Illustrated Physics
physics, drawn
Depth

Series — page 2

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
Ways to arrange 10 coins. The number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Entropy

  1. 1 Entropy is a count, and the arrow of time is arithmetic
  2. 2 The entropy that lives on a surface
  3. 3 The exponential that decides everything
  4. 4 What a system actually minimises
  5. 5 Mixing what is already mixed
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11 essays · thermodynamics 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.

Equipartition

  1. 1 Half a kT for every way of moving
  2. 2 The share that is not half a kT
  3. 3 The temperature a molecule does not have
  4. 4 The energy that refuses to be shared
  5. 5 The condition three modes never meet
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11 essays · thermodynamics Malus's law. The fraction of polarised light passing a filter, against the angle between the light's own direction of shaking and the filter's axis. It is the cosine squared: half at 45 degrees, nothing at 90.

Polarisation

  1. 1 The direction of the shaking, and the filter that only asks about it
  2. 2 The angle at which reflection picks a side
  3. 3 The crystal that answers twice
  4. 4 The rotation a return trip doubles
  5. 5 The phase that is only a shape
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11 essays · optics The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.

Radiating charge

  1. 1 A charge that turns must glow
  2. 2 The force a charge exerts on itself
  3. 3 Whether a charge on a table glows
  4. 4 The bill that arrives when the pushing stops
  5. 5 The flash a circling charge sends once a turn
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11 essays · astrophysics Refraction from n = 1 into n = 1.5. A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires.

Refraction

  1. 1 The bend at the boundary, and what it is really about
  2. 2 The law that only asks about one component
  3. 3 The angle that is two angles
  4. 4 The ray on the wrong side of the normal
  5. 5 The reflection that needs no surface
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11 essays · optics How much each colour is scattered. Scattering strength against wavelength, as the inverse fourth power, normalised to one at 550 nanometres. Light at 450 nanometres is scattered 4.35 times as strongly as light at 650 nanometres — which is the whole reason the sky is the colour it is.

Scattering

  1. 1 Why the sky is blue and the sunset is not, from one exponent
  2. 2 When the particle is the size of the wave
  3. 3 Why a litre of water is not blue for the reason the sky is
  4. 4 Everything a scatterer removes, from one direction
  5. 5 The cloud light has to walk through
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11 essays · optics The straight line between two plates. A layer of water 10 mm deep with its top plate drawn along at 1 m/s. In the steady state the velocity is a straight line from zero at the fixed surface to the plate's speed at the moving one, and the stress needed to keep it going is μ times that slope: 0.100 Pa. The fluid at each wall is at rest with respect to it, which is an experimental fact rather than a consequence of anything above.

Viscosity

  1. 1 Momentum going sideways
  2. 2 The fourth power in a pipe
  3. 3 The thickness that goes both ways
  4. 4 The shear that only reaches so far
  5. 5 The drift a sound leaves behind
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11 essays · fluids Wavefronts from a moving source. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

Doppler

  1. 1 The note that changes on approach, and the two ways of getting it
  2. 2 The shift that survives at right angles
  3. 3 The sky that crowds into a cone
  4. 4 The cone the source leaves behind
  5. 5 The brightness that is not the same for everyone
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10 essays · waves The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Field transformation

  1. 1 Magnetism is electricity seen sideways
  2. 2 The field nobody can transform away
  3. 3 The one quantity a boost leaves alone
  4. 4 Charge and current are one thing
  5. 5 Six numbers, one object
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10 essays · relativity The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw.

Guided waves

  1. 1 The pipe that will not carry a low note
  2. 2 The channel with no walls
  3. 3 The wave a surface is enough to hold
  4. 4 The mode that will not turn a corner
  5. 5 Two tails that swap everything
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10 essays · waves The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

Osmosis

  1. 1 The pressure that comes from counting
  2. 2 What it costs to take the salt out
  3. 3 The membrane that almost holds
  4. 4 The swelling a membrane cannot stop
  5. 5 The counterions that never leave the chain
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10 essays · fluids The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.

Pendulum

  1. 1 The pendulum, and the small lie that makes it simple
  2. 2 The period that depends on the swing, computed exactly
  3. 3 The three ways of coming to rest
  4. 4 The length nobody has to measure
  5. 5 Held up by a force that averages to nothing
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10 essays · mechanics Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Radiation pressure

  1. 1 Light has a pressure
  2. 2 The size the light cannot blow away
  3. 3 The brightness a mass cannot exceed
  4. 4 The light that pulls rather than pushes
  5. 5 The friction made of light
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10 essays · astrophysics A light clock at β = 0.6. The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.

Time dilation

  1. 1 The clock that has to slow, and why no clock can refuse
  2. 2 The twin who comes back younger
  3. 3 The clock that is wrong in two directions
  4. 4 The two clocks that flew in opposite directions
  5. 5 The longest way round is the shortest clock
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10 essays · relativity 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.

Velocity-addition

  1. 1 Speeds that refuse to add, and the quantity that does
  2. 2 The turn that two pushes leave behind
  3. 3 The motion that measures faster than light
  4. 4 The space that speeds live in
  5. 5 The drag that was only an addition
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10 essays · relativity 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.

Capillarity

  1. 1 How high water will climb
  2. 2 The angle a liquid makes with what it sits on
  3. 3 The column that is pulled, not pushed
  4. 4 The pore that fills from dry air
  5. 5 The angle a voltage can set
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9 essays · fluids Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve.

Chaos

  1. 1 The error that doubles on a schedule
  2. 2 The last curve to go
  3. 3 The fold that has to be there
  4. 4 The map a dripping tap turns out to be
  5. 5 The calm that is the ghost of a cycle
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9 essays · mechanics Velocity and acceleration in uniform circular motion. Velocity drawn tangent to a circular path and acceleration drawn toward its centre, at eight points around the circle. The speed never changes and the acceleration is never zero; the two vectors are perpendicular everywhere.

Circular motion

  1. 1 Turning is an acceleration, and constant speed does not help
  2. 2 The forces that are not there
  3. 3 The deflection that closes on itself
  4. 4 The ratio that decides whether the planet is turning
  5. 5 The wave that can only travel west
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9 essays · mechanics Two sources 4 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.

Coherence

  1. 1 Why two lamps never interfere
  2. 2 How far a wave can remember
  3. 3 The fringe that measures a star
  4. 4 The grain that is in the light
  5. 5 The correlation that survives what the phase does not
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9 essays · optics A conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

Conductors

  1. 1 The inside of a conductor, where the field is exactly nothing
  2. 2 How much charge a shape will hold, before anything is charged
  3. 3 The charge that has to be somewhere else
  4. 4 How far a field gets into metal
  5. 5 The pressure a charge puts on its own metal
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9 essays · electromagnetism

All essays