Sign Convention and Focal Length of Spherical Mirrors

Setting Up a Common Language: The Cartesian Sign Convention

Before you can use any formula for mirrors or lenses, you need a consistent set of rules for assigning positive and negative signs to distances and heights. Without such rules, you would need a different equation for every arrangement of object, mirror, and image. The system adopted throughout optics is called the Cartesian sign convention, and it borrows the familiar idea of a coordinate axis.

Here is how it works:

• Origin: All distances are measured from the pole of the mirror (or the optical centre of the lens). This point acts as the origin of a coordinate system.
• Direction of light as the positive x-axis: Distances measured in the same direction as the incoming light are positive. Distances measured in the direction opposite to the incoming light are negative.
• Heights follow the y-axis: Heights measured upward from the principal axis (perpendicular to it) are positive. Heights measured downward are negative.

Fig 9.2: The Cartesian Sign Convention

In a typical setup, the object sits to the left and light travels from left to right. The mirror is on the right. So:

• The object distance is measured from the pole to the object, which is to the left. Since that is against the direction of light, the object distance comes out negative.
• For a concave mirror, the focus and centre of curvature are also on the same side as the object (in front of the mirror), so their distances from the pole are negative too.
• For a convex mirror, the focus and centre of curvature are behind the mirror (on the right), which is along the direction of light, so their distances are positive.

The beauty of this system is that once you set it up, a single mirror formula and a single lens formula cover every possible case, whether the mirror is concave or convex, whether the image is real or virtual, magnified or diminished. You just plug in the signed values and the algebra handles the rest.

What Happens When Parallel Light Hits a Spherical Mirror

Now that distances have clear signs, the next question is: where exactly does a mirror bring light to a focus? To answer this, picture a beam of rays all travelling parallel to the principal axis and striking the mirror.

The Principal Focus

There is one important condition to keep in mind. The rays must be paraxial (from the Greek para, meaning near, and axis): they hit the mirror at points close to the pole and make small angles with the principal axis. Under this condition:

• Concave mirror: The reflected rays all converge and actually meet at a definite point on the principal axis. This point is called the principal focus, labelled $F$. It lies between the pole $P$ and the centre of curvature $C$.
• Convex mirror: The reflected rays spread apart after bouncing off the surface. However, if you trace them backward (behind the mirror), they all appear to come from a single point on the principal axis. That point is also called the principal focus $F$, but since the rays never actually pass through it, the focus of a convex mirror is virtual.

Fig 9.3: Focus of a concave and convex mirror

The distance between $F$ and $P$ is called the focal length, written as $f$.

The Focal Plane

What if the parallel beam comes in at a slight angle to the principal axis instead of exactly along it? In that case, the reflected rays still converge (or appear to diverge from) a single point, but that point is no longer $F$ itself. It lies on a flat surface that passes through $F$ and stands perpendicular to the principal axis. This surface is called the focal plane of the mirror (Fig. 9.3(c)). Any parallel beam, regardless of the angle it makes with the axis, comes to a focus somewhere on this plane.

Deriving the Focal Length: Why $f = R/2$

One of the most important results in mirror geometry is that the focal length equals exactly half the radius of curvature. Let us see where this comes from.

Setting Up the Geometry

Consider a single ray arriving parallel to the principal axis and hitting the concave mirror at a point $M$. The centre of curvature of the mirror is $C$, so the line $CM$ is a radius of the sphere and therefore perpendicular to the mirror surface at $M$. In other words, $CM$ is the normal at $M$.

Fig 9.4: Geometry of reflection of an incident ray on (a) concave and (b) convex spherical mirror

Let $\theta$ be the angle between the incoming ray and the normal $CM$. By the law of reflection, the reflected ray also makes angle $\theta$ with $CM$.

Drop a perpendicular from $M$ to the principal axis and call the foot of that perpendicular $D$. Now look at the angles:

• The angle $\angle MCP = \theta$ (the angle at $C$ in triangle $MCP$, which is the same as the angle of incidence because the incident ray is parallel to the axis).
• The angle $\angle MFP = 2\theta$. Why? Because $\angle MFP$ is the exterior angle of triangle $MCF$, so it equals the sum of the two non-adjacent interior angles: $\theta$ (at $C$) $+ \theta$ (the angle of reflection at $M$ measured from $CM$).

Writing the Trigonometric Relations

From the right triangle $MCD$:

$\tan\theta = \frac{MD}{CD} \qquad \text{(9.1a)}$

From the right triangle $MFD$:

$\tan 2\theta = \frac{MD}{FD} \qquad \text{(9.1b)}$

Here $MD$ is the height of the point $M$ above the axis, $CD$ is the horizontal distance from $C$ to $D$, and $FD$ is the horizontal distance from $F$ to $D$.

Applying the Paraxial Approximation

Because the rays are paraxial, $\theta$ is very small. For small angles (measured in radians):

$\tan\theta \approx \theta \quad \text{and} \quad \tan 2\theta \approx 2\theta$

Substitute these into Equations (9.1a) and (9.1b):

$\theta \approx \frac{MD}{CD} \quad \text{and} \quad 2\theta \approx \frac{MD}{FD}$

Divide the second relation by the first:

$\frac{2\theta}{\theta} = \frac{MD/FD}{MD/CD}$

The $MD$ terms cancel and $2\theta / \theta = 2$, so:

$2 = \frac{CD}{FD}$

Rearranging:

$FD = \frac{CD}{2} \qquad \text{(9.2)}$

From FD to Focal Length

There is one last step. Since the ray is paraxial, $M$ is very close to the pole $P$, which means $D$ is also nearly at $P$. So:

• $FD \approx FP = f$ (the focal length)
• $CD \approx CP = R$ (the radius of curvature)

Substituting into Equation (9.2):

$f = \frac{R}{2} \qquad \text{(9.3)}$

This is the result: the focal length of a spherical mirror is half its radius of curvature. Although we derived it for a concave mirror, the same geometry applies to a convex mirror, so the relation $f = R/2$ holds for both types.

Why This Result Matters

This formula connects two physical quantities, the focal length and the radius of curvature, through a simple factor of two. It means that if you know the curvature of a mirror (which is set during manufacturing), you immediately know where it will bring parallel light to a focus. It also means a mirror with a shorter radius curves more sharply and focuses light closer to its surface.

Keep in mind that $f = R/2$ relies on the paraxial approximation. Rays that strike far from the pole do not obey this relation exactly: they focus at slightly different points, producing a blurring effect called spherical aberration (the inability of a spherical mirror to bring all parallel rays to a single sharp focus). For most textbook problems, however, the paraxial condition is assumed, and $f = R/2$ is treated as exact.