A loop leaving the field
At its defaults it draws a loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.
induction-loop is one function in lib/figures/fields.js —
charge, current, flux and the lines drawn between them. 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.
A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.
How much of μ₀N²A/ℓ a real coil actually has
The options are the ones The circuit that fights its own change 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 inductance of a 140-turn coil of radius 10 mm, divided by the long-solenoid formula μ₀N²A/ℓ, against how long the coil is compared with its diameter. The inductance is computed as a sum of Maxwell's mutual inductances between every pair of turns plus each turn's own, so the long-coil formula appears nowhere in it. At a length equal to the diameter the real coil has 68% of what the formula promises, and at a quarter of the diameter about a third. The reason is that the formula assumes every turn is threaded by the full interior field, and near the ends of a short coil the field has already begun to spread. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computed points agree with the table.
How much of μ₀N²A/ℓ a real coil actually has
The options are the ones The circuit that fights its own change 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 inductance of a 60-turn coil of radius 25 mm, divided by the long-solenoid formula μ₀N²A/ℓ, against how long the coil is compared with its diameter. The inductance is computed as a sum of Maxwell's mutual inductances between every pair of turns plus each turn's own, so the long-coil formula appears nowhere in it. At a length equal to the diameter the real coil has 67% of what the formula promises, and at a quarter of the diameter about a third. The reason is that the formula assumes every turn is threaded by the full interior field, and near the ends of a short coil the field has already begun to spread. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computed points agree with the table.
Where the supply's power goes while the current is arriving
The options are the ones The circuit that fights its own change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A 156.0 microhenry coil in series with 12 ohms across 12 volts, from the instant the switch closes. Time is in units of the circuit's own constant L/R, which is 13.0 microseconds here. At the first instant the current is zero, so the resistor takes nothing and every watt the supply delivers goes into the field. As the current approaches 1.00 amps the field stops taking any and the resistor takes it all. The area under the field's curve is 77.98 microjoules, which is ½LI² — and it is the work done against the coil's own back emf rather than against anything external.
Closing the switch, and the 327× voltage that opening it makes
The options are the ones The circuit that fights its own change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A 156.0 microhenry coil across 12 volts through 12 ohms. The current rises with a time constant of 13.0 microseconds while the voltage across the coil falls from the full supply to nothing — the coil opposes the change and not the current. At 4 time constants the circuit is broken into 4000 ohms, which might be the resistance of an opening contact or of the air beside it. The current cannot change instantaneously, so it continues through the new resistance, and the voltage it develops there is 3927 volts — 327 times the supply, from a circuit containing no source of that size. That is why a coil is switched with a diode across it and why an ignition coil works.
Closing the switch, and the 3311× voltage that opening it makes
The options are the ones The circuit that fights its own change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A 512.1 microhenry coil across 24 volts through 6 ohms. The current rises with a time constant of 85.4 microseconds while the voltage across the coil falls from the full supply to nothing — the coil opposes the change and not the current. At 5 time constants the circuit is broken into 20000 ohms, which might be the resistance of an opening contact or of the air beside it. The current cannot change instantaneously, so it continues through the new resistance, and the voltage it develops there is 79461 volts — 3311 times the supply, from a circuit containing no source of that size. That is why a coil is switched with a diode across it and why an ignition coil works.
What checks it
physicscheck asserts something about induction-loop 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.
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.
ElectromagnetismThe coupling a resonance rescues
Two coils a hand's breadth apart share a few thousandths of their magnetic flux, and as a transformer they deliver almost nothing. Tune each with a capacitor to the same frequency and the same pair can carry most of the power across. Resonance does not strengthen the coupling, which is fixed by the geometry and falls as the cube of the distance. It changes how many times the energy goes round before it is lost. The efficiency then depends on one number, the coupling times the quality factor, and that one number explains a toothbrush on its stand, a phone on its pad, a car parked over a plate in the road and a light bulb lit two metres away.
ElectromagnetismThe coupling that is the same both ways
A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.
ElectromagnetismThe drag that falls as the magnet speeds up
Drop a magnet down a copper pipe and it creeps, because the currents it stirs up in the metal pull back on it in proportion to its speed. Push it faster and the proportion fails: the drag rises to a peak and then falls away, while a second force, pointing straight up, grows to replace it. A magnet moving fast enough over a sheet of aluminium is not braked but held up, with lift thirty times its drag at the speed of a fast train, and the reason is that at high speed the metal has no time to forget the magnet has passed.
ElectromagnetismThe 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.
ElectromagnetismThe magnet that falls slowly
Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.
ElectromagnetismThe pan a better conductor will not heat
An induction hob heats a pan with currents the pan carries itself, driven by a field oscillating twenty-five thousand times a second, and it will heat a cast-iron skillet furiously while an aluminium saucepan, which conducts electricity six times better, barely warms. The better conductor is the worse load. What a pan presents to the coil is a surface resistance that goes as the square root of permeability times resistivity, and on that measure iron beats aluminium twenty times over. The same arithmetic explains why kitchen foil on the same hob can catch fire, and why a steel tip heated this way switches itself off at a fixed temperature.
ElectromagnetismThe ring that jumps because its current is late
Slip an aluminium ring over the iron core of a coil, switch on the alternating current, and the ring leaps off the top. The usual explanation is Lenz's law: the ring's induced current opposes the change, so the ring is repelled. But an alternating field changes both ways every cycle, and an induced current that simply opposed the change would be pushed and pulled equally and stay put. The ring jumps only because its current runs late, held back by its own inductance — which is why a cooled ring jumps far higher and a ring with a saw cut through it does not move.
ElectromagnetismThe rule that is two laws wearing one coat
The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.