Grade by Grade
E. Physics: Light and Geometric Optics · Light and Geometric Optics

Curved Mirrors: Concave, Convex and the Focal Point

E2.2 E2.1 E3.3 A1.5About 65 minutes
Learning goals
  • I can identify concave (converging) and convex (diverging) mirrors.
  • I can label the centre of curvature, vertex, principal axis, focal point and focal length, and use f = R ÷ 2.
  • I can explain, using normals, why parallel rays converge (concave) or diverge (convex).
  • I can state the special rays used to draw curved-mirror ray diagrams.

Words to know

concave mirror
A mirror that curves inward; it converges parallel rays (converging mirror).
convex mirror
A mirror that bulges outward; it diverges parallel rays (diverging mirror).
centre of curvature (C)
The centre of the sphere of which the mirror is a part.
vertex (V)
The centre point of a curved mirror's surface.
principal axis
The line through C and V.
focal point (F)
The point where reflected parallel rays meet (concave) or appear to come from (convex).
focal length (f)
The distance from V to F; f = R ÷ 2.
spherical aberration
Blurring caused because rays far from the axis of a spherical mirror do not pass exactly through F.

Look into a spoon

Pick up a shiny metal spoon and look into the inside of the bowl. Your face is upside down! Turn the spoon over and look at the back: now you are right side up, but tiny, with a huge nose. A spoon is a pair of curved mirrors, and in the next three lessons you will learn to predict exactly what each kind of curved mirror does.

Two kinds of curved mirror

Most curved mirrors are pieces of a hollow sphere (a spherical mirror).

  • A concave mirror is curved inward, like the inside of a spoon or a cave. ("Con-cave": it caves in.) It reflects parallel rays so they come together, so it is also called a converging mirror.
  • A convex mirror bulges outward, like the back of a spoon. It reflects parallel rays so they spread apart, so it is also called a diverging mirror.

Remember from Grade 4 and earlier lessons

In Grade 4 you saw that shiny curved surfaces, like spoons, make funny-looking images. In this unit you already learned that every reflection obeys θi = θr, measured from the normal. Curved mirrors follow exactly the same law. The only new idea is that the normal points in a different direction at each point on the curve.

The parts of a curved mirror

  • Centre of curvature (C): the centre of the sphere the mirror is part of.
  • Radius of curvature (R): the distance from C to the mirror.
  • Vertex (V): the centre point of the mirror's surface.
  • Principal axis: the line through C and V.
  • Focal point (F): the point where rays parallel to the principal axis meet (concave), or appear to come from (convex), after reflecting.
  • Focal length (f): the distance from V to F. For mirrors that are small compared with their sphere, f = R ÷ 2: F is halfway between C and V.

Any line from C to the mirror is a radius, and a radius always meets a circle at 90°. So the normal at any point on a spherical mirror is the line from C to that point. That one fact explains everything curved mirrors do.

normal (radius)CFVradius of curvature Rfocal length f = R ÷ 2principal axis
Six rays parallel to the principal axis, each reflected exactly by the law of reflection from a real circular mirror (the dashed line is one normal). The rays meet near F, halfway between C and V. Rays far from the axis cross slightly closer to the mirror: this blurring is called spherical aberration.
Worked example

Example 1: Finding the focal length

A concave makeup mirror is part of a sphere with a radius of 36 cm. Where is its focal point?

  1. The focal length is half the radius of curvature: f = R ÷ 2.
  2. f = 36 cm ÷ 2 = 18 cm.
  3. F lies on the principal axis, in front of the mirror, 18 cm from the vertex (halfway between V and C).

f = 18 cm, in front of the mirror.

Convex mirrors have a virtual focus

When parallel rays hit a convex mirror, the normals (lines from C, which is behind the mirror) tilt outward, so the reflected rays spread out. If you extend the reflected rays backwards behind the mirror (dashed lines), they appear to come from one point: the virtual focal point F, halfway between V and C, behind the mirror. No light really passes through it.

VFCvirtual focus behind the mirror
Parallel rays reflect off a convex mirror and diverge. Their dashed extensions meet behind the mirror at the virtual focus F, halfway between V and C.

Parabolic mirrors: perfect focus

A spherical mirror focuses parallel rays only approximately. A mirror shaped like a parabola sends every parallel ray exactly through F. And because light paths are reversible, a light source placed at F of a parabolic mirror produces a strong parallel beam.

Parabolic reflectors are used in car headlights, flashlights, satellite dishes (which focus microwaves onto a receiver at F), solar cookers and large reflecting telescopes.

FV
A parabolic mirror. Every ray parallel to the axis reflects exactly through F. With the arrows reversed, a light bulb at F produces a parallel beam, as in a flashlight or headlight.

The four rules for drawing rays (concave mirrors)

To draw ray diagrams quickly, physicists use four special rays whose paths are easy to predict. Each one follows from the law of reflection:

  1. A ray parallel to the principal axis reflects through F.
  2. A ray through F reflects parallel to the principal axis. (This is rule 1 run backwards: light paths are reversible.)
  3. A ray through C hits the mirror along a normal (a radius), so θi = 0°, and it reflects straight back on itself.
  4. A ray aimed at V reflects at an equal angle on the other side of the principal axis (at V the principal axis is the normal).

For convex mirrors, the same rules work with "through F" replaced by "as if from F" or "towards F", because F and C are behind the mirror. You will use these in the convex-mirror lesson.

Drawing convention: In ray diagrams the mirror is drawn only gently curved, and rays are reflected at the vertical line through V. This is accurate for mirrors that are small compared with their radius of curvature, which is the case for almost all real mirrors.

