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Linear charge density is electric charge divided by length along a charged line, or locally $dQ/d\ell$. Its SI unit is coulombs per metre. For a uniformly charged long cylindrical conductor, it equals the total surface charge density integrated around the cylinder's circumference; it need not represent a charge literally supported on a one-dimensional line.

A coefficient describing the locally absorbed fraction of an incident photon energy flow per path length, under a stated averaging convention. It is distinct from the total linear attenuation coefficient, which counts removal from an uncollided beam. Dividing by mass density gives an energy-absorption coefficient per mass.

Linear mass density is mass per unit length, measured in $\mathrm{kg}{\mathrm{m}}^{-1}$. A uniform string of mass $m$ and length $L$ has $\mu =m/L$; with constant tension $T$, its transverse wave speed is $\sqrt{T/\mu }$. Mass density instead measures mass per unit volume.

For $\tau =t_{\ast }-t$, balance $\mu a_t/a\sim \gamma /a$ to obtain radius proportional to $\tau$. Axial lubrication conservation then makes the axial length proportional to $\tau$ as well. The displayed variables reduce the lubrication equation for a bubble with viscous exterior to $\zeta A{A}^{\prime }-A^2=\frac{1}{16}\{A^4[(2\zeta A^{\prime }+1)/A{]}^{\prime }{\}}^{\prime }$. The physical interface slope is $\sqrt{\lambda }{A}^{\prime }$, compatible with a slender neck when $\lambda \ll 1$ and the dimensionless profile varies on order-one scales.

The linear thermal expansion coefficient is the fractional change in a specified length per change in temperature, at fixed external pressure for a sample free to expand. For small changes, $\Delta L/L\simeq {\alpha }_L\Delta T$. An isotropic sample with the same response in all directions has volumetric coefficient of thermal expansion $\alpha =3{\alpha }_L$, since differentiating $V\propto L^3$ gives $dV/V=3dL/L$. Direction-dependent materials require a specified measurement direction. ^[ref].

The line mass is mass per unit length along a filament. For an axisymmetric density profile $\rho (R)$,with the upper limit replaced by the filament radius when the density has compact support.

In a planar incompressible flow, a line source injects volume flux $Q$ per unit length perpendicular to the plane. Its radial velocity is $u_r=Q/(2\pi r)$, since integrating the outward velocity around a circle gives $2\pi r{u}_r=Q$. Its complex potential of a two-dimensional potential flow is $w=Q\log (z-b)/(2\pi )$ and its complex velocity is $w^{\prime }=Q/[2\pi (z-b)]$. Here $Q>0$ means outward flow; a line sink has negative $Q$. Away from $b$, this is an irrotational flow as well as an incompressible flow.

Line tension is energy per unit length of a boundary, such as the edge between two phases of a lipid bilayer. Positive line tension tends to minimize the perimeter at fixed domain area. Its dimensions and geometry differ from surface tension, which is energy per unit area.

When Majorana zero modes have disjoint exponentially localized supports, every local parity-even observable acts as a scalar on the fixed-parity ground space. Finite overlap changes this by terms exponentially small in separation divided by localization length.

For the density jump of a magnetic pressure discontinuity in an isothermal atmosphere, with $0<|B_{+}|<|B_{-}|$, the fastest Rayleigh-Taylor instability has $m=0$ and $|k|=K_{\ast }=(B_{-}^2-B_{+}^2)/(4B_{+}^{2}H)$. Its vertical decay length is $K_{\ast }^{-1}$. Hence localization relative to both common scale heights requires $4B_{+}^2\ll B_{-}^2-B_{+}^2$. This also makes its growth speed small compared with the isothermal sound speed. If $B_{+}=0$, the ideal sharp-interface dispersion relation instead has unbounded growth as $|k|\to \infty$: there is no finite maximizing mode.

The localization length $\xi$ measures the decay scale of a localized eigenstate's wave amplitude envelope. Here $a$ is the site spacing and $n_0$ its centre. With this amplitude convention, probability decays on half that scale:Individual random tails fluctuate; this notation describes their typical envelope, not exact equality at every site. A localization length much greater than a finite sample's length makes a state look extended inside that sample. The participation ratio below is an effective site count, not an independent fitted value of $\xi$.

When the mass is constant $V_{+}>0$ above the wall and $V_{-}<0$ below it, amplitude decay lengths are ${\xi }_{+}=\hslash c/V_{+}$ and ${\xi }_{-}=\hslash c/|V_{-}|$. A scalar transverse envelope multiplying a unit spinor has normalization $\sqrt{2/({\xi }_{+}+{\xi }_{-})}$. Density decays twice as fast as amplitude.

The electric field does work on matter at local power per volumeHere $\mathbf{J}$ is current density and the dot product includes the relative direction. Positive $\mathbf{J}\cdot \mathbf{E}$ means energy passes from the field to matter; negative means matter supplies energy to the field. In general this is a work rate, not automatically an irreversible heating rate.^[ref]

For a stationary isotropic Ohmic conductor with real scalar electrical conductivity $\sigma >0$, local Ohm's law gives $\mathbf{J}=\sigma \mathbf{E}$. Thus the local Joule heating rate isThe electrical resistivity ${\rho }_{\mathrm{el}}$ has units $\Omega \mathrm{m}$, and electrical conductivity has units $(\Omega \mathrm{m}{)}^{-1}$. These formulas use the material's rest frame and dissipative conduction current density. Moving media, complex conductivity, and additional storage or transport processes require their appropriate energy balance rather than this scalar substitution.

