Illustrated Physics
physics, drawn
Depth

Series — page 4

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.
Hydrogen's emission lines, where they are actually seen. Every transition down to level 1, 2, 3 in hydrogen, drawn at the wavelength it emits, on a logarithmic axis in nanometres. The Lyman series begins at 121.5 nm and crowds toward its limit at 91.1 nm; The Balmer series begins at 656.1 nm and crowds toward its limit at 364.5 nm; The Paschen series begins at 1874.6 nm and crowds toward its limit at 820.1 nm. Only the Balmer series has lines in the visible band, which is why it was the one found first.

Atomic spectra

  1. 1 The spectrum is a subtraction, not a list of values
  2. 2 The line that is really two
  3. 3 The field an atom calls strong
  4. 4 The line that takes eleven million years
  5. 5 The atom the size of a bacterium
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8 essays · quantum An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

Attenuation

  1. 1 The distance that takes the treble out
  2. 2 The answer that cannot come first
  3. 3 Where the loudness goes
  4. 4 The steps in an absorption curve
  5. 5 The ripple that counts the neighbours
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8 essays · waves One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

Bands

  1. 1 What happens when the wells get close
  2. 2 The mass a curve decides
  3. 3 The crossing that never happens
  4. 4 One level, and the field that bends the bands
  5. 5 The barrier the metal cannot choose
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8 essays · quantum 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.

Blackbody

  1. 1 The curve that would not come down
  2. 2 The glow that says nothing about the surface
  3. 3 The gas that nobody counted
  4. 4 The height a planet is seen from
  5. 5 The spectrum that is a thermometer at every height
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8 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
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8 essays · quantum The field a polarised sphere makes inside itself. A uniformly polarised sphere, its bound surface charge drawn at the size the cosine gives it, the uniform field that charge makes inside, and the exact dipole field it makes outside. The internal field is the same everywhere and points against the polarisation — that is what makes it a depolarising field — and its size is P/3ε₀, the third being the sphere's share of the one unit the three axes divide between them.

Dielectrics

  1. 1 The field the matter takes away
  2. 2 The constant that depends on how fast it is asked
  3. 3 A refraction with no wave in it
  4. 4 The force that lives where the model is not
  5. 5 The force whose sign a frequency chooses
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8 essays · electromagnetism The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

Etendue

  1. 1 The brightness no lens can increase
  2. 2 The cone a fibre will accept
  3. 3 The cone light has to find to get out
  4. 4 The invariant that is a count
  5. 5 The work a diluted beam will not do
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8 essays · optics Snell's law, found by searching. Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.

Fermat

  1. 1 The path that does not change
  2. 2 The ray that bends without a surface
  3. 3 The path that takes the longest time
  4. 4 The surface that images one point exactly
  5. 5 The medium that images every point
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8 essays · optics 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
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8 essays · astrophysics A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Gauss's law

  1. 1 Counting what comes out, and never looking inside
  2. 2 The shape decides the falloff, and the force law never changes
  3. 3 The pull that grows on the way down
  4. 4 The field outside that cannot find the core
  5. 5 The number of dimensions a planet can orbit in
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8 essays · electromagnetism Where a clock gains, and where it loses. The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.

Gravitational redshift

  1. 1 The clock that runs slow lower down
  2. 2 The parallelogram that will not close
  3. 3 The clock that measures a height
  4. 4 The clocks that must all slow together
  5. 5 The radius a mass adds that no circumference shows
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8 essays · astrophysics 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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8 essays · astrophysics 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
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8 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
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8 essays · relativity Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.

Light deflection

  1. 1 The bend Newton got half right
  2. 2 The delay that is not a bend
  3. 3 The lens with no focal length
  4. 4 The half of the bend a slow body never feels
  5. 5 The fifth image a galaxy hides
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8 essays · astrophysics 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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8 essays · electromagnetism Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

Potential

  1. 1 One number for every point, and nothing at all is lost
  2. 2 Nothing can be held still by a static field
  3. 3 The potential is where the wanderers stop
  4. 4 The field that keeps Greenwich time
  5. 5 The voltage that bends an electron like glass
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8 essays · electromagnetism Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

Self-gravity

  1. 1 The size at which a body becomes round
  2. 2 The ball of gas that heats up as it cools
  3. 3 The mass no cold matter can hold up
  4. 4 The disturbance that grows instead of travelling
  5. 5 The distance that forgets the moon
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8 essays · astrophysics 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.

Simultaneity

  1. 1 Now is a choice of slicing
  2. 2 Which came first, and who decides
  3. 3 The ring where the two beams disagree
  4. 4 The speed that cannot be measured one way
  5. 5 How big now is
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8 essays · relativity How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

Spin

  1. 1 The angular momentum that is not a rotation
  2. 2 Four states, and one of them is odd
  3. 3 The turn that has to be made twice
  4. 4 Two states where the counting says three
  5. 5 The experiment that defines spin and cannot be done on it
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8 essays · quantum

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