Energy conservation — where it appears
Named by 21 essays across 7 fields — each of them below, with the objects they name alongside it.
The angle that throws furthest, and why nobody notices
Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.
Collisions are easier than forces, and momentum is the reason
Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.
The hill that gives it back, and the forces that do not
Potential energy turns a question about motion into a picture of a landscape. It works for gravity and springs, it fails for friction, and the difference between those two cases is the whole of what makes energy useful.
One number for every point, and nothing at all is lost
The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.
The mass, and where it sits, which is what decides the race
Release a hoop and a marble together on a slope and the marble wins, whatever they weigh and whatever their size. Neither mass nor radius survives the arithmetic; only the arrangement does.
The field that makes the other, and only while it is changing
A magnet sitting next to a coil does nothing at all. Move it and a current flows. The law is not about the field but about its rate of change, and everything electrical since 1831 rests on that distinction.
Mass is a form of energy, which is not the same as a source of it
The famous equation is usually read as a promise that mass can be turned into energy. It says something stricter and stranger — that a mass is an energy already, sitting there, whether or not anything ever releases it.
Light arrives in lumps, and brightness only changes how many
Shine dim blue light on a metal and electrons come out. Shine intense red light and none do, however long the wait. The frequency decides whether anything happens; the intensity decides only how much.
A photon with a momentum, and a collision that proves it
X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.
What happens where the medium changes
Two conditions at a join — the displacement is continuous, and so is the transverse force — fix the reflected and transmitted amplitudes completely. What decides them is one quantity, the impedance, and not the stiffness or the density separately: two quite different media with the same impedance are, to a wave, the same medium.
The point that keeps moving as if nothing had happened
Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.
Where the energy of a field actually is
A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W.
The invariant that survives a boost
Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.
Five balls, and the law that does not choose
The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.
The circuit that fights its own change
Every circuit is threaded by the field its own current makes, so every circuit resists having that current altered. It is the reason a coil takes time to start and the reason breaking one makes a voltage a thousand times the supply's.
What adding does to the energy
Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.
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.
The energy that did not all arrive
A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.
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 reflection that needs no surface
Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.
The rocket that leaves its fuel at home
A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.
Named alongside it
The objects these essays reach for when they reach for this one.
Momentum conservationE = mc²Centre of massConservation lawsElastic collisionPhotonSpectrumElectric fieldElectric potentialElectromagnetic inductionEmfField energy