The Physics
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Opus in profectus

# Spherical Mirrors

## Discussion

### introduction

Curved mirrors come in two basic types: those that converge parallel incident rays of light and those that diverge parallel incident rays of light.

One of the easiest shapes to analyze is the spherical mirror. Typically such a mirror is not a complete sphere, but a spherical cap — a piece sliced from a larger imaginary sphere with a single cut. Although one could argue that this statement is quantifiably false, since ball bearings are complete spheres and they are shiny and plentiful. Nonetheless as far as optical instruments go, most spherical mirrors are spherical caps.

Start by tracing a line from the center of curvature of the sphere through the geometric center of the spherical cap. Extend it to infinity in both directions. This imaginary line is called the principal axis or optical axis of the mirror. Any line through the center of curvature of a sphere is an axis of symmetry for the sphere, but only one of these is a line of symmetry for the spherical cap. The adjective "principal" is used because its the most important of all possible axes. Compare this with the principal of a school, who is in essence the most important or principal teacher. The point where the principal axis pierces the mirror is called the pole of the mirror. Compare this with the poles of the Earth, the place where the imaginary axis of rotation pierces the literal surface of the spherical Earth.

Imagine a set of rays parallel to the principal axis incident on a spherical mirror (paraxial rays as they are sometimes called). Let's start with a mirror curved like the one shown below — one where the reflecting surface is on the "inside", like looking into a spoon held correctly for eating, a concave mirror. Rays of light parallel to the principal axis of a concave mirror will appear to converge on a point in front of the mirror somewhere between the mirror's pole and its center of curvature. That makes this a converging mirror and the point where the rays converge is called the focal point or focus. Focus was originally a Latin word meaning hearth or fireplace — poetically, the place in a house where the people converge or, analagously, the place in an optical system where the rays converge. With a little bit of geometry (and a lot of simplification) it's possible to show that the focus lies approximately midway between the center and pole. I won't try this proof.

Positions in the space around a spherical mirror are described using the principal axis like the axis of a coordinate system. The pole serves as the origin. Locations in front of a spherical mirror (or a plane mirror, for that matter) are assigned positive coordinate values. Those behind, negative. The distance from the pole to the center of curvature is called (no surprise, I hope) the radius of curvature (r). The distance from the pole to the focal point is called the focal length (f). The focal length of a spherical mirror is then approximately half its radius of curvature.

 f ≈ r 2

It is important to note up front that this is an approximately true relationship. We will assume it to be exactly true until becomes a problem. For many mundane applications, it's close enough to the truth that we won't care. It's not until we encounter situations requiring extreme precision that we'll deal with this aberration (as it is literally called). Astronomical telescopes should not be built with spherical mirrors. Real telescopes are made with parabolic or hyperbolic mirrors, but as I said earlier, we'll deal with this later.

Now, imagine a mirror with the opposite curvature — one where the reflecting surface is on the "outside", like looking into a spoon that's been flipped upside down from its useful orientation, a convex mirror. Let's shine paraxial rays onto this mirror and see what happens. Convex mirrors are diverging mirrors. Instead of converging onto a point in front of the mirror, here rays of light parallel to the principal axis appear to diverge from a point behind the mirror. We'll also call this location the focal point or focus of the mirror even though its disagrees with the original concept of the focus as a place where things meet up. In your best Russian reversal voice say, "In convex house, people go away from hearth" (or something like that, but funnier).

Locations in front of a diverging mirror have positive position values, since points in front of any mirror are always positive. The distance from the pole to the center of curvature is still the radius of curvature (r) but now its negative. The distance from the pole to the focus is still the focal length (f), but now it's also negative. With two sign switches, the rule that focal length is half the radius of curvature is still true in the same approximate way as before.

 f ≈ r 2

We have just discussed the basic and important concepts associated with spherical mirrors. Let's now talk about how they're used.

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### equations

Geometric derivation of the magnification equation. Similar triangles. The magnification equation.

 M = hi = di ho do

Geometric derivation of the spherical mirror equation. Magnification equation, plus new similar triangles.

 M = hi = di = f ho do do − f

Cross multiply, distribute, collect like terms.

 di(do − f) = dof dido − dif = dof dido = dif + dof

Divide by didof.

 dido = dif + dof didof didof didof

Simplfy. The spherical mirror formula.

 1 = 1 + 1 f do di

Uh huh, fried taters.