1. Parallel to the axis → reflects through FCFV2. Through F → reflects parallelCFV3. Through C → reflects back on itselfCFV4. To V → reflects at an equal angleCFV
The four special rays for a concave mirror. Any two of them are enough to locate an image; a third ray is a useful check.
Worked example

Example 2: Which rule?

For a concave mirror, describe the reflected ray in each case: (a) a ray travels along the line from C to the mirror; (b) a ray passes through F before hitting the mirror; (c) a ray from a distant star travels parallel to the principal axis.

  1. (a) A line from C is a radius, which is a normal. θi = 0°, so θr = 0°.
  2. (b) Rule 2: through F → reflects parallel to the principal axis.
  3. (c) Rule 1: parallel to the principal axis → reflects through F.

(a) It reflects straight back along its own path. (b) It reflects parallel to the axis. (c) It reflects through F.

Worked example

Example 3: Measuring a focal length safely

Nadia holds a concave mirror so that it reflects light from a window on the far side of a large room onto a sheet of paper. A sharp, upside-down picture of the window appears when the paper is 22 cm from the mirror. Estimate f and R.

  1. Light from a very distant object arrives almost parallel to the principal axis.
  2. Parallel rays meet at F, so the sharp image forms (almost) at F.
  3. f ≈ 22 cm; R = 2f ≈ 44 cm.

f ≈ 22 cm and R ≈ 44 cm. (The window is not infinitely far away, so f is really a little less than 22 cm.)

Just for fun

A spoon is the most useful physics lab in your kitchen: one side is a concave mirror, the other side is a convex mirror, and both sides are excellent at holding ice cream.

Hands-on activity

Spoon and makeup-mirror investigation

You need: a large shiny metal spoon (a ladle is even better), a concave makeup or shaving mirror (the kind that magnifies), white paper, a ruler, a room with a bright window.

  1. Look into the inside of the spoon from about 30 cm away. Describe your image (upright or inverted? larger or smaller?). Slowly bring the spoon very close to your eye and see what changes.
  2. Look at the back of the spoon. Describe your image at different distances.
  3. Find f. Stand at the far side of a room from a bright window. Hold the makeup mirror so it faces the window, and hold the paper between you and the mirror (slightly off to one side so you don't block the light). Move the paper until you see a sharp, upside-down image of the window. Measure the distance from the mirror to the paper. That distance is about f. Calculate R = 2f.
  4. Repeat step 3 two more times and average your results.

What to look for: the inside of the spoon gives an inverted image from far away and a blurry, then upright, image very close; the back always gives a small, upright image. A typical magnifying makeup mirror has f of roughly 15 cm to 50 cm.

Safety (grown-up note): Never point a concave mirror at the Sun or let it reflect sunlight onto anything: it can focus enough energy to burn skin, damage eyes or start a fire.

Your turn

1

What is the difference between a concave and a convex mirror?

2

A concave mirror has a radius of curvature of 50 cm. What is its focal length?

3

Why is a line from C to the mirror a normal?

4

Describe how a ray parallel to the principal axis reflects from (a) a concave mirror, (b) a convex mirror.

5

Why do car headlights use parabolic reflectors with the bulb at F?

6

What happens to a ray that passes through C before hitting a concave mirror?

More practice: warm-up

1

Is the inside of a spoon concave or convex?

2

What letter represents the vertex of a mirror?

3

Another name for a concave mirror is a ___ mirror.

4

f = 12 cm. What is R?

5

Is the focal point of a convex mirror real or virtual?

More practice: on your own

1

Explain why rays far from the principal axis of a spherical mirror do not meet exactly at F.

2

Explain how rule 2 can be worked out from rule 1.

3

A satellite dish is a parabolic reflector. Where should the receiver be placed, and why?

4

A ray hits a concave mirror exactly at V, making 25° with the principal axis. Describe the reflected ray.

5

A student finds a sharp image of a distant tree 15 cm from a concave mirror. What are f and R?

Challenge

1

For a spherical concave mirror with R = 240 mm, a ray parallel to the axis hits the mirror where the normal makes 30° with the axis. The reflected ray crosses the axis at a distance R ÷ (2 cos 30°) from C. How far from V does it cross, and how does this compare with f?

2

Large astronomical telescopes use parabolic, not spherical, mirrors. Explain why.

3

Explain why a concave mirror aimed at the Sun can start a fire but a convex mirror cannot.

Check your understanding

Quiz

1.

A concave mirror has R = 40 cm. Its focal length is:

2.

Which mirror is also called a diverging mirror?

3.

For a concave mirror, a ray parallel to the principal axis reflects:

4.

The normal at any point on a spherical mirror:

5.

Why do flashlights use a parabolic reflector with the bulb at F?

6.

A ray passing through C hits a concave mirror. Its angle of incidence is:

For parents and teachers

Big idea: Curved mirrors still obey θi = θr; the normal at each point is the radius from C. From this come the focal point (f = R ÷ 2) and the four special rays students will use to draw ray diagrams (E2.2, E3.3).

Common difficulties:

  • Mixing up concave and convex. "Concave caves in." The inside of a spoon is concave.
  • Thinking F is the centre. F is halfway between C and the mirror.
  • Memorizing the rules without reasons. Ask your teen to explain each rule using the normal; understanding makes them easier to remember.

How to help: Do the spoon investigation together; it takes five minutes and makes the next two lessons much easier. Keep concave mirrors out of direct sunlight: they can start fires.