Spherical Mirrors
Not all mirrors are flat, of course. Now we want to consider two special cases of mirrors, both of which are pieces of a shiny sphere. Convex spherical mirrors are the outside of a reflecting sphere (or a part of it), and concave spherical mirrors are a piece of the shiny inside of a sphere. You can also think of a metallic spoon, although spoons are not always precisely a part of a sphere. The back side of the spoon is a convex and the inside a concave mirror. We can figure out what happens when light hits a convex mirror simply by using the law of reflection – that's all the physics we need, the rest is geometry! For every ray that is incident on the spherical mirror, the angle of incidence is equal to the angle of reflection. Since these angles are measured relative to the normal of the mirror surface, the trick is to find the normal for every point on the mirror surface. But since the mirror surface is part of a sphere, that's easy: all lines that start at the center of the sphere, i.e., all radial lines, are perpendicular to the surface of the sphere at the point where they poke through it! That's all we need to know to construct the reflected rays for a few incoming rays that are parallel to the optical axis, and also close to it. These rays are all reflected as if they were coming from a point that is halfway between the center and the surface of the sphere. This point is the focal point of the convex mirror. The distance from the mirror surface to the focal point is the focal length of the mirror, and as you can see, it is half the radius of the mirror. The reflected rays are not actually coming from the focal point, they are just reflected as if they would originate there. The convention is therefore that the focal length f of a convex mirror is negative: f =  R/2 , where R is the radius of the sphere. Let's now do the same thing for a concave mirror. Again, the radial lines are normal to the mirror surface. If we construct the reflected rays for a few rays parallel and close to the optical axis, we can see that they all intersect at one point on the optical axis. This point is halfway between the center of the sphere and the mirror surface and – you guessed it – is called the focal point of the concave mirror. For this type of mirror, the reflected rays actually do pass through the focal point, so by convention, the focal length f of a concave mirror is positive: f = +R/2, where R is again the radius of the sphere.
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