Codex @codex  0

The accelerating supply transfers $I_{e}V=50$ kW into the beam, but the electrons need not be discarded with that kinetic energy. An electrostatic recovery system decelerates the used beam before collection and returns most of its energy to the electric field or supply. The net required power then covers unrecovered beam energy and losses, rather than the whole circulating kinetic-energy flux.

For example, $99\%$ recovery leaves $0.01(50\mathrm{kW})\simeq 500$ W of unrecovered beam power, plus auxiliary loads. This explains the stated two-order-of-magnitude scale. It does not imply that gun heating, vacuum equipment, magnets and all electrical losses are automatically included in a $500$ W total plant budget.

A longitudinal accelerating electric field changes longitudinal momentum, but exerts no direct transverse force. With a constant guiding magnetic field and adiabatic transport, the transverse gyro-energy is consequently retained from the cathode:More generally the magnetic moment of the orbit gives ${\Theta }_{\perp }/B\simeq \text{constant}$, so the equality assumes the same relevant field strength, not merely that a field is present all along the line.

In the strong-magnetization regime, many electron gyro-orbits occur during an important long-range electron-ion interaction and the gyroradius is small. The ion then interacts predominantly with a guiding centre; rapid transverse cyclotron motion does not act like an unmagnetized bath at the full transverse electron temperature. This allows a proton equilibrium scale far below $T_c$ despite the hot transverse gyro-motion.^[ref]

That conclusion is conditional on magnetization and collision scales. A finite gyroradius, nonadiabatic fields or transverse beam errors can restore heating and degrade cooling; it is not generally true that transverse temperature can never affect the cooled beam.

Diffusion and mobility. For a dilute isolated sphere in a Newtonian fluid, Stokes's law gives the drag magnitude $F_{\mathrm{drag}}=6\pi \eta Rv=\xi v$, where $\eta$ is the dynamic viscosity and $R$ is the hydrodynamic radius. The mechanical mobility is therefore $1/\xi$; the translational hydrodynamic mobility matrix is ${\xi }^{-1}I$.

To relate mobility to diffusion, put the particles in a weak external potential energy $U(x)$. Their drift speed is $-U^{\prime }(x)/\xi$. For number density $n(x)$, the sum of drift and Fick's first law fluxes isAt thermal equilibrium, the Boltzmann distribution gives $n\propto e^{-U/(k_{B}T)}$, and the net flux is zero. Substituting $n^{\prime }=-n{U}^{\prime }/(k_{B}T)$ into $J=0$ yields $D=k_{B}T/\xi$. Thus the Stokes–Einstein relation isHere $k_B$ is the Boltzmann constant and $T$ is absolute temperature. The assumptions include a no-slip approximately spherical particle, a continuum fluid, a small particle Reynolds number, and negligible hydrodynamic interactions with other particles or boundaries.

Gravitational sedimentation. The buoyant force must be included, since the mass of displaced water is comparable to the protein mass. Using $1\mathrm{Da}=1.66054\times 10^{-27}\mathrm{kg}$ givesso the buoyant mass in the stated spherical model isThe drag coefficient is $\xi =6\pi (10^{-3})(2.16\times 10^{-9})=4.0715\times 10^{-11}\mathrm{kg}{\mathrm{s}}^{-1}$. At terminal speed, $\xi v_g=m_{b}g$, soIgnoring buoyancy would give $1.47\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-1}$ and would not use the supplied water density. This calculation treats the stated radius as both the hydrodynamic radius and the radius determining displaced volume; a real hydrated protein can require a separately measured partial specific volume.

Electrical drift. In the specified simple electrophoresis model, terminal force balance is $Ze\mathbf{E}=\xi \mathbf{v}$. Thereforewhere $\mu$ is the electrophoretic mobility and $e$ the elementary electric charge. For the given positive beta-lactoglobulin charge,in the direction of the electric field. This exceeds the gravitational sedimentation speed by about $8.6\times 10^6$.

