Illustrated Physics
physics, drawn
Concept

Energy flux — where it appears

Energy crossing unit area per unit time, the quantity that falls with distance according to how the area of the enclosing surface grows. A spherical wave's flux falls as the inverse square, a cylindrical one's as the inverse first power, and a guided wave's not at all — so the geometry alone fixes the exponent.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

How a wave thins out

A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.

waves · Wave motion Where the light that was sideways ends up. The direction a photon is seen to travel in the laboratory, against the direction it was emitted in the frame of the source, for a source moving at 0.5c, 0.9c, 0.99c. The straight diagonal is what would happen if a boost only changed frequencies; every curve lies well below it, which is aberration. The number that matters is where the emitted right angle lands, because half of everything emitted is on that side of it: 60.0° at 0.5c, 25.8° at 0.9c, 8.1° at 0.99c. The usual shorthand for that angle is 1/γ, which gives 49.6°, 25.0°, 8.1° — good to a few per cent only once the source is genuinely relativistic, and wrong by 17% at 0.5c. Nothing is emitted differently in any of these cases: the source is radiating exactly as it always did, and it is the map from its angles to ours that has changed.

The sky that crowds into a cone

A boost does not only shift frequencies. It remaps directions, so half of everything a fast traveller can see is squeezed into a forward cone of half-angle about 1/γ — and because brightness carries four powers of the Doppler factor, what lies ahead is overwhelming and what lies behind has effectively gone.

relativity · Doppler The self-force matters at 6.27 × 10⁻²⁴ s, and nowhere a charge has ever been. The ratio of the radiation reaction to the applied force, which is τ divided by the time the force takes to change, on a logarithmic axis. τ = μ₀q²/6πmc = 6.266e-24 s for an electron, and light crosses 1.879 femtometres in that time — two thirds of the classical electron radius. a 3 GHz accelerating cavity: 1.9e-14; a 500 nm optical field: 3.8e-9; an electron orbiting a proton, ground state: 4.1e-8; an X-ray at 0.1 nm: 1.9e-5; over a classical electron radius: 6.7e-1. The largest of them, "over a classical electron radius", is still 1.5e+0 times too slow. So the correction is never large for any force anybody can apply, and the only regime where it would be is one in which the charge's own structure has already made the whole description meaningless.

The force a charge exerts on itself

Larmor's formula says how much an accelerating charge radiates and says nothing about who pays. Conservation says the charge does, so there is a force on it — and the equation that force produces has a free particle accelerating for ever with nothing pushing it, or else beginning to move before it is pushed. Both solutions are absurd, and the interval over which they are absurd is smaller than the electron the equation was written for.

astrophysics · Radiating charge Minima that are not zeros. The amplitude along a line carrying a wave towards a load and its reflection back, for reflection magnitudes of 0, 0.35, 0.7, 1. With everything reflected the pattern touches zero and is a standing wave in the strict sense. With less than everything it does not: the minima sit at one minus the reflection and the maxima at one plus it, so the pattern is a partial standing wave sitting on a travelling one. The spacing is half a wavelength in every case, and the depth is the only thing that changes — which is why one number, the ratio of the maximum to the minimum, is enough to report the whole pattern.

The node that is not standing still

A wave meeting a perfect reflector makes a standing wave with real nodes. A partial reflection makes something that looks the same and is not: the minima are not zeros, energy flows steadily through them, and the depth of the pattern is a measurement of the load that caused it.

waves · Standing waves One lever, two frames, two different torques. A right-angled lever pivoted at its corner, with arms of length L, held still by a force F at the end of each arm, perpendicular to it. Left, in the lever's own frame: torque FL anticlockwise from the horizontal arm's force, FL clockwise from the vertical arm's, and they cancel. Right, the same lever seen moving to the right at 0.8 of the speed of light: the horizontal arm, along the motion, is contracted to L/γ = 0.60L, and the force on it, which is across the motion, is reduced to F/γ = 0.60F; the vertical arm and its force, along the motion, are unchanged. The torques are now 0.36FL and FL, and the lever is twisted clockwise by 0.64FL — the square of its speed as a fraction of light's. It does not turn.

The lever that is twisted and does not turn

A right-angled lever held still by two equal forces has no net torque on it. Watch the same lever go by at high speed and one arm is shorter, one force is weaker, and the torques no longer cancel — yet the lever goes by without turning. Nothing is wrong with either frame's arithmetic. The forces are doing work on the moving lever, the energy they put in is flowing through its arms, and a flow of energy carries momentum that points across the lever's motion: a body that moves straight with a momentum that does not.

relativity · Mass-energy

Named alongside it

The objects these essays reach for when they reach for this one.

Dipole radiationIntensityAberrationAbsorptionAccelerationAmplitudeAngular momentumAttenuationBoundary conditionsCausalityClassical electron radiusConservation of energy

All concepts