Rectilinear Propagation of Light

Lessons

Senior 1 · Physics

Rectilinear Propagation of Light

5 lessons

Lesson 4 of 5

Reflection of Light at a Plane Surface

24 min

Why can you see your face in a mirror but not in a wall?

A white wall and a bathroom mirror both send back most of the light that falls on them. Yet the mirror shows your face and the wall shows nothing but white. On a calm morning Lake Kivu shows a perfect upside-down copy of the hills; when the wind gets up, the copy breaks into dancing glitter. The difference lies in HOW each surface sends the light back, and it follows one simple law.

Definition

Reflection is the bouncing back of light when it meets a surface.

The incident ray is the ray arriving at the surface; the reflected ray is the ray leaving it.
The point of incidence is where the incident ray meets the surface.
The normal is a line drawn at right angles ( $90^\circ$ ) to the surface at the point of incidence.
The angle of incidence $i$ is the angle between the incident ray and the normal; the angle of reflection $r$ is the angle between the reflected ray and the normal.

Experiment: measuring i and r

On a sheet of white paper, draw a straight line and stand a plane mirror upright on it. Draw a normal at right angles to the line, and mark angles of $20^\circ$ , $30^\circ$ , $40^\circ$ , $50^\circ$ and $60^\circ$ from the normal with a protractor.
Shine a narrow beam from a ray box along the $20^\circ$ line so that it hits the mirror where the normal meets it.
Mark two points on the reflected beam with a sharp pencil, remove the mirror, join the points to the point of incidence, and measure the angle of reflection.
Repeat for the other angles. (Without a ray box, use optical pins: look at the images of two pins in the mirror and line up two more pins with them.)

Angle of incidence $i$	20°	30°	40°	50°	60°
Angle of reflection $r$	20°	31°	40°	49°	60°

Within the accuracy of the protractor (about $1^\circ$ ), the angle of reflection always equals the angle of incidence.

Figure 4.1: The law of reflection. Both angles are measured from the normal, not from the mirror.

Important

Safety. Tape the edges of glass mirrors so they cannot cut you. If you use a laser pointer instead of a ray box, never point it at anyone's eyes, even after a reflection.

Definition

The laws of reflection

The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.
The angle of incidence equals the angle of reflection: $i = r$ .

Worked example 1: a ray at 30° to the mirror

A ray of light strikes a plane mirror, making an angle of $30^\circ$ with the surface of the mirror. Find the angle of incidence, the angle of reflection, and the angle between the incident and reflected rays.

The normal is at $90^\circ$ to the mirror, so $i = 90^\circ - 30^\circ = 60^\circ$ . By the law of reflection, $r = 60^\circ$ .

\text{angle between the rays} = i + r = 60^\circ + 60^\circ = 120^\circ

Worked example 2: working backwards

The angle between an incident ray and its reflected ray is $70^\circ$ . What is the angle of incidence?

The normal bisects the angle between the rays, and $i = r$ :

i + r = 70^\circ \quad\Rightarrow\quad 2i = 70^\circ \quad\Rightarrow\quad i = 35^\circ

Regular and diffuse reflection

Figure 4.2: Every ray obeys the law of reflection in both cases. On a rough surface the normals point in many directions, so the reflected rays scatter.

	Regular (specular) reflection	Diffuse (irregular) reflection
Surface	smooth and polished	rough (bumpy on a tiny scale)
Parallel incident rays	are reflected as a parallel beam	are scattered in many directions
Image formed?	yes, a clear image	no image
Examples	a mirror, polished metal, calm water, a shop window	paper, a painted wall, cloth, wood, rippled water
Why it is useful	mirrors, periscopes, headlamp reflectors	lets us see objects from every direction; a matt board or paper page is easy to read because it does not glare

Diffuse reflection is how we see almost everything around us: light scattered from each point of a book or a face reaches our eyes wherever we stand.

Worked example 3: turning the mirror

A ray strikes a plane mirror with $i = 25^\circ$ . The mirror is then turned through $10^\circ$ , so that the angle of incidence becomes $35^\circ$ , while the incident ray stays fixed. Through what angle does the reflected ray turn?

Before: the angle between the incident and reflected rays is $25^\circ + 25^\circ = 50^\circ$ . After: $35^\circ + 35^\circ = 70^\circ$ .

70^\circ - 50^\circ = 20^\circ

The reflected ray turns through $20^\circ$ , twice the angle the mirror turned.

The periscope

Figure 4.3: A periscope. At each mirror the angle of incidence is 45°, so the angle of reflection is 45° and the ray turns through 90°.

A periscope uses two parallel plane mirrors, each at $45^\circ$ to the tube, to let you see over a wall or a crowd. Light from the object turns $90^\circ$ down the tube at the top mirror and $90^\circ$ again, into your eye, at the bottom mirror. Submarines, army tanks and some spectators at football matches use periscopes.

A soldier in a trench at Gallipoli in 1915 uses a periscope to watch over the top of the trench while his head stays safely below it. Photo: from The War Illustrated, 1915, public domain, via Wikimedia Commons

Warning

Common mistakes. (1) Measuring the angles from the MIRROR instead of from the NORMAL. A ray at $30^\circ$ to the mirror has $i = 60^\circ$ . (2) Drawing the normal anywhere other than at the point of incidence. (3) Thinking diffuse reflection breaks the law of reflection: every ray still obeys $i = r$ ; only the normals point in different directions.

Tip

Normal means "at right angles". Always draw the normal first, as a dashed line; then measure both angles from it. For a mirror turned through $\theta$ , the reflected ray turns through $2\theta$ .

Reflection (Cognito). Watch how the normal is drawn, how the angles of incidence and reflection are measured from it, and the difference between specular and diffuse reflection.

✦

Practice

State the two laws of reflection.
Draw a ray striking a plane mirror with an angle of incidence of $35^\circ$ . Label the incident ray, reflected ray, normal, $i$ and $r$ .
A ray makes an angle of $50^\circ$ with the surface of a mirror. Find (a) the angle of incidence; (b) the angle of reflection; (c) the angle between the incident and reflected rays.
The angle between an incident ray and the reflected ray is $90^\circ$ . Find the angle of incidence.
Explain why you can see your face in a mirror but not in a sheet of white paper, although both reflect most of the light falling on them.
A ray hits a mirror along the normal. What is the angle of reflection, and where does the reflected ray go?
Draw a periscope and trace a ray through it. Why must the mirrors be at $45^\circ$ ?
Keza turns a mirror through $15^\circ$ while the incident ray stays fixed. She predicts that the reflected ray turns through $15^\circ$ too. Evaluate her prediction.

Show answersHide answers

(1) The incident ray, the reflected ray and the normal lie in the same plane. (2) The angle of incidence equals the angle of reflection.
Normal drawn at $90^\circ$ to the mirror at the point of incidence; incident ray at $35^\circ$ to the normal on one side, reflected ray at $35^\circ$ on the other.
(a) $90^\circ - 50^\circ = 40^\circ$ ; (b) $40^\circ$ ; (c) $80^\circ$ .
$2i = 90^\circ$ , so $i = 45^\circ$ .
The mirror is smooth, so parallel rays from each point of your face stay together after reflection (regular reflection) and form an image. The paper's surface is rough on a tiny scale, so it scatters the light in all directions (diffuse reflection), and no image forms.
$i = 0^\circ$ , so $r = 0^\circ$ : the ray is reflected straight back along its own path.
Two parallel mirrors, each at $45^\circ$ to the tube: the ray turns down at the top mirror and out to the eye at the bottom. At $45^\circ$ , $i = r = 45^\circ$ , so each reflection turns the ray through exactly $90^\circ$ .
Wrong. Turning the mirror by $15^\circ$ changes $i$ by $15^\circ$ and $r$ by $15^\circ$ , so the angle between the rays changes by $30^\circ$ : the reflected ray turns through $30^\circ$ , twice the mirror's turn.

Key points

Laws of reflection: the incident ray, reflected ray and normal lie in one plane, and $i = r$ , both measured from the normal.
Smooth surfaces give regular reflection and images; rough surfaces give diffuse reflection, which lets us see objects from every direction.
A periscope uses two parallel mirrors at $45^\circ$ , each turning the light through $90^\circ$ .

Sources and credits

25.2: The Law of Reflection, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: British trench periscope Cape Helles 1915.jpg by an unknown photographer for The War Illustrated (1915), Wikimedia Commons, in the public domain.
Video: GCSE Physics - Reflection, Cognito, embedded from YouTube.

The experiment write-up and results, the ray diagrams, the worked examples and the practice exercises are written by Ganzaa.

1Why can you see your face in a mirror but not in a wall?

2Experiment: measuring i and r

3Regular and diffuse reflection

4The periscope

Why can you see your face in a mirror but not in a wall?

Experiment: measuring i and r

Regular and diffuse reflection

The periscope