Illustrated Physics
physics, drawn
Depth

Series — page 3

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.
Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

Buoyancy

  1. 1 The weight of the water that is not there
  2. 2 Why a ship comes back upright
  3. 3 The depth past which it must sink
  4. 4 The block the water does not lift
  5. 5 The body that displaces two things
5 essays · fluids 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
5 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
5 essays · mechanics Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises.

Correspondence

  1. 1 Where the quantum picture hands back the old one
  2. 2 The average that obeys Newton
  3. 3 The state that swings like a pendulum
  4. 4 The return a classical cloud never makes
  5. 5 The probability that goes below zero
5 essays · quantum Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.

Decay

  1. 1 A nucleus with no clock
  2. 2 The exponential that is only true in the middle
  3. 3 The energy that did not all arrive
  4. 4 The chain that runs at its slowest member's rate
  5. 5 The half-life that chemistry can change
5 essays · quantum Rays through a raindrop. Parallel rays entering a spherical drop at different heights, refracting in, reflecting once from the back, and refracting out. The outgoing rays crowd together near one particular direction, and that crowding is the bow.

Dispersion

  1. 1 The angle the rainbow has to be, and why nobody chose it
  2. 2 Two glasses that cancel a derivative
  3. 3 The wavelength a fibre does not smear
  4. 4 The ring at twenty-two degrees
  5. 5 The delay that is a random variable
5 essays · optics A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand.

Energy

  1. 1 The hill that gives it back, and the forces that do not
  2. 2 The energy that depends on the observer
  3. 3 The speed at which grip hands over to power
  4. 4 The floor that does no work
  5. 5 The wall that moves while the ball is in flight
5 essays · mechanics The correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

Entanglement

  1. 1 The correlation no instructions can produce
  2. 2 The disagreement that one run settles
  3. 3 What two have they cannot give a third
  4. 4 A link between two that never met
  5. 5 The corner of Hilbert space that is ever visited
5 essays · quantum 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
5 essays · relativity What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

Flux freezing

  1. 1 The field that cannot get out
  2. 2 The knot the field cannot untie
  3. 3 The same force whichever way the surface faces
  4. 4 The wave that does not know what the gas is made of
  5. 5 The twist that outlives the turbulence
5 essays · astrophysics What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis.

Friction

  1. 1 The force that takes what it needs
  2. 2 The chatter a stiffer holder removes
  3. 3 The grip that is not a coefficient
  4. 4 The part of the wrap that is actually gripping
  5. 5 The grip that needs a little slipping
5 essays · mechanics What a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

Gravitational waves

  1. 1 The wave that stretches one way and squeezes the other
  2. 2 The orbit that has to shrink
  3. 3 What the instrument actually hears
  4. 4 A few cycles that are only mass and spin
  5. 5 The ring that does not come back
5 essays · astrophysics Why a straight front stays straight. A plane wavefront, with 9 points on it treated as sources and a wavelet of radius vt drawn about each. The envelope of those circles — the curve touching all of them — is a second straight line, parallel to the first and displaced by exactly vt. That is the whole of straight-line propagation: nothing else has to be assumed, and in particular nothing has to be said about rays, which are afterwards defined as the normals to these fronts. The construction is drawn with the wavelets left in, because they are the part that does the work.

Huygens

  1. 1 Every front is a source
  2. 2 The spiral that says how much light arrives
  3. 3 The wave that comes from the rim
  4. 4 The backward wave Huygens had to remove
  5. 5 The fan of plane waves inside every beam
5 essays · waves Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

Hydrostatics

  1. 1 The pressure that only knows depth
  2. 2 Force multiplied, and nothing gained
  3. 3 The surface a spin decides
  4. 4 The height a siphon cannot pass
  5. 5 The push that has no direction
5 essays · fluids A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.

Induction

  1. 1 The field that makes the other, and only while it is changing
  2. 2 The magnet that falls slowly
  3. 3 The rule that is two laws wearing one coat
  4. 4 The coupling that is the same both ways
  5. 5 The circuit that fights its own change
5 essays · electromagnetism The action along a family of paths. On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

Least action

  1. 1 Least action, except that it is not least
  2. 2 The conservation law a symmetry hands over
  3. 3 The force a coordinate cannot see
  4. 4 The principle that fixes the energy instead of the clock
  5. 5 The action that knows where every path ends
5 essays · mechanics 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.

Length contraction

  1. 1 The length that depends on when, and is not really about length
  2. 2 The pole that fits and does not fit
  3. 3 The contraction no photograph shows
  4. 4 The string that breaks between two rockets
  5. 5 The disc that cannot be spun
5 essays · relativity The field of a current loop, seen edge on. Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.

Magnetism

  1. 1 The field with no ends, and the force that does no work
  2. 2 The force that does no work
  3. 3 The loop that behaves like a needle
  4. 4 The drift that does not care what the charge is
  5. 5 The current no particle carries
5 essays · electromagnetism A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

Momentum

  1. 1 Collisions are easier than forces, and momentum is the reason
  2. 2 The point that keeps moving as if nothing had happened
  3. 3 The push that needs nothing to push against
  4. 4 The pile that lands heavier than it weighs
  5. 5 Five balls, and the law that does not choose
5 essays · mechanics The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is.

Periodic media

  1. 1 The gap a repeat opens
  2. 2 The mode that lives in the mistake
  3. 3 The frequency a lattice cannot carry
  4. 4 The end that knows how the middle was cut
  5. 5 The mirror that works from every direction
5 essays · waves

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