# Is the focal length of a mirror always shorter than the radius of curvature?

Is the focal length always shorter than the radius of curvature? If so, is there a reason for that?

This is to help me when drawing ray diagrams.

I am asking for spherical and plane mirrors.

I would be happy to find out for lenses as well, though do lenses have a centre of curvature ?

## 3 Answers

For the spherical mirror, the focal length is half the radius curvature.

Technically, a spherical mirror doesn't really have a focal length. For small arc lengths, circular mirrors can be approximated by parabolic mirrors, which do have a fixed focal length (for large arc lengths, this approximation breaks down, a phenomenon known as "spherical aberration"). The focal length of the parabola that approximates the sphere is half the radius of the sphere, a fact that is often simplified to "the focal length of a spherical mirror is half its radius". Conversely, the focal length of a parabola is half the radius of the sphere that approximates it at its extremum. As for why that is, there are mathematical explanations, but I'm not sure there's a simple explanation.

If the source of light was at the center of the curvature of a spherical mirror, i.e., at the distance $R$ from the mirror, the light would be reflected right back, because the incident and reflected angles will be $90^\circ$, i.e., it would converge at the source.

From the above, it should be obvious that, if the light source is moved further away from the center, the reflected light would be converging closer to the mirror. Following this logic, it should be clear that the focus of the mirror, which is a converging point of the reflected light coming from infinity, should be even closer to the mirror, i.e., the focal length should be less than $R$ (actually, about $\frac R 2$).