Why is radius of curvature equal to 2f for a lens or mirror?

[mkbh_10]

What is radius of curvature and why is it equal to 2f where f is the focal length of a lens or a mirror ?

 

[pixel01]

You can check the definition in this : ttp://mathworld.wolfram.com/RadiusofCurvature.html

The radius of a spherical mirror is equal to 2f : that's right. But the lense : I am not sure.
You can see in the attached picture: AOF is an isosceles triangle.

 

[whydoesitwork]

but looking at th diagram I don't see where the centre of that arc is.

 

[jtbell]

What is radius of curvature and why is it equal to 2f where f is the focal length of a lens or a mirror ?

For a concave mirror see

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/mireq.html#c1

For a lens, the radius of curvature (it can be different for the front and back surfaces of a lens) is generally not equal to 2f. For the focal length of a lens see

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/foclen.html#c2

and

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html#c1

 

[mkbh_10]

What is the significance of radius of curvature ?

 

[jtbell]

Imagine that the surface is part of a circle or sphere. The radius of curvature is the radius of that circle/sphere.

 

[pixel02]

but looking at th diagram I don't see where the centre of that arc is.

In my picture, the arc center is O because it is a part of a cicle. Remember that diagram and thurs formula only fits for a spherical concave mirror.

PS. Sorry I can not login my nick pixel01 because I forgot the pw and unfortunately the email address has been expired.