Illustrated Physics
physics, drawn
Concept

Permeability — where it appears

The constant relating a magnetic field to the current producing it, which together with the permittivity fixes the speed of light. That the two electrical constants combine into a speed, and that the speed is the measured one, is the argument by which light was identified as an electromagnetic wave.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

The field that wraps a current

Ampère's law says that going once round a closed path and adding up the field counts the current threaded through it, and nothing else about the path survives into the answer. Four different loops round one wire return the same number to five decimal places — which is exactly why the law is both easy and treacherous.

electromagnetism · Ampere law The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

The term that made light

Ampère's law contradicts itself the moment a current stops being steady, and the contradiction is visible in one picture: two surfaces on the same loop, with a current through one of them and nothing through the other. The term that repairs it turns four equations into a wave, whose speed is two constants measured in a room with the lamps off.

electromagnetism · Maxwell equations The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

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.

electromagnetism · Field energy Skin depth against frequency, over eleven decades. The skin depth of copper, stainless steel, seawater, mu-metal against frequency, both axes logarithmic. Every line has the same slope, −½, because the depth goes as the inverse square root of the frequency for all of them; what separates them is conductivity and permeability, which enter the same way. Three points are marked and each is a practical fact. Copper at mains frequency has a skin depth of 9.22 mm, so a busbar of that thickness carries current throughout and a thicker one does not. Copper at a gigahertz has 2.1 µm, so the current in a microwave component runs in a layer thinner than the plating and the surface finish becomes the conductor. Seawater at the frequency navies use to signal submerged submarines has 28.9 m, which is why that link is measured in tens of hertz and carries a few characters a minute. Mu-metal is the outlier and the reason it exists: its conductivity is forty times worse than copper's and its permeability twenty thousand times better, so at low frequency it is the one material on the chart that is thin.

How far a field gets into metal

Every static argument about conductors says the field inside one is zero, and every one assumes the electrons have had time to move. Give them less time and the field gets in — 9.2 millimetres into copper at mains frequency, 2.1 microns at a gigahertz — and the metal box that silences a radio does almost nothing about the cable running past it.

electromagnetism · Conductors Two fields of the same magnet, and inside they point opposite ways. A uniformly magnetised sphere, with the field lines of B on the left and of H on the right, both computed from the exact solution — uniform inside, a dipole outside. Outside the sphere the two pictures are identical up to a constant, because there B is μ₀ times H and nothing else. Inside they are opposite: B is 0.67 tesla pointing along the magnetisation and H is 267 kiloamps a metre pointing against it. The B lines close on themselves and never end; the H lines begin on the top face and end on the bottom, which is what a field with sources looks like. Nothing about the magnet changed between the two panels — only which currents the circulation is allowed to count.

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

electromagnetism · Ampere law The thickness fixes the area, and the design only moves it. How much a metal-backed resistive sheet 7.5 mm from the metal absorbs, measured as ln(1/|Γ|), against wavelength, for sheets of 150, 250, 377, 800 Ω per square. Rozanov's bound says the area under any such curve, over all wavelengths, can be at most 2π² times the thickness: 147.9 mm here. The areas computed are 147.9 mm for 150 Ω, 147.9 mm for 250 Ω, 147.9 mm for 377 Ω, 69.7 mm for 800 Ω. Every sheet up to the impedance of free space meets the bound exactly and only moves its area between wavelengths; the more resistive sheet wastes part of it.

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

waves · Impedance Why silt heaves and sand and clay do not. The relative rate at which water is drawn to a growing ice lens, against grain diameter on a logarithmic scale, for three loads on the soil above it, in a capillary model: the rate is the soil's permeability, which grows as the grain diameter squared, times the suction the lens can sustain less the load. The suction is capped both by the supercooling at the lens, here 2 K, and by the Gibbs–Thomson pressure at which ice would break into the pores. Coarse soil has open pores that ice simply invades, so the suction never exceeds the load: under 10 kPa, half a metre of soil, nothing heaves above 58 μm. Fine clay can sustain the full suction but passes water too slowly to feed it. Between lies a fastest grain size, 29 μm under 10 kPa, 10 μm under 30 kPa and 2.9 μm under 100 kPa: silt, the soil engineers have long called frost-susceptible. Real heave pressures exceed this model's, and the fit is to the shape, not the scale.

The frost that lifts by drinking

Frozen ground rises in winter, sometimes by tens of centimetres, and the usual explanation — water expands by nine per cent when it freezes — is wrong: soil soaked in benzene, which shrinks as it freezes, heaves too. The lift comes from layers of clear ice that grow by pulling liquid water up through the soil from below. Ice and water touching each other need not be at the same pressure, and when the water is free to flow, ice a degree below its melting point can push with 1.12 megapascals while it drinks — twelve times less than freezing water in a sealed pipe, and enough to lift a house at a tenth of a degree.

thermodynamics · Chemical potential The force to pull a foot out of packed sand. The suction that resists pulling a foot (a sole of 250 cm², 10 cm long) straight up out of sand, as water is drawn through the sand into the space the foot leaves — Darcy's law with a Kozeny–Carman permeability — against the pulling speed, on logarithmic axes, for compacted sand (fraction 0.6) of three grain sizes and for loose sand (0.41) of 0.1 mm; the dotted line is a 70 kg person's weight. The force is proportional to the speed and inversely to the square of the grain size. Pulled at a centimetre a second from compacted 0.1 mm sand, a foot resists with 2.5 kN, 4 times the person's weight; the speed at which the suction equals that weight is 2.71 mm/s. In the loose sand before it compacted, the same pull would need 15 per cent of the force; with clay filling the pores of the compacted sand, as in natural quicksand, a thousand times more, and the speed a body's weight can pull at falls to microns a second.

The sand that lets a body in and will not let it out

Quicksand cannot swallow anyone: liquefied, it is nearly twice as dense as water, and a person in it floats with their chest clear. What makes it dangerous is the opposite asymmetry. Going in is easy, because a step breaks the gel holding the grains apart and the sand turns liquid. Coming out is hard, because the sand settles and packs around whatever sank, and a trapped foot can only rise as fast as water can seep through the packed grains to fill the space it leaves. Pulling harder barely helps; letting water in does.

fluids · Rheology Where the two theories part: the magnetic permeability. The voltage across the spinning wall as a fraction of a metal's, against the material's relative permeability μ, for a permittivity ε = 6. Lorentz's electron theory (flat) polarises the material only through the force on its charges, so μ does not enter: (ε − 1)/ε = 0.833 at every μ. Relativity gives (εμ − 1)/εμ, which agrees exactly at μ = 1 — so no ordinary insulator, such as the ebonite H. A. Wilson spun in 1904, can tell them apart — and climbs towards a metal's value as μ grows. At μ = 3 (dot) the relativistic voltage is 13.3 per cent larger. That is why Wilson and Wilson needed an insulator that was also magnetic, and made one from steel spheres set in wax.

The voltage a moving magnet adds to an insulator

Spin a hollow cylinder of insulator in a magnetic field along its axis and a voltage appears between its inner and outer surfaces, because every piece of it is moving through the field. How large that voltage is turns out to depend on whether the insulator is magnetic, and the dependence was predicted by one theory and missed by the other. Maxwell's equations hold in a moving material only if the material's response is written in its own frame and carried into the laboratory by relativity — and then a moving magnet is also an electric dipole.

electromagnetism · Maxwell equations The resistance a pan offers the coil. The surface resistance — the square root of πfμρ — of five pan materials against frequency, on logarithmic axes, with the band where induction hobs work, 20 to 100 kHz, shaded. All five rise as the square root of the frequency. At 25 kHz carbon steel offers 9.9·10⁻⁴ Ω and ferritic stainless 0.0024 Ω, while austenitic stainless, which is not magnetic, offers 2.7·10⁻⁴ Ω, aluminium 5.1·10⁻⁵ Ω and copper 4.1·10⁻⁵ Ω. For the same coil current the heating goes as this resistance, so steel heats 19 times as hard as aluminium. Matching steel with aluminium by frequency alone would need 377 times the frequency — 9 MHz, where the hob's own electronics and coil would be the problem.

The 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.

electromagnetism · Induction

Named alongside it

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

Magnetic fieldAmperes lawConstitutive lawMagnetisationPermittivityAmpere lawAnalyticityArchimedes principleBandwidthBound chargeBoundary conditionsBuoyancy

All concepts