Generator

Where the radiation goes

One function in the extremes library, called 32 times across 5 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

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.

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.

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.

The classical atom, and how long it lasts. 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.

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.

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 0.3 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 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.

The classical atom, and how long it lasts. 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.

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.

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.

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.

Four powers of the Doppler factor. 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.

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.

The 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.

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.

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.

Whether 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.

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