Illustrated Physics
physics, drawn
Series

Dielectrics — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The field a polarised sphere makes inside itself. A uniformly polarised sphere, its bound surface charge drawn at the size the cosine gives it, the uniform field that charge makes inside, and the exact dipole field it makes outside. The internal field is the same everywhere and points against the polarisation — that is what makes it a depolarising field — and its size is P/3ε₀, the third being the sphere's share of the one unit the three axes divide between them.

    The field the matter takes away

    Put a piece of glass in an electric field and the field inside it is smaller. How much smaller is not a property of glass. A needle of it keeps almost the whole field, a sphere keeps three-quarters, a slab across the field keeps a seventh — same material, same applied field, three answers, and the difference is arithmetic about shape.

    part 1 · electromagnetism
  2. Water's permittivity across six decades of frequency. The real and imaginary parts of water's permittivity against frequency, on logarithmic axes, from 10 to 16 in powers of ten hertz. ε′ begins at 80.1 — the number a textbook prints — falls through the Debye relaxation near twenty gigahertz, is dragged down again by the librational, bending and stretching bands of the molecule, and settles at 1.777 in the visible, whose square root is 1.3330: water's refractive index. The identity n = √ε_r is exact and is an identity between two numbers taken at the same frequency; using the static permittivity in it predicts an index of 8.9 and is the most instructive wrong answer in the subject. The curve is a Debye term and four Lorentz oscillators, with two parameters solved so that the two plateaus are the measured ones rather than fitted by eye.

    The constant that depends on how fast it is asked

    Water's relative permittivity is 80.1. Its refractive index is 1.333, and the square of 1.333 is 1.777. The identity n = √ε_r is exact, the two numbers differ by a factor of forty-five, and nothing is wrong with either — because a permittivity is a function of frequency and the two measurements were made eight orders of magnitude apart.

    part 2 · electromagnetism
  3. A refraction with no wave in it. Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 4, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 4.000, 4.000, 4.000, 4.000, 4.000, against 4. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

    A refraction with no wave in it

    A field line crossing from one dielectric into another bends, by a law that looks exactly like Snell's with the sines replaced by tangents. Nothing is oscillating, nothing is travelling, and no speed appears anywhere in the derivation — only the two conditions that say what a boundary may and may not do to a field.

    part 3 · electromagnetism
  4. The only part of the field that can pull. A dielectric slab part-way into a parallel-plate capacitor. Everywhere except at the slab's edge the field is perpendicular to the plates and therefore perpendicular to the direction the slab can move, so it exerts no force along that direction at all: inside the parallel-plate model, which is uniform between the plates and zero outside them, nothing pulls the slab anywhere. The force lives in the bowed lines drawn at the edge, where the field leaks past the end of the dielectric and acquires a component along the plates. Those lines are what the model throws away as a small correction near the boundary, and they are the entire mechanism. The energy method sidesteps the drawing altogether: differentiate the total energy with respect to the insertion and the answer is 1.328e-3 newtons, inward, without ever asking where on the slab the force is applied.

    The force that lives where the model is not

    A slab of glass held at the mouth of a charged capacitor is pulled in. Inside the parallel-plate model there is no force at all — the field is perpendicular to the slab's motion everywhere — and the same model's energy nevertheless gives the pull exactly right. The mechanism is entirely in the part of the field the model throws away.

    part 4 · electromagnetism
  5. A living cell and a dead one, told apart by a frequency. The real part of the dielectrophoretic factor for a living cell and a dead one, ten micrometres across, in a liquid of conductivity 0.01 S/m, against frequency. The cell is a conducting interior inside a membrane five nanometres thick. The living cell's intact membrane blocks slow fields, so at low frequency it behaves as an insulator and is pushed out of strong field; above 42 kHz the membrane is short-circuited by its own capacitance and the conducting interior is felt, so the cell is pulled in; above 140 MHz the permittivities take over and it is pushed again. The dead cell's membrane leaks, its interior has lost ions, and it changes sign only at 8.8 MHz. In 2 bands — below 42 kHz and from 8.9 MHz to 140 MHz — the same field pushes the two in opposite directions.

    The force whose sign a frequency chooses

    A neutral particle in a non-uniform field is pulled towards strong field if it polarises more than the liquid around it and pushed away if it polarises less. With conduction in the picture, which of the two it does is decided by how fast the field alternates — and a living cell, a conductor wrapped in an insulator five nanometres thick, changes sign twice, where a dead one changes once. The same electrodes can then send the living cells one way and the dead ones the other.

    part 5 · electromagnetism
  6. A field strings a suspension into chains. 48 identical polarisable spheres, two radii apart at least, scattered at random through a periodic cell twenty radii square — 38 per cent of its area — and the same spheres after the field (vertical) has been on long enough for the viscous liquid to let them move. Each moves in proportion to the dipolar force from all the others and is stopped only by contact. At the start no two are touching; at the end 45 are bonded end to end along the field and 48 of the 48 spheres belong to chains, lying along the field lines and kept apart sideways because side-by-side dipoles repel. Chains that reach from one electrode to the other are what make the liquid resist shear, and they form in the milliseconds it takes a sphere to travel its own diameter.

    The liquid a field turns solid

    Stir fine starch or glass beads into silicone oil and the result pours like cream. Put a few kilovolts across it and within milliseconds it holds its shape and will not flow until it is pushed hard. Nothing in it is charged. Each particle becomes a dipole in the field, dipoles pull on each other end to end and push apart side by side, and so the particles string themselves into chains from one electrode to the other — a solid made of forces that exist only while the voltage is on.

    part 6 · electromagnetism
  7. The voltage a gas gap can hold, on one curve. The breakdown voltage of a gap in air between flat electrodes against the product of pressure and gap width, on logarithmic scales — Paschen's law, in which pressure and gap enter only as their product because both count how many collisions an electron makes on the way across. Townsend's model, with γ = 0.01 electrons freed per ion at the cathode, has its minimum of 305 V at 1.12 Pa·m — a gap of 11 μm at atmospheric pressure, or 112 Pa across a centimetre. Measured minima in air are about 330 V. To the left the gap is too short, or the gas too thin, for an electron to make enough collisions, and the voltage climbs steeply; below about 0.41 Pa·m there is no gas breakdown at any voltage. To the right, collisions are so frequent that an electron gains too little energy between them, and the voltage rises nearly in proportion to pd. Dashed, the empirical law for air at larger pd: 31.4 kV for a centimetre at one atmosphere, against the model's 35.5.

    The spark that needs room to start

    Air is an excellent insulator until it is not, and the usual statement of when — three megavolts a metre — is true only for gaps of about a centimetre. A gap breaks down when one electron's avalanche of ionisation sends back enough ions to the cathode to free one more electron, and that condition depends on the pressure and the gap only through their product. The result is a curve with a floor: about 330 volts in air, below which no gap of any width will spark, and above which both a thinner gas and a denser one insulate better.

    part 7 · electromagnetism
  8. The field round a thin wire. The electric field against distance from the axis, on a logarithmic axis, for a wire inside a coaxial cylinder of radius 10 cm at 60 kV, for wire radii of 0.2, 1 and 5 mm; the dashed line is 30 kV/cm, the field at which air breaks down across a gap of a few centimetres, and each curve starts at its wire's surface. The field falls as 1/ρ, so it is concentrated at a thin wire: 483 kV/cm at the surface of the 0.2 mm wire, 130 at the 1 mm wire and 40.1 at the 5 mm one. The field exceeds 30 kV/cm only within 3.2 mm of the axis for the thinnest wire, 4.3 mm for the middle one and 6.7 mm for the thickest; across the remaining nine centimetres the air is far below breakdown, so ionisation can start at the wire and has nowhere to go.

    The glow a thin wire wears before it sparks

    Raise the voltage on a thin wire inside a metal tube and, long before anything sparks across, the wire starts to glow faintly violet and to hiss. The air right next to it is ionising, and the air a millimetre further out is not. The field round a wire falls away so fast that an avalanche of ionisation can start at the surface and find nowhere to go. That partial breakdown, corona, costs power lines energy in the rain, sets the size of the wires they are made from, cleans the smoke from chimneys and counts particles in detectors — and the field it needs is not a property of air but of the wire's radius.

    part 8 · electromagnetism

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