The relationship between an object, its image, and a spherical mirror isn't magic. It's geometry. To derive the formulas that govern mirrors, we rely on a key simplification called the paraxial ray approximation or small-angle approximation. This assumes that light rays striking the mirror are very close to the principal axis and make small angles with it. For these rays, the curved surface of the mirror behaves almost like a flat plane at the point of reflection.

This assumption is crucial. It allows us to use simple geometry and trigonometry, like the property of similar triangles, without getting bogged down in complex curves. Without it, we'd have to deal with spherical aberration, where rays hitting different parts of the mirror don't all converge at a single focal point.

Let's derive the mirror formula using the concave mirror above. An object of height $h_o$ is placed at a distance $u$ from the pole (P). A real, inverted image of height $h_i$ is formed at a distance $v$. The center of curvature is C, and the focal point is F.

From the geometry, we can see two pairs of similar triangles:

1. Triangle ABC is similar to A'B'C.
2. Triangle MDF is similar to A'B'F. (We assume M is at the pole P for paraxial rays, so MD has height $h_o$.)

From the first pair, we get the ratio:

$\frac{A'B'}{AB} = \frac{CB'}{CB}$ which translates to $\frac{h_i}{h_o} = \frac{R-v}{u-R}$

From the second pair, we get:

$\frac{A'B'}{MD} = \frac{B'F}{PF}$ which translates to $\frac{h_i}{h_o} = \frac{v-f}{f}$

Equating these two expressions and simplifying leads us to the mirror formula. But to use it correctly, we need a consistent way to handle signs for distances.

The New Cartesian Sign Convention provides a standard set of rules for optical measurements. It's essential for getting the right answer every time, whether you're dealing with mirrors or lenses.

Applying these rules, for a real image formed by a concave mirror, the object distance $u$, image distance $v$, and focal length $f$ are all negative because they are measured to the left of the pole, against the incident light.

After applying the sign convention and a bit of algebra, the geometric relationships simplify into a single, elegant equation.

A simple but important related formula connects the focal length ($f$) to the radius of curvature ($R$).

The mirror formula tells us where an image will be, but not how large it is. For that, we need the linear magnification formula, which is the ratio of the image height ($h_i$) to the object height ($h_o$).

Let's work through an example. An object is placed 30 cm in front of a concave mirror with a focal length of 20 cm. Where is the image formed, and what is its nature?

Let's review the key terms and formulas we've derived.

Now, test your understanding of these derivations and calculations.

Understanding these derivations is the key to solving more complex problems involving combinations of mirrors and lenses.