Concave Mirror Ray Diagram

Mirrors and lenses share a surprising amount of physics — both form images by redirecting light, both use the same $1/f = 1/d_{o} + 1/d_{i}$ equation (with slightly different sign rules), and both have the same five regimes depending on where the object sits relative to $F$ and $2F$. But there's one crucial difference: with a mirror, the image forms on the same side as the object (because light bounces back), whereas with a lens, the image forms on the opposite side. This flips the geometry, and it trips up students who try to use lens intuition directly.

A concave mirror — sometimes called a converging mirror — curves inward, like the inside of a spoon. Parallel light rays hitting it all reflect to meet at the focal point $F$, which is in front of the mirror (on the same side as the incoming light). The center of curvature $C$ is at twice the focal distance: $C = 2f$. For our problem, $f = 10$ cm, so $C = 20$ cm.

The mirror equation and sign convention

The mirror equation looks identical to the thin-lens equation:

$\dfrac{1}{f} = \dfrac{1}{d_{o}} + \dfrac{1}{d_{i}}$

For mirrors, the sign convention most commonly used is:

• Distances are measured from the mirror's surface
• Distances on the reflecting side (in front of the mirror) are positive
• Distances behind the mirror are negative
• For a concave mirror, $f > 0$; for a convex mirror, $f < 0$
• A positive $d_{i}$ means a real image (in front of the mirror); a negative $d_{i}$ means a virtual image (behind the mirror)

The magnification is:

$m = -\dfrac{d_{i}}{d_{o}}$

Negative $m$ means inverted; positive means upright. Same rule as lenses.

Solving the problem (method 1 — mirror equation)

$d_{o} = 25$ cm, $f = 10$ cm.

$\dfrac{1}{d_{i}} = \dfrac{1}{f} - \dfrac{1}{d_{o}} = \dfrac{1}{10} - \dfrac{1}{25} = \dfrac{5}{50} - \dfrac{2}{50} = \dfrac{3}{50}$

$d_{i} = \dfrac{50}{3} \approx 16.67 \text{ cm}$

Positive $d_{i}$ → the image is real, in front of the mirror.

$m = -\dfrac{16.67}{25} = -0.667$

Negative $m$ → inverted. $|m| = 0.667$ → diminished (67% of the object's height). If the object is 6 cm tall, the image is 4 cm tall and upside-down.

Since $d_{o} = 25 > 2f = 20$, the object is beyond $C$ (the center of curvature, which plays the same role as $2F$ for lenses). This places us in the "beyond 2F" regime: image is real, inverted, diminished, and located between $F$ and $C$.

The three principal rays for a concave mirror (method 2)

The ray diagram logic is analogous to a converging lens, but the rays reflect instead of refracting:

Ray 1 — parallel, reflects through F. A ray from the object tip, parallel to the principal axis, hits the mirror and reflects through the focal point $F$.

Ray 2 — through C, reflects back on itself. A ray aimed at the center of curvature $C$ hits the mirror surface perpendicularly (since $C$ is the center of the curved surface), so it reflects straight back along the same path. This is the mirror analog of the "through the center" ray for lenses.

Ray 3 — through F, reflects parallel. A ray from the object tip that passes through $F$ hits the mirror and reflects back parallel to the principal axis. This is the reverse of Ray 1.

These three reflected rays converge at a point in front of the mirror. That's where the real image forms. For our numbers, they converge at roughly 16.7 cm from the mirror, confirming the equation.

Verification with similar triangles (method 3)

Ray 2 (through $C$) travels in a straight line from the object tip through $C$ and back. It creates two similar triangles with the principal axis: one with height $h_{o}$ and base $d_{o} - R$ (where $R = 2f$), and one with height $h_{i}$ and base $R - d_{i}$.

For Ray 1 (parallel, then through $F$): the triangle from the mirror to $F$ with height equal to the incoming ray's distance from the axis, and the triangle from $F$ to the image give:

$\dfrac{h_{o}}{f} = \dfrac{h_{i}}{d_{i} - f}$

Combining with $m = h_{i}/h_{o} = d_{i}/d_{o}$ and doing the algebra reproduces the mirror equation. The geometry is the proof.

The five regimes of a concave mirror

Just like a converging lens, a concave mirror has five regimes based on where the object is:

• Beyond C ($d_{o} > 2f$): Image is real, inverted, diminished, between $F$ and $C$. (Our problem.)
• At C ($d_{o} = 2f$): Image is real, inverted, same size, at $C$. $m = -1$.
• Between F and C ($f < d_{o} < 2f$): Image is real, inverted, magnified, beyond $C$.
• At F ($d_{o} = f$): Image at infinity — reflected rays are parallel.
• Inside F ($d_{o} < f$): Image is virtual (behind the mirror), upright, magnified.

The last regime — inside $F$ — is how a makeup or shaving mirror works. Your face is closer to the mirror than $F$, so the mirror creates a magnified, upright, virtual image. That's why those mirrors have a short focal length (a deeply curved surface).

Concave mirror vs converging lens: a comparison

The physics is almost identical, but the geometry is mirrored (pun intended):

• For a lens, the real image forms on the opposite side from the object. For a mirror, it forms on the same side.
• For a lens, a virtual image (inside $F$) is on the same side as the object. For a mirror, a virtual image is behind the mirror (opposite side from the object).
• The same equation, the same five regimes, the same magnification formula.

This means if you truly understand convex lenses (Problems 146–147), you already understand concave mirrors — you just need to flip the image side.

Real-world applications

• Satellite dishes & radio telescopes: The dish is a large concave mirror (for radio waves) with the receiver at $F$. Parallel radio waves from a distant source focus at $F$.
• Car headlights: The bulb sits at $F$ of a concave reflector. By the same ray logic, light from $F$ reflects outward as a parallel beam — a headlight beam.
• Shaving/makeup mirrors: Object inside $F$ → magnified, upright, virtual image.
• Reflecting telescopes: The primary mirror is a large concave mirror (often parabolic to reduce aberration) that focuses starlight at $F$, where a secondary mirror or sensor captures it. Newton invented this design.
• Solar concentrators: Large concave mirrors focus sunlight onto a small point, producing intense heat. Solar furnaces at Odeillo, France, reach 3,500°C this way.

Common mistakes

• Getting the image side wrong. Real images from a concave mirror are in front of the mirror, not behind it. Behind the mirror is where virtual images go — the opposite of a lens.
• Confusing $C$ and $F$. $C = 2f$ always. Some problems give $R$ (radius of curvature) instead of $f$. Remember $f = R/2$.
• Applying lens sign conventions to mirrors. Some textbooks use the "new Cartesian" convention where all distances measured opposite to incident light are negative. Check which convention your course uses — the equation is the same, but the signs differ.
• Forgetting that mirrors reverse left-right. A plane mirror shows a laterally inverted image. Concave and convex mirrors do this too, on top of any vertical inversion from $m < 0$.

Try it in the visualization

Start with $d_{o} = 25$ cm and confirm the image at ~16.7 cm. Then drag the object toward $C = 20$ cm and watch the image grow to same-size. Keep dragging past $C$ into the $F$-to-$C$ zone — the image blows up and moves far away. At $F = 10$ cm the rays go parallel (image at infinity). Inside $F$ the image jumps behind the mirror and becomes virtual and upright (the makeup-mirror regime). Toggle rays on/off to see which two are enough to locate the image.