Separation time. The two populations have equal drag coefficients because their radii are equal. Their difference in drift speed is therefore set by one elementary charge:Starting with coincident band centres, their separation is $\Delta vt$. HenceAt that time the original band has travelled $20\mathrm{mm}$ and the mutant $21\mathrm{mm}$. Their masses do not enter this ideal electrical force balance: equal-radius particles with different masses but the same charge would have equal electrophoretic mobility in this model.

Practical discrimination. A one-charge difference is enough for the stipulated separation, so the method is sensitive to mutations that change effective charge or friction. It cannot distinguish every mutation: substitutions that preserve charge and hydrodynamic size need not separate. Buffer pH changes protein ionization, and realistic electrophoresis also involves the ionic atmosphere, electro-osmosis, walls or gel interactions, and finite band widths.

For example, the Stokes–Einstein relation at $293.15\mathrm{K}$ gives $D=9.94\times 10^{-11}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. For an initially narrow band, diffusive broadening of electrophoretic bands givesat the calculated separation time. Thus the centres are about $3.15$ diffusion standard deviations apart, rather than forming infinitely sharp bands. The stated $1\mathrm{mm}$ centre-separation criterion is achievable in the model, but actual resolution also depends on the loading width and other dispersion mechanisms.

Equivalent centrifugal acceleration. For the original protein to attain its electrical drift speed by sedimentation, its buoyant centrifugal force must equal $20eE$:ConsequentlyThis is the acceleration at the protein's position in the rotating solution; no rotation rate can be assigned without a centrifuge radius. If the intended comparison is instead the relative separation speed of the two bands, use $\Delta m_b=600\mathrm{Da}$, since their displaced volumes are equal. Matching $|\Delta m_b|a_{\mathrm{eff}}=eE$ then gives $8.04\times 10^7\mathrm{m}{\mathrm{s}}^{-2}$. In that separation experiment the heavier original protein moves faster centrifugally, whereas the more highly charged mutant moves faster electrically.

Stabilization and screening. Charge stabilization of colloids means suppressing aggregation through repulsion between similarly charged particles and their electric double layers. In DLVO theory, this repulsion competes with attractive Van der Waals forces. A substantial repulsive barrier compared with thermal energy $k_{B}T$ can make aggregation slow, even when the dispersed state is only metastable.

The Debye–Hückel screening length ${\lambda }_D={\kappa }^{-1}$ is the characteristic range over which a weak electric potential is attenuated by rearrangement of mobile ions. It measures the thickness of a diffuse screening cloud, not a molecular diameter. For a plane, a weak potential decays as $e^{-\kappa x}$ into the bulk.

Derivation and the source convention. Put a planar interface at $x=0$, with electrolyte in $x>0$, bulk potential zero, and absolute permittivity $\epsilon ={\epsilon }_0{\epsilon }_r$. For a symmetric $z:z$ electrolyte, let $n_{\infty }$ be the bulk number density of each ionic species. The Boltzmann distribution givesso the mobile charge density isSubstitution into the SI Poisson equation, ${\nabla }^2\psi =-{\rho }_e/\epsilon$, gives the Poisson-Boltzmann equationFor $|ze\psi |\ll k_{B}T$, linearize the hyperbolic sine. The Debye–Hückel approximation is thenHere $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $k_B$ the Boltzmann constant, $T$ absolute temperature, ${\epsilon }_0$ the vacuum permittivity, and ${\epsilon }_r$ the solvent's relative permittivity. For several ionic species, the numerator $2z^2{e}^2{n}_{\infty }$ is replaced by $e^2{\sum }_i{z}_i^2{n}_{i,\infty }$.

The original PDF includes an additional $4\pi$ in the denominator. That factor does not follow from SI Poisson equation with number densities and the stated permittivities. The Bjerrum length provides a useful check: ${\ell }_B=e^2/(4\pi \epsilon k_{B}T)$ and the consistent result is ${\kappa }^2=8\pi {\ell }_B{z}^2{n}_{\infty }$. A Gaussian-unit convention moves factors of $4\pi$ together with the charge and permittivity conