Focus and focal length

The idea

Rays parallel to the principal axis meet, after reflection from a concave mirror, at its principal focus F, or appear to come from it for a convex mirror; the pole-to-focus distance is the focal length f.

Sunlight on a sheet of paper

Sunlight arrives as parallel rays. A concave mirror faces the Sun, and the light it reflects falls on a sheet of paper. You slide the paper back and forth.

What will you see on the paper?

• A bright spot that is smallest at one particular place
• A spot of the same size wherever the paper is held
• No light at all: a mirror only shows pictures

Find the focus

Concave mirror: slide the paper along the axis until the spot on it is the smallest and brightest. Convex mirror: the rays spread out, so slide the pin behind the mirror to the point they seem to come from.

Where the rays meet

A number of rays parallel to the principal axis fall on a concave mirror. After reflection, they all meet at a point on the principal axis. This point is called the principal focus, F, of the concave mirror. A convex mirror reflects the rays so that they appear to come from a point on the principal axis. This point is the principal focus of the convex mirror. The distance between the pole and the principal focus is the focal length, f.

The bright spot on the paper is the image of the Sun. Its distance from the mirror gives the focal length, f = PF. This is a classroom activity: never look at the Sun, and never focus sunlight on paper at home.

• Mirror | Parallel rays after reflection | Where F is
• Concave | Meet at one point | In front of the mirror, on the axis
• Convex | Spread out, and seem to come from one point | Behind the mirror, on the axis

Notes

Rays parallel to the principal axis meet, after reflection from a concave mirror, at its principal focus F, or appear to come from it for a convex mirror; the pole-to-focus distance is the focal length f.

Check yourself

Parallel rays fall on this concave mirror. Which point is its principal focus?

A student holds a concave mirror towards the Sun. A sharp, bright spot forms on a sheet of paper 15 cm from the mirror. What is the focal length of the mirror?

Answer: 15 cm

The sharp spot is at the principal focus F. Its distance from the mirror is the focal length, so f = 15 cm.

Parallel rays fall on this convex mirror. Which point is its principal focus?

A sheet of paper held at the principal focus of a concave mirror in strong sunlight starts to burn. Why?

• Point X. X is the centre of curvature C, farther from the mirror. The reflected rays meet nearer the mirror, at Y.
• Point Z. Z is the pole, on the mirror itself. The rays meet in front of the mirror.
• Point Y — correct. Yes! The reflected rays meet there, on the principal axis.
• Point Y — correct. Yes! The reflected rays seem to come from Y, behind the mirror.
• Point X. X is in front of the mirror, but the reflected rays spread out there. They do not meet.
• Point Z. Z is the centre of curvature, farther behind the mirror. The dotted lines meet at Y.
• The mirror makes heat of its own. A mirror does not make heat. It only gathers the sunlight that falls on it into one spot.
• The mirror gathers the Sun's light into one small spot, so much heat falls on that spot — correct. Yes! The heat of the concentrated sunlight ignites the paper.
• The mirror spreads the Sun's light over the whole paper. That would not burn the paper. At the focus, the light is gathered into one small spot.