Pour milk into a shining steel tumbler on a sunny morning, and look down into it. A bright curved line lies on the milk, with a sharp point at its middle. The sunlight reflects from the curved inside wall of the tumbler. Rays that strike the wall at different places cross at different places. Where they crowd together, the milk glows.
The tangent and the radius
A tangent to a circle is a straight line that touches the circle at one point and does not cut it. At the point of contact, the tangent is at right angles to the radius. The reason is short. Every point of the tangent except the point of contact lies outside the circle, farther than \(R\) from the centre. So the point of contact is the point of the line nearest to the centre. The shortest line from a point to a straight line is the perpendicular.
The tangent to a circle at any point is perpendicular to the radius drawn to that point. For a sphere, the tangent plane at any point is perpendicular to the radius drawn to that point.
The laws of reflection and refraction measure every angle from the normal, the line at right angles to the surface. For a curved surface, the normal at a point is at right angles to the tangent there. By the theorem, it lies along the radius. So at any point of a spherical mirror, a lens face or a raindrop, the normal runs to the centre of curvature.
Now take a ray parallel to the axis, at a height \(h\) above it, and let it meet the surface at N. The radius CN is the normal at N. The ray and the axis are parallel. So, by alternate angles, the angle between the ray and CN equals the angle between the axis and CN. The height of N is then \(h = R\sin i\). So the angle of incidence obeys \(\sin i = h/R\).
Move the ray to the axis and \(i\) shrinks to zero. Move it to \(30\,\text{cm}\) and \(\sin i = 0.75\), so \(i = 48.6°\). The normal turns with the point of incidence, always along the radius. That is why every ray at a curved mirror has its own normal, and its own angle of reflection.
A ray parallel to the principal axis strikes a concave mirror of radius \(50\,\text{cm}\) at a height of \(25\,\text{cm}\). What is its angle of incidence, in degrees?
Light meets the curved face of a glass hemisphere at a point N. Which line is the normal at N?
Back to physics: a ray far from the axis
The chapter Reflection: Plane and Spherical Mirrors asks where a ray far from the axis crosses it after reflection. The answer needs only the normal and one isosceles triangle. It also explains the bright curve in the tumbler.
A ray parallel to the axis strikes a concave mirror of radius \(40\,\text{cm}\) at a height of \(20\,\text{cm}\). After reflection it crosses the axis at a point X. How far is X from the pole P?
The normal at the point M is the radius CM. So \(\sin i = 20/40 = 0.5\), and \(i = 30°\).
In triangle CMX, the angle at C is \(i\), by alternate angles. The angle at M is \(i\) too, since the reflected ray makes \(r = i\) with CM. Two equal angles make CX equal to XM. The perpendicular from X to CM then bisects CM, so \(CX\cos i = R/2\).
\(CX = 20/\cos 30° = 20/0.866 = 23.1\,\text{cm}\). The pole is \(40\,\text{cm}\) from C, so \(PX = 40 - 23.1 = 16.9\,\text{cm}\).
X is \(16.9\,\text{cm}\) from the pole, not \(20\,\text{cm}\). A ray close to the axis has \(\cos i \approx 1\) and crosses at \(R/2 = 20\,\text{cm}\), the focus.
The tumbler has its answer. Rays near the middle of the curved wall cross close to one point, the focus. Rays striking farther out cross nearer the wall, and each pair of neighbours crosses at a slightly different place. The crossings crowd along a curve, and the curve is brightest where it comes to a point, at the focus.
Arcs, chords and sectors
Two radii cut a circle into an arc and a sector, the slice of the disc between them. Let the angle between the radii be \(\theta\), in radians. The arc has the length \(s = R\theta\). The sector is the fraction \(\theta/2\pi\) of the whole disc, so its area is \(\tfrac{\theta}{2\pi} \times \pi R^2 = \tfrac12 R^2\theta\).
The chord is the straight line joining the two ends of the arc. Drop the perpendicular from the centre to the chord. It cuts the chord and the angle in half. Each half of the chord is then \(R\sin(\theta/2)\), so the chord is \(2R\sin(\theta/2)\). A chord is always shorter than its arc, but for a small angle the two are almost equal.
