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.
Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption.

Photon

  1. 1 Light arrives in lumps, and brightness only changes how many
  2. 2 A photon with a momentum, and a collision that proves it
  3. 3 The experiment a wave cannot pass
  4. 4 The outcomes identical photons refuse
  5. 5 The noise pushed below the floor
5 essays · quantum Four answers to one push. Shear stress against shear rate for four fluids. The straight line through the origin is the Newtonian definition and is the only one of the four for which the word viscosity names a number. The Bingham fluid does not move at all until the stress passes 0.4, which is why toothpaste holds a shape on a brush and why wet concrete can be stood in a heap.

Rheology

  1. 1 The fluid that answers back
  2. 2 The liquid that remembers
  3. 3 The paste that holds up its own hill
  4. 4 The liquid that climbs the rod
  5. 5 The stretch a chain cannot outrun
5 essays · fluids 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
5 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
5 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
5 essays · quantum The screening that survives to zero frequency. How far a magnetic field gets into copper against how fast it is changing, drawn beside the London depth of a superconductor with a carrier density of 6.0e+28 per cubic metre. The metal's curve falls as one over the square root of the frequency — measured on the drawn curve as -0.5000 against −½ — and it is a straight line on these axes with no bottom: at a hundred hertz the field reaches nine millimetres in, and at zero frequency it reaches all the way through, because a normal metal screens by dissipating and a steady field dissipates nothing. The superconductor's line is flat at 21.7 nanometres. The frequency does not appear in the expression for it, so there is nothing for it to depend on, and the screening is as complete at zero frequency as at any other. The two lines cross at 9.0e+12 hertz, in the far infrared, and above that the ordinary metal is actually the better screen — which is a useful corrective, because a superconductor's advantage is not that it screens harder but that it does not need the field to be changing. What the picture cannot show is where the flat line stops: above the energy gap the pairs break, the superconductor becomes an ordinary metal, and the flat line turns into a sloping one.

Superconductivity

  1. 1 The field that is pushed out
  2. 2 The two in the flux quantum
  3. 3 Two lengths, and which one is longer
  4. 4 The voltage that is a frequency
  5. 5 The circuit that forgets its charge
5 essays · electromagnetism A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

Tunnelling

  1. 1 The wall that is not quite a wall
  2. 2 A wall that a factor of two makes impassable
  3. 3 The last atom does all the seeing
  4. 4 Two walls that let more through than one
  5. 5 How long the crossing takes
5 essays · quantum 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
5 essays · relativity A horizon 0.97 light years behind, made by nothing but the motion. Position across and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.

Accelerated frames

  1. 1 The wall of silence behind a rocket that never stops
  2. 2 The ship that never arrives at c
  3. 3 The temperature of an acceleration
  4. 4 The clock that does not feel the turn
4 essays · relativity 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
4 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
4 essays · quantum 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
4 essays · mechanics 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
4 essays · electromagnetism Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size.

Equivalence principle

  1. 1 The floor that cannot be told from gravity
  2. 2 The term free fall cannot remove
  3. 3 The fall that does not depend on what is falling
  4. 4 The binding energy that has to fall too
4 essays · astrophysics 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
4 essays · optics The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 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 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ 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.

Field energy

  1. 1 Where the energy of a field actually is
  2. 2 The angular momentum that is in nothing at all
  3. 3 The momentum of something that is not moving
  4. 4 The force read off a surface that touches nothing
4 essays · electromagnetism 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
4 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
4 essays · astrophysics One parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Harmonic approximation

  1. 1 Every minimum is a parabola
  2. 2 The two pendulums that will not stop swapping
  3. 3 Why heating a perfect spring changes nothing
  4. 4 The oscillator that answers at three times the question
4 essays · mechanics Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy.

Impedance

  1. 1 What happens where the medium changes
  2. 2 The layer that makes a reflection vanish
  3. 3 The taper that matches every note
  4. 4 The mismatch no network can remove
4 essays · waves

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