f ) be derived experimentally?
    Answer: Yes, the relationship ( R = 2f ) can be verified experimentally by measuring the radius of curvature and the focal length of a spherical mirror.
24. Question: How does the radius of curvature affect the shape of a mirror?
    Answer: The radius of curvature determines the degree of curvature; a smaller radius of curvature results in a more curved mirror, while a larger radius of curvature results in a flatter mirror.
25. Question: In which type of spherical mirror is the focal point real?
    Answer: In a concave mirror, the focal point is real and located in front of the mirror.