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