# F = R/2

1. Apr 5, 2009

### amanara

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

2. Apr 5, 2009

### Bob S

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

3. Apr 5, 2009

### amanara

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

4. Apr 6, 2009

### Staff: Mentor

Does this help?

Mirror Equation (in particular the bottom section)

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

5. Apr 6, 2009

### Bob S

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: Apr 6, 2009