For uniform current in a straight homogeneous wire of length $L$, wire cross-sectional area $A$ and constant electrical conductivity, $E=V/L$, $J=I/A$ and $R=L/(\sigma A)$. The volume integral recovers terminal power:As a numerical check, the wire already used in electrical resistance has $L=1\mathrm{m}$, $A=10^{-6}{\mathrm{m}}^2$, $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$ and $V=5\mathrm{V}$. It gives $J=2.5\times 10^{6}A/{\mathrm{m}}^2$, $p_{\mathrm{vol}}=1.25\times 10^{7}W/{\mathrm{m}}^3$ and $P=12.5\mathrm{W}$, agreeing with $I=2.5\mathrm{A}$ and $R=2\Omega$. Concentrating the same electric current in a smaller wire cross-sectional area therefore raises the local heating rate if electrical resistivity stays fixed.

With constant amplitude outside a core, the planar nematic divergence-elasticity identity reduces the elastic density to $K{\lambda }_0^2|\nabla \chi {|}^2/2$. A topological charge of a two-dimensional nematic disclination $q$ has $\chi =q\theta +{\chi }_0$ and $|\nabla \chi {|}^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\mathrm{pair}}=2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ plus core terms, since its far-field gradients cancel.

For scalar kinetic term $|D\phi {|}^2$ and condensate magnitude $|\phi |=v$ without a factor $1/\sqrt{2}$, the canonical vector mass is $m_A^2=2e^2{v}^2$. In a constant-phase gauge the current is $\mathbf{j}=-2e^2{v}^2\mathbf{A}$. A static magnetic field obeys ${\nabla }^2\mathbf{B}=m_A^2\mathbf{B}$, so its penetration length is $1/(\sqrt{2}|e|v)$. This normalization differs from defining the vacuum amplitude as $v/\sqrt{2}$.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.

For constant electron density, common path length $L$, and a solenoidal isotropic spectral tensor, the rotation-measure correlation contains the window $|{\int }_0^L{e}^{i{k}_{z}z}dz{|}^2$. As $L$ becomes large compared with the correlation length, this window acts as $2\pi L\delta (k_z)$ within the integral. The transverse projector then has $zz$ component one, yielding the displayed two-dimensional Fourier transform. For finite $L$, the windowed integral retains longitudinal modes and endpoint corrections. The method of using such rotation measure correlations to infer magnetic spectra is developed by Enßlin and Vogt.

In magnetostatics, an ideal infinitely long solenoid is a cylindrical sheet of circumferential surface current density $K=nI$. It has negligible winding advance, no magnetic core, and no imposed external magnetic field. Choose positive electric current to produce magnetic field along $+z$. In this ideal model the interior magnetic field is uniform and axial, $B={\mu }_{0}nI$, and the exterior magnetic field is zero; ${\mu }_0$ is the vacuum permeability. For a finite, closely wound air-core coil, the same result is an approximation well inside the coil, several radii from either end. Air is treated as having the magnetic permeability of vacuum. ^[ref].

For a short proof, the translational and rotational symmetries of the ideal cylindrical current sheet, together with the Biot-Savart law, give an axial magnetic field. Ampère's law on rectangles wholly outside makes that axial exterior value independent of radius; the condition of no imposed magnetic field makes it zero at infinity and hence everywhere outside. Rectangles wholly inside similarly show that the interior value is independent of radius. Now choose an Amperian loop with axial length $s$, one long side inside and the other outside. The transverse sides are perpendicular to the magnetic field, so their contributions to the line integral vanish. The loop crosses $ns$ turns of the winding, givingHere $s$ is the integration-path length, not the whole coil length $\ell$. An Amperian loop surrounding both sides of the winding encloses equal and opposite electric currents; zero net circulation alone would not prove zero magnetic field at every exterior point.

For the $1000$-turn, $0.50\mathrm{m}$ coil with electric current $I=0.50\mathrm{A}$, the long-solenoid approximation givesThis illustrative calculation uses vacuum permeability ${\mu }_0\approx 1.256637\times 10^{-6}H/m$, a rounded value, not an exact constant. Since the 2019 redefinition of SI, vacuum permeability is determined experimentally.^[ref] Doubling $I$ doubles the ideal magnetic field; doubling $N$ at fixed $\ell$ does the same. Increasing $\ell$ while keeping $N$ fixed decreases $n$ and hence the ideal magnetic field.

A long-wave approximation assumes that a layer thickness $H$ is much smaller than its streamwise length $L$. The small aspect ratio $H/L$ makes transverse gradients dominate longitudinal gradients and is the geometric basis of lubrication theory.

For gap scale $H$, length $L$, lower-wall tangential speed $U$, upper-wall normal speed $V$, and induced horizontal scale $W=|U|+VL/H$, the lubrication limit requiresThe last condition includes both $\rho |U|H^2/(\mu L)\ll 1$ and $\rho VH/\mu \ll 1$.