The mirror equation

From a ray diagram to an equation

Narrated video. Transcript:

Ray diagrams tell us roughly where an image is. To get exact numbers we need an equation linking the object distance u, the image distance v and the focal length f.

Here is an object AB in front of a concave mirror, beyond C. A ray parallel to the axis hits the mirror at M and reflects through F. A ray to the pole P reflects symmetrically. They meet at A′, and the image is A′B′.

For paraxial rays the mirror is almost flat near P, so we treat M as lying on the line through P, perpendicular to the axis. Then PM equals the object height AB.

Triangles A′B′F and MPF are similar. So B′A′ over PM, which is B′A′ over BA, equals B′F over FP.

Triangles A′B′P and ABP are also similar, because the angles of incidence and reflection at P are equal. So B′A′ over BA equals B′P over BP.

Put the two together: B′F over FP equals B′P over BP. And B′F is B′P minus FP, so B′P minus FP, over FP, equals B′P over BP.

Now apply the signs. B′P is −v, FP is −f and BP is −u, because all three lie in front of the mirror. Substituting, v minus f over f equals v over u.

Divide through by v and rearrange: 1/v + 1/u = 1/f. This is the mirror equation. Although we drew one case, with signs it holds for every position and for convex mirrors too.

The magnification m is the image height over the object height, h′ over h. From the second pair of triangles, h′ over h equals minus v over u, so m = −v/u.

A negative m means an inverted image; a positive m means an upright one. The size of m tells us how much bigger or smaller the image is.

Using the mirror equation

Narrated video. Transcript:

Example one. A 2 centimetre tall object stands 25 centimetres in front of a concave mirror of radius 30 centimetres. Where is the image, and how big is it?

Write the signs first. u equals −25 centimetres. R equals −30, so f equals −15 centimetres.

1/v equals 1/f minus 1/u: minus 1 over 15, plus 1 over 25. That is minus 5 over 75 plus 3 over 75, which is minus 2 over 75. So v equals −37.5 centimetres.

The minus sign says the image is in front of the mirror, so it is real. Magnification m equals −v/u: minus (−37.5), divided by −25, which gives −1.5.

So the image is 1.5 times taller and inverted: h′ = −1.5 × 2 = −3 centimetres. A real, inverted, magnified image, 37.5 centimetres from the mirror.

Example two. A car has a convex rear-view mirror of focal length 20 centimetres. A scooter 1.5 metres tall is 2.8 metres behind it. How big does the scooter look?

Here u is −280 centimetres and f is +20 centimetres. 1/v equals 1/20 plus 1/280, which is 14 over 280 plus 1 over 280, or 15 over 280.

So v is about +18.7 centimetres: behind the mirror, a virtual image. And m equals −v/u, 18.7 over 280, which is one-fifteenth.

The 1.5 metre scooter appears just 10 centimetres tall. That is why the mirror says: objects in mirror are closer than they appear.

Always follow the same three steps: put in the signs, solve 1/v + 1/u = 1/f, then read the nature of the image from the signs of v and m.

Try it: check the mirror equation

Interactive simulation. Things to try:

• This is Example one: f = −15 cm, u = −25 cm. Check that v = −37.5 cm and m = −1.5 in the readouts.
• The card under the readouts puts your numbers into 1/v + 1/u = 1/f. Watch the sum stay equal to 1/f as you drag.
• Move the object to u = −15 cm, exactly at F. Where does the image go?
• Switch to Convex and drag the object anywhere. v is always positive and m is always between 0 and +1.

Find the image

A 2 cm tall object is placed 20 cm in front of a concave mirror of focal length 15 cm. Where is the image and how tall is it?

1. 60 cm in front of the mirror, 6 cm tall, inverted
2. 60 cm behind the mirror, 6 cm tall, upright
3. 8.6 cm in front of the mirror, 0.86 cm tall, inverted
4. 60 cm in front of the mirror, 6 cm tall, upright

The convex mirror

An object is placed 10 cm in front of a convex mirror of focal length 10 cm. What is the magnification?

1. +0.5
2. −0.5
3. +2
4. Infinite: the object is at the focus

Summary

• Mirror equation: 1/v + 1/u = 1/f
• Magnification: m = h′/h = −v/u
• Focal length: f = R/2

• The mirror equation comes from two pairs of similar triangles in the ray diagram, for paraxial rays.
• Put in the signs before substituting: objects in front of the mirror have u < 0; a concave mirror has f < 0, a convex mirror f > 0.
• For a mirror, v < 0 means a real image in front; v > 0 means a virtual image behind.
• m < 0: inverted; m > 0: upright. |m| > 1: magnified; |m| < 1: diminished.