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

[amanara]

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

 

[Bob S]

The quickest way (only approximate) is to use thin lens formulas:
Use f_o = object distance; f_i = image distance; and f_fl = focal length.
So 1/f_o + 1/f_i = 1/f_fl [basic thin lens formula]
If both f_o and f_i = R (radius of curvature)
then 1/R + 1/R = 1/f_fl = 2/R
So f_fl = R/2

 

[amanara]

but what if the object distance is not equal to image distance?

 

[jtbell]

Does this help?

Mirror Equation (in particular the bottom section)

(I found this by a Google search for "focal length of a spherical mirror".)

 

[Bob S]

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 f_o = f_i = R

Using the thin lens formula 1/f_o + 1/f_i = 1/f_fl
Substituting R we get
1/R + 1/R = 1/f_fl
So 2/R = 1/f_fl
Now suppose the object is at infinity
1/inf. + 1/f_i = 1/f_fl = 2/R
then f_i = R/2