Physics uses all three. In a short time, a planet's line to the Sun sweeps a thin sector of area \(\tfrac12 r^2\,\Delta\theta\). Kepler's law of areas is about that sector. A rim point of a rolling wheel, \(\theta\) round from the contact, is a chord \(2R\sin(\theta/2)\) from the contact, so it moves at \(2v\sin(\theta/2)\).
An auto-rickshaw wheel rolls without slipping, with its centre moving at \(10\,\text{m/s}\). A point on the rim is \(120°\) round from the point of contact. How fast is it moving, in m/s?
A satellite moves on a circle of radius \(7000\,\text{km}\) round the Earth and turns through \(0.001\,\text{rad}\) every second. What area does the line from the Earth's centre to the satellite sweep in one second, in units of \(10^{10}\,\text{m}^2\)?
How far a curve stands off a flat plate
Rest a curved surface of radius \(R\) on a flat plate, so that it touches the plate at one point O. At a distance \(r\) from O, the curve stands a small height \(t\) above the plate. This height is called the sagitta, the Latin word for an arrow. The radius from the centre C to the curve, the upright line through O and the level line at height \(t\) make a right triangle.
\[ r^2 + (R - t)^2 = R^2 \]
Why. Pythagoras' theorem in the right triangle with the hypotenuse \(R\), the upright side \(R - t\) and the level side \(r\).
\[ r^2 = 2Rt - t^2 \]
Why. expand \((R - t)^2 = R^2 - 2Rt + t^2\) and cancel \(R^2\) on both sides.
\[ r^2 \approx 2Rt \]
Why. for a shallow arc \(t\) is much smaller than \(R\), so \(t^2\) is tiny beside \(2Rt\).
\[ t \approx \dfrac{r^2}{2R} \]
Read it. divide both sides by \(2R\).
For a real lens the rule is excellent. A lens face of radius \(1\,\text{m}\) resting on a glass plate stands \(t = (2 \times 10^{-3})^2/2 = 2 \times 10^{-6}\,\text{m}\) above it at \(2\,\text{mm}\) from the contact. That thin film of air, \(2\,\mu\text{m}\) thick, makes Newton's rings. The spherometer of the chapter Measurement and Error uses the same rule backwards: it measures \(t\) and finds \(R = r^2/2t\).
A lens face of radius \(80\,\text{cm}\) rests on a flat glass plate. How thick is the air film \(4\,\text{mm}\) from the point of contact, in micrometres?
The tangent is perpendicular to the radius only for a circle or a sphere. A parabolic mirror has a normal at every point too, but the normals do not all pass through one centre. The rule \(t \approx r^2/2R\) needs \(r\) much smaller than \(R\). At \(r = R/2\) it gives \(0.125R\), while the exact height is \(0.134R\), about \(7\,\%\) more.
Eratosthenes looked after the great library of Alexandria. He read that at Syene, far to the south, the Sun shone straight down a deep well at noon on the longest day. On that day at Alexandria, a stick's shadow showed the Sun \(7.2°\) from overhead, a fiftieth of a full circle. The arc from Syene to Alexandria, about 5000 stadia, was therefore a fiftieth of the Earth's circumference. So the whole circumference was about 250 000 stadia. The idea was simply \(s = R\theta\), read on the Earth.
The chapter Reflection: Plane and Spherical Mirrors uses the radius as the normal to find the focus and the spread of wide rays. It also places the images of two inclined mirrors on one circle. Refraction, Total Internal Reflection and Prisms finds the circle of light above a lamp under water. Spherical Surfaces and Lenses derives the refraction formula at a curved surface from the radius as the normal. Thin Films, Newton's Rings and Other Interference measures the air film under a lens by \(t = r^2/2R\). Rolling Motion and Kinematics of Circular Motion use chords and the circle that best fits a curved path. Relative Velocity in Two Dimensions finds the least drift of a boat from a tangent to a circle. Magnetic Force on Moving Charges follows a charge along an arc across a strip of field. Temperature and Thermal Expansion bends a bimetallic strip into an arc.
- Measuring the angle of incidence from the axis.Measure it from the normal, which is the radius through the point of incidence.
- Using the arc length where the chord is needed.The straight distance between two points of a circle is the chord \(2R\sin(\theta/2)\), shorter than the arc \(R\theta\).
- Using \(t \approx r^2/2R\) for a deep curve.The rule needs \(r\) much smaller than \(R\). For a deep curve use \(t = R - \sqrt{R^2 - r^2}\).