Illustrated Physics
physics, drawn
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The figure library — page 2

Every picture here is generated from code at build time. This page lists each family of figures in its plainest form, with every essay that draws on it.

interferometer

9 essays

Two ports, and one of them dark. The two outputs of a balanced two-path interferometer against the phase added to one arm, in turns. With the arms equal, every photon leaves by the same port and the other receives nothing at all — not a little, nothing, to 6.7e-16 across the whole sweep. The photon has not chosen a path and then been redirected; the two amplitudes for reaching the second port cancel, and cancellation is only available because both paths were taken. Half a turn moves every photon to the other port. This is the apparatus the interaction-free measurement is built on, and the dark port is the whole of the mechanism: a detector at a place where nothing ever arrives is an instrument of enormous sensitivity, because anything at all that arrives there is news.

Every branch interferometer draws, and what checks it

redshift-tower

9 essays

A photon climbing a tower. A photon emitted at the foot of a tower 22.5 m high and received at the top. It arrives with its frequency lower by gh/c² = 2.455·10⁻¹⁵ — two and a half parts in a thousand million million. Nothing was done to the photon on the way up; the two ends of the tower disagree about how fast time passes, and the frequency is the evidence. The same fraction says a clock at the foot loses 0.21 nanoseconds a day against one at the top.

Every branch redshift-tower draws, and what checks it

two-source-interference

9 essays

Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

Every branch two-source-interference draws, and what checks it

velocity-addition

9 essays

Composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.

Every branch velocity-addition draws, and what checks it

chain-fall

8 essays

What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

Every branch chain-fall draws, and what checks it

gradient-reflection

8 essays

Reflection against the thickness of the boundary. How much of the wave comes back, against the width of the transition between media of wavenumber 1 and 2.2, on a logarithmic scale. At zero width the curve reaches 1.406e-1, which is the abrupt-interface value computed from the two end values and nothing else — so the smooth calculation contains the sharp one as a limit rather than contradicting it. Past a width of about a wavelength the fall is a straight line on this scale, of measured slope -12.551 per unit width against the -12.566 the asymptotic form gives, which is 4π times the smaller of the two wavenumbers. Exponential, not merely small: three wavelengths of transition costs eight decades of reflection. The circles are the same quantity computed by slicing the profile into uniform layers, agreeing with the curve to a factor of 1.0115.

Every branch gradient-reflection draws, and what checks it

horizon-thermo

8 essays

Hotter as it shrinks. The temperature of a horizon and its evaporation time, against mass, on logarithmic axes. A solar-mass horizon is at 6.17·10⁻⁸ K — far colder than the coldest thing anybody has made — and takes 2.1·10⁶⁷ years to evaporate. Both lines are straight, with slopes of exactly −1 and +3, and the first slope is the whole difficulty: losing mass makes the object hotter, so the process accelerates and ends in a burst rather than fading out.

Every branch horizon-thermo draws, and what checks it

planck-spectrum

8 essays

The blackbody spectrum, against what classical physics predicted. Spectral exitance against wavelength for a blackbody at 3000, 4000, 5000 kelvin, in kilowatts per square metre per nanometre. Each curve peaks at the wavelength Wien's displacement law gives — 966 nm at 3000 K, 724 nm at 4000 K, 580 nm at 5000 K — and falls to nothing at short wavelengths.

Every branch planck-spectrum draws, and what checks it

simultaneity

8 essays

Simultaneity at β = 0.5. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

Every branch simultaneity draws, and what checks it

stern-gerlach

8 essays

A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

Every branch stern-gerlach draws, and what checks it

wave-packet

8 essays

A packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

Every branch wave-packet draws, and what checks it

birefringence

7 essays

The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

Every branch birefringence draws, and what checks it

capillary-rise

7 essays

Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

Every branch capillary-rise draws, and what checks it

field-energy

7 essays

The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 100 square centimetres — 0.0100 square metres — against the separation of its plates, drawn twice. Held at 10 nC the energy rises in proportion to the separation; held at 113 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 1.00 mm, where they cross at 565 nJ, and there their slopes are equal and opposite: the attraction between the plates is 565 µN, or 5.647·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Every branch field-energy draws, and what checks it

granular-column

7 essays

The stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

Every branch granular-column draws, and what checks it

heat-capacity

7 essays

The staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant.

Every branch heat-capacity draws, and what checks it