During the lunch break, a student catches the sunlight in the shiny lid of a steel tiffin box. A bright patch of light appears on the far wall of the classroom. A tiny twist of the wrist sends the patch racing across the wall, from the blackboard to the window.
How far a mirror turns a ray
Without the mirror, a ray would go straight on. The mirror turns it back into the room. The angle between the direction the ray would have had and the direction it now has is the deviation \(\delta\). In Fig 2.1 the straight-on direction is dashed. The incident and reflected rays make an angle \(i + r = 2i\) with each other. The turn is what is left of a straight angle.
The formula has three useful cases. At normal incidence, \(i = 0°\), the deviation is \(180°\), and the ray goes straight back. At \(i = 45°\) it is \(90°\), a right-angled turn, as in a periscope. A grazing ray, \(i = 90°\), is not turned at all. The second form, \(\delta = 2g\), is often quicker: a ray at \(40°\) to a mirror is turned through \(80°\).
Rank these rays by the deviation a plane mirror gives them, largest first.
Turn the mirror, and the ray turns twice as far
Keep the incident ray fixed and turn the mirror through a small angle \(\theta\). The axis of turning lies in the mirror, perpendicular to the plane of incidence. The normal is fixed to the mirror, so the normal also turns through \(\theta\). The angle of incidence therefore changes by \(\theta\), and the deviation changes by twice that.
\[ \delta_1 = 180^\circ - 2i \]
Why. the deviation before the turn, with angle of incidence \(i\).
\[ i' = i + \theta \]
Why. the normal turns with the mirror through \(\theta\) while the incident ray stays put, so the angle between them changes by \(\theta\).
\[ \delta_2 = 180^\circ - 2(i + \theta) \]
Why. the same formula after the turn.
\[ \delta_1 - \delta_2 = 2\theta \]
Read it. the incident ray did not move, so the whole change of deviation is a turn of the reflected ray: \(2\theta\).
In Fig 2.2 the ray starts with \(i = 30°\) and a deviation of \(120°\). After the mirror turns through \(15°\), the angle of incidence is \(45°\) and the deviation is \(90°\). The reflected ray has turned through \(120° - 90° = 30°\). The readouts show the factor of two at every instant.
The other case is different. Keep the mirror fixed and turn the incident ray through \(\theta\). The normal does not move, so \(i\) changes by \(\theta\) and \(r\) changes by \(\theta\). The reflected ray therefore turns through \(\theta\), not \(2\theta\), in the opposite sense to the incident ray.
A mirror spinning at a steady rate sweeps its reflected ray at twice that rate. Suppose a mirror turns at \(2\,\text{rad/s}\). The reflected ray turns at \(4\,\text{rad/s}\). Its spot on a screen \(10\,\text{m}\) away moves at \(10 \times 4 = 40\,\text{m/s}\) at the instant the ray is normal to the screen.
A ray falls on a plane mirror. The ray is kept fixed and the mirror is turned through \(10°\) about an axis in its plane, perpendicular to the plane of incidence. Through what angle does the reflected ray turn, in degrees?
A plane mirror is fixed. The incident ray is turned anticlockwise through \(12°\). What happens to the reflected ray?
In 1858 the first telegraph cable was laid across the Atlantic Ocean. Its signals arrived as tiny currents, too weak to move an ordinary needle visibly. William Thomson fixed a small mirror to a light magnet hung inside a coil. A lamp's beam reflected from the mirror onto a scale some distance away. The smallest turn of the magnet swung the spot of light through twice that angle. The distance of the scale made the swing easy to read.
Two mirrors at an angle
Two plane mirrors \(M_1\) and \(M_2\) meet at an angle \(\theta\). A ray reflects once from each. Each reflection turns the ray through twice its angle with that mirror, and both turns are in the same sense. So the angles simply add. Call \(\alpha\) the angle the ray makes with \(M_1\), and \(\beta\) the angle it makes with \(M_2\).
\[ \delta = 2\alpha + 2\beta \]
Why. each mirror turns the ray through twice the glancing angle, \(\delta = 2g\), and the two turns are in the same sense.
\[ \alpha + \beta + \theta = 180^\circ \]
Why. the two mirrors and the ray between them form a triangle whose angles are \(\theta\) at the corner, \(\alpha\) at \(M_1\) and \(\beta\) at \(M_2\).
\[ \delta = 2(180^\circ - \theta) = 360^\circ - 2\theta \]
Read it. substitute \(\alpha + \beta\). The angle of incidence has dropped out.
Take mirrors at \(70°\) and a ray meeting \(M_1\) with \(i_1 = 30°\). After \(M_1\) the ray makes \(90° - 30° = 60°\) with \(M_1\). The triangle at the corner leaves \(180° - 70° - 60° = 50°\) with \(M_2\), so \(i_2 = 40°\). The deviations are \(120°\) and \(100°\), a total of \(220°\). The formula agrees: \(360° - 2 \times 70° = 220°\).
At \(\theta = 90°\) the total deviation is \(180°\) for every angle of incidence. The ray returns antiparallel to itself, along a line close to the one it came in on. Turning the pair of mirrors about their line of meeting changes the angle of incidence, but not the deviation. So the returning ray does not move at all.
Two plane mirrors are inclined at \(60°\). A ray is reflected once from each. Find the total deviation, in degrees.
A ray falls on a pair of plane mirrors fixed at right angles to each other and reflects once from each. The pair is now turned through \(5°\) about the line where the mirrors meet. Through what angle does the final ray turn?
The red reflector at the back of a bicycle is a moulded sheet of many tiny corners. Each corner has three faces at right angles to one another. Such a corner sends every ray straight back towards its source, whatever the angle. The headlight of a car behind is therefore reflected back to the driver, and the reflector shines brightly. The studs that mark the lanes of a highway at night and the stripes on a traffic policeman's jacket work the same way.
Two mirrors at \(90°\) return rays that stay in one plane. Three mutually perpendicular mirrors, a corner reflector, return every ray in space. The astronauts of Apollo 11 left a panel of such corner reflectors on the Moon in 1969. Laser pulses from the Earth still bounce off it, and their return time measures the distance to the Moon to within centimetres.
Now back to the tiffin lid. Tilting the lid through \(5°\) turns the reflected beam through \(10°\). If the beam first met the wall at right angles, \(8\,\text{m}\) away, the patch moves \(8 \tan 10° \approx 1.4\,\text{m}\). The factor two comes from the mirror. The long throw across the room comes from the distance.
- Writing the deviation of a plane mirror as \(2i\).The deviation is measured from the straight-on direction: \(\delta = 180° - 2i\). It is twice the angle with the mirror, not with the normal.
- Doubling the turn when the incident ray, not the mirror, is turned.The factor two belongs to a turning mirror. A turned incident ray turns the reflected ray through the same angle, the other way.
- Expecting the deviation by two mirrors to depend on the angle of incidence.\(\delta = 360° - 2\theta\) depends only on the angle between the mirrors.