Deriving the Focal Length of a Concave Mirror

[wiz0r]

Homework Statement

Ok, so my profesor asked my to prove this;

Prove that f = R/2. To prove it geometrically.

.....)
.....)
.... F ...)
_________._____)___________
......)
......)
....)

Ok, so I have that curved convave mirror, now I need to prove that f = R/2, where R is the radius of the circle, and f is the focal lenght.

The Attempt at a Solution

Um, well I can't understand why is it R/2, and not just R, I'll post whatever I'll find while doing research. But, please, can anyone help me get started? Or just explain it to, I'd be really grateful. Thanks!

 

[wiz0r]

ok, I've got a better picture;

http://en.wikipedia.org/wiki/Image:Concave_mirror.svg

I can see it better now, why f = r/2, but, why? how can i prove it?

the center of curvature is the center of the circle, so obviously the focus = r/2, but how can i prove this geometrically?

 

[wiz0r]

Ok, well I found this;

http://www.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/mirror-calcs.htm

That explains it, I suppose, but do you think that's what my professor is looking for? Can you help me explain it better?

 

[wiz0r]

anyone?

 

[Doc Al]

I suspect that your professor wants you to apply the law of reflection (and a bit of geometry) to derive the focal length of a concave mirror. That last link does it for you, but I suggest you rederive it for yourself.