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.
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.

Granular matter

  1. 1 The silo that does not weigh what it holds
  2. 2 The angle that does not know the size of the heap
  3. 3 The heap that becomes a solid
  4. 4 The big one comes to the top
  5. 5 The hourglass that keeps time
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6 essays · fluids 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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6 essays · waves How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical.

Horizons

  1. 1 The surface that only lets things in
  2. 2 Two clocks that disagree about the fall
  3. 3 The horizon that nothing marks
  4. 4 The circle light cannot leave
  5. 5 The hole that outlives everything and then does not
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6 essays · astrophysics A converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

Imaging

  1. 1 What a lens is doing, and why three rays are enough
  2. 2 The mirror that cannot focus, and the shape that can
  3. 3 The image that is a diffraction pattern twice
  4. 4 The condition a lens must meet
  5. 5 The focus that is a slab, not a plane
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6 essays · optics Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently.

Magnetisation

  1. 1 The magnet that has to fight its own field
  2. 2 The magnetism classical physics forbids
  3. 3 The curve that is really a staircase
  4. 4 What holds a magnet together is not magnetism
  5. 5 The first length that belongs to the substance
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6 essays · electromagnetism Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it.

Mass-energy

  1. 1 Mass is a form of energy, which is not the same as a source of it
  2. 2 The invariant that survives a boost
  3. 3 The mass that is missing
  4. 4 The box of light that weighs something
  5. 5 The fuel a starship needs
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6 essays · relativity 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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6 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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6 essays · mechanics One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round.

Plasma oscillation

  1. 1 The frequency below which nothing gets in
  2. 2 The whistle that arrives sorted
  3. 3 The long-range force that does not reach
  4. 4 The wave that dies with nothing to rub against
  5. 5 The wall a plasma builds against itself
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6 essays · astrophysics Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place.

Projectile

  1. 1 The angle that throws furthest, and why nobody notices
  2. 2 The angle that drag moves
  3. 3 Everywhere a throw can reach
  4. 4 One curve answers every slope
  5. 5 The best throw is a tangency
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6 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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6 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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6 essays · optics Response against driving frequency. The steady-state amplitude of a driven oscillator against driving frequency, at three damping ratios. Lighter damping gives a taller and narrower peak, and the peak sits slightly below the natural frequency.

Resonance

  1. 1 The frequency that gets an answer, and the quarter cycle nobody mentions
  2. 2 The swing that is pumped, not pushed
  3. 3 The mass that makes another stand still
  4. 4 The width that is a lifetime
  5. 5 The resonance with a zero in it
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6 essays · waves 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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6 essays · optics A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Spacetime diagram

  1. 1 Two axes, one speed, and a diagram that does the arguing
  2. 2 The quantity nobody argues about
  3. 3 The diagram a ruler cannot read
  4. 4 The transformation that never mentions light
  5. 5 What the light cones alone can decide
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6 essays · relativity Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

Stratification

  1. 1 The layer a parcel cannot leave
  2. 2 The wave that picks an angle
  3. 3 The wave that is required to stand still
  4. 4 The reflection that changes the wavelength
  5. 5 The latitude past which a tide cannot split
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6 essays · fluids Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

Third law

  1. 1 The staircase that never reaches the floor
  2. 2 The entropy that is still there at zero
  3. 3 Hotter than any temperature there is
  4. 4 The entropy that depends on how fast it was cooled
  5. 5 A law about spectra, not about heat
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6 essays · thermodynamics 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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6 essays · relativity 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.

Wave packets

  1. 1 The packet that moves at another speed than its own crests
  2. 2 The speed that depends on the length
  3. 3 The packet that will not keep its shape
  4. 4 The speed that carries no signal
  5. 5 The pulse two failures keep alive
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6 essays · waves Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

Ampere law

  1. 1 The field that wraps a current
  2. 2 The field outside the solenoid, which is not zero
  3. 3 The law that is always true and rarely useful
  4. 4 The field that points against the magnet it is in
  5. 5 A potential that does not come back to itself
5 essays · electromagnetism

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