The focal length f is related to the radius of curvature r of the spherical convex mirror by
(a) f =

Q: Why is the focal length positive for a convex mirror?

According to the Cartesian sign convention, distances measured in the direction of incident light are positive. For a convex mirror, the focus lies behind the mirror, so we measure from the pole in the direction of light, making it positive.

Q: Does $f = r/2$ hold for all mirrors?

It holds strictly for paraxial rays, which means it is valid for mirrors with a small aperture where the rays strike close to the principal axis.

Q: What if it was a concave mirror?

For a concave mirror, the focus and center of curvature are in front of the mirror (against the direction of incident light). Therefore, the focal length would be negative, $f = -r/2$.

Q: Why do we measure from the pole?

The pole is chosen as the origin $(0,0)$ in the Cartesian sign convention system to standardize all measurements across different optical components.

Q: What does the '+' sign physically mean?

It indicates that the focus is virtual. Real light rays do not actually converge there; they only appear to diverge from that point after reflection.

Question

The focal length $f$ is related to the radius of curvature $r$ of the spherical convex mirror by

Options:

(A) $f = r$

(B) $f = -r$

(C) $f = -\frac{r}{2}$

(D) $f = +\frac{r}{2}$

Sigma Percentile · Accepted Answer

### The Geometry of Spherical Mirrors
When we talk about spherical mirrors, we are essentially looking at a sliced portion of a hollow glass sphere. The center of this original sphere is what we call the **Center of Curvature ($C$)**, and the radius of this sphere is the **Radius of Curvature ($r$)**. The geometric center of the mirror's reflecting surface is known as the **Pole ($P$)**.

For paraxial rays—light rays that strike the mirror close to its principal axis—they all converge at (or appear to diverge from) a single point after reflection. This magical point is the **Principal Focus ($F$)**.

### The Paraxial Approximation
Through simple geometry and the law of reflection, it can be proven that for small aperture mirrors, the focus lies exactly halfway between the pole and the center of curvature. This gives us the fundamental magnitude relationship:
$$|f| = \frac{|r|}{2}$$

### The Cartesian Sign Convention
In optics, we use a strict coordinate system to avoid chaos. The Pole ($P$) acts as the origin $(0,0)$. The direction of incident light is taken as the positive x-axis.

For a **convex mirror**, the reflecting surface bulges outwards towards the light source. This means the "inside" of the sphere—where the Focus and Center of Curvature reside—is *behind* the mirror. Since light travels from left to right, and we measure distances from the pole to the right to reach $F$ and $C$, both the focal length and the radius of curvature fall on the positive side of the axis.

Thus, applying the sign convention:
$$f = +\frac{r}{2}$$

This positive sign is not just a mathematical formality; it physically signifies that the convex mirror has a **virtual focus**, meaning real light rays never actually pass through it, but only appear to diverge from it!

Animated Solution for Physics - Optics: The focal length $f$ is related to the radius of curvature $r$ of the spherical convex mirror by

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Visualized Solution

Convex Mirror Geometry

Focal Length and Radius

Sign Convention

Final Relation

The Sigma Insight: Spherical Mirror

The Geometry of Spherical Mirrors When we talk about spherical mirrors, we are essentially looking at a sliced portion of a hollow glass sphere

The Paraxial Approximation Through simple geometry and the law of reflection, it can be proven that for small aperture mirrors, the focus lies exactly halfway between the pole and the center of curvature

The Cartesian Sign Convention In optics, we use a strict coordinate system to avoid chaos

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Animated Solution for Physics - Optics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The focal length f is related to the radius of curvature r of the spherical convex mirror by