Where the radiation goes
At its defaults it draws where the radiation goes. The angular distribution of the power radiated by an accelerating charge. On the left the charge is slow: the pattern is sin²θ about the acceleration, with nothing radiated along it and the maximum at right angles. On the right the same charge is moving at 0.9 of the speed of light, and aberration sweeps the whole pattern forward into a narrow cone — the peak here is at 13.4°, against the 1/2γ = 12.5° the usual estimate gives, inside a cone of half-angle 1/γ = 25.0°. A synchrotron is a searchlight for this reason and no other.
larmor-radiation is one function in lib/figures/extremes.js —
radiation, cross-sections and self-gravity. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
The angular distribution of the power radiated by an accelerating charge. On the left the charge is slow: the pattern is sin²θ about the acceleration, with nothing radiated along it and the maximum at right angles. On the right the same charge is moving at 0.9 of the speed of light, and aberration sweeps the whole pattern forward into a narrow cone — the peak here is at 13.4°, against the 1/2γ = 12.5° the usual estimate gives, inside a cone of half-angle 1/γ = 25.0°. A synchrotron is a searchlight for this reason and no other.
The classical atom, and how long it lasts
The options are the ones A charge that turns must glow passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.
Where the radiation goes
The options are the ones A charge that turns must glow passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The angular distribution of the power radiated by an accelerating charge. On the left the charge is slow: the pattern is sin²θ about the acceleration, with nothing radiated along it and the maximum at right angles. On the right the same charge is moving at 0.3 of the speed of light, and aberration sweeps the whole pattern forward into a narrow cone — the peak here is at 53.7°, against the 1/2γ = 27.3° the usual estimate gives, inside a cone of half-angle 1/γ = 54.7°. A synchrotron is a searchlight for this reason and no other.
The classical atom, and how long it lasts
The options are the ones A charge that turns must glow passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
An electron in a circular orbit of 1.06·10⁻¹⁰ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.245·10⁻¹⁰ seconds — 1.2e+2 picoseconds. It completes about 5.78·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.
Where the radiation goes
The options are the ones A charge that turns must glow passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The angular distribution of the power radiated by an accelerating charge. On the left the charge is slow: the pattern is sin²θ about the acceleration, with nothing radiated along it and the maximum at right angles. On the right the same charge is moving at 0.9 of the speed of light, and aberration sweeps the whole pattern forward into a narrow cone — the peak here is at 13.4°, against the 1/2γ = 12.5° the usual estimate gives, inside a cone of half-angle 1/γ = 25.0°. A synchrotron is a searchlight for this reason and no other.
Four powers of the Doppler factor
The options are the ones A charge that turns must glow passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
How bright a moving source looks, against the direction it is looked at from, for speeds of 0.5c, 0.9c, 0.99c and on a logarithmic scale. The source radiates the same total power in its own frame at every one of these speeds and radiates it evenly; what changes is the Doppler factor, which enters the received intensity four times over — once for each photon's energy, once for the rate they arrive at, and twice for the solid angle they are squeezed into. Forward against backward, that is a factor of 9 at 0.5c, 361 at 0.9c, 3.96·10⁴ at 0.99c. The consequence is that anything relativistic pointed away is not merely dimmed but effectively deleted, and anything pointed at the observer is over-represented in every catalogue by the same factor — which is a statement about the sample rather than about the source.
What checks it
physicscheck asserts something about larmor-radiation that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A charge that turns must glow
An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.
AstrophysicsThe bill that arrives when the pushing stops
A charge accelerating steadily radiates at the full Larmor rate while the radiation reaction force on it is exactly zero, so for as long as the push holds, nothing about the charge's motion pays a single watt. The energy is lent by the field that travels with the charge, the loan is called the Schott term, and it is repaid the moment the acceleration changes.
AstrophysicsThe circling that comes in quanta
An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.
AstrophysicsThe cross-section that forgets the colour
An electron on a spring, shaken by light, scatters it — weakly and very blue below its resonance, enormously at it, and above it by exactly the same amount whatever the colour. That flat stretch is the Thomson cross-section, 6.65 × 10⁻²⁹ m², and it is why an X-ray picture of a crystal is a map of electrons and why the corona is white.
AstrophysicsThe flash a circling charge sends once a turn
A slow electron going round a magnet broadcasts one radio frequency. The same electron at nearly the speed of light, on a circle of the same kind, sends X-rays. Nothing about the mechanism changes between the two. The charge chases its own light, and the chase compresses one power of γ worth of emission into three.
AstrophysicsThe 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.
AstrophysicsThe light a charge makes by changing medium
A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.
AstrophysicsThe radio that only heavier electrons can make
Above every aurora the Earth broadcasts a gigawatt of radio that no one on the ground can hear. It comes out just below the frequency at which electrons circle the field there, and it is a maser — amplified by the electrons themselves. The surprise is what makes the amplifier work. Without relativity it could not: summed over every electron, a wave at the circling frequency is always absorbed. The few per cent by which a fast electron is heavier, and circles more slowly, is not a correction to the mechanism. It is the mechanism.
AstrophysicsThe rainbow a passing electron leaves on a grating
An electron moving steadily in a straight line through empty space does not radiate, at any speed. Send it skimming over a ruled metal grating, without touching it, and it lights up — every colour at its own angle, blue-green ahead and red further out, from an electron that is neither turning nor slowing. The grating hands the electron's field the one thing it lacked: a wavenumber that lets part of it escape. It is the same mechanism as the blue glow in a reactor pool, with a ruled surface standing in for the slow light in the water.
RelativityThe 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.
AstrophysicsWhether a charge on a table glows
The equivalence principle says a charge at rest in a gravitational field is a charge accelerating in empty space, and an accelerating charge radiates. Nothing is supplying the energy. The argument has run for eighty years, and its resolution is that radiation is not something a single observer can define.
AstrophysicsWhy the glow of a fast charge is blue
A charge moving in a straight line at constant speed does not radiate — in a vacuum. In water, if it moves faster than light does in water, it glows, and the glow is blue. The cone of the light is the familiar half of the story. The other half is a count: Frank and Tamm's formula says each centimetre of water yields about two hundred visible photons, in numbers that rise as the inverse square of the wavelength, which is why a reactor pool shines blue and why the light keeps getting stronger into the ultraviolet until water stops being transparent. The charge never accelerates. What radiates is the water.