Prove how focus of a spherical mirror = Radius of curvature/2

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Discussion Overview

The discussion revolves around the relationship between the focal length of a spherical mirror and its radius of curvature. Participants explore the derivation of the formula that states the focal length equals half the radius of curvature, considering various scenarios including object and image distances.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using the thin lens formula to derive the relationship, proposing that if both object distance (fo) and image distance (fi) equal the radius of curvature (R), then the focal length (ffl) can be calculated as R/2.
  • Another participant questions the validity of the derivation when the object distance is not equal to the image distance, indicating a potential limitation in the initial approach.
  • A later reply references the mirror equation and attempts to clarify the situation by reiterating the case where the object is at the center of curvature, leading to the same conclusion about the focal length.
  • Further, it is noted that if the object is at infinity, the image distance can also be derived to be R/2, reinforcing the earlier claim but under different conditions.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the applicability of the thin lens formula under varying conditions of object and image distances. Multiple viewpoints are presented, and the discussion remains unresolved as to the general applicability of the derived relationship.

Contextual Notes

Limitations include the assumption that the object distance equals the image distance in certain cases, and the implications of varying object distances on the validity of the derived relationships are not fully explored.

amanara
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Prove how focus of a spherical mirror = Radius of curvature/2
 
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The quickest way (only approximate) is to use thin lens formulas:
Use fo = object distance; fi = image distance; and ffl = focal length.
So 1/fo + 1/fi = 1/ffl [basic thin lens formula]
If both fo and fi = R (radius of curvature)
then 1/R + 1/R = 1/ffl = 2/R
So ffl = R/2
 
but what if the object distance is not equal to image distance?
 
Does this help?

Mirror Equation (in particular the bottom section)

(I found this by a Google search for "focal length of a spherical mirror".)
 
amanara said:
but what if the object distance is not equal to image distance?
We know that if the object is at the center of radius of curvature R, then the image is also at the center of radius of curvature. So fo = fi = R

Using the thin lens formula 1/fo + 1/fi = 1/ffl
Substituting R we get
1/R + 1/R = 1/ffl
So 2/R = 1/ffl
Now suppose the object is at infinity
1/inf. + 1/fi = 1/ffl = 2/R
then fi = R/2
 
Last edited:

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