Illustrated Physics
physics, drawn
Concept

Resistivity — where it appears

A material's opposition to electric current per unit length and cross-section, the reciprocal of conductivity. In plasmas it is tiny but decisive: it sets how fast field lines can slip through the fluid and where reconnection and magnetic energy dissipation can occur.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter.

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

astrophysics · Flux freezing The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

astrophysics · Flux freezing Three shapes of film, one sheet resistance. Three thin films of the same material, of sheet resistance one ohm per square, with four small contacts on each edge labelled A to D in order. For each, the two readings van der Pauw's method takes — the voltage between D and C per unit current from A to B, and between A and D per unit current from B to C — computed by solving for the current flow on a fine grid. For a square, contacts at the corners: 0.221 and 0.221 Ω, giving 1.000 Ω per square; a disc, contacts unevenly spaced: 0.089 and 0.450 Ω, giving 0.999 Ω per square; an irregular lamina: 0.209 and 0.232 Ω, giving 0.999 Ω per square. The readings change with the shape and with where the contacts are; the sheet resistance recovered from them by exp(−πR₁/Rₛ) + exp(−πR₂/Rₛ) = 1 does not.

The sheet resistance that forgets the shape

To measure how well a thin film conducts, the obvious method is to cut it into a neat strip and pass a current along it. In 1958 Leo van der Pauw, an engineer at Philips, showed that the strip is unnecessary. Put four small contacts anywhere on the edge of a film of any shape, take two readings, and one equation returns the film's resistance per square — the shape and the spacing of the contacts drop out entirely. The proof is a single fact about logarithms and the maps that preserve angles, and it fails for exactly one kind of film: one with a hole in it.

electromagnetism · Conductors

Named alongside it

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

DissipationFlux freezingMagnetic energyMagnetic reconnectionPlasmaConductivityConformal mapContact resistanceCurrent sheetEquilibriumForce-free fieldFour terminal measurement

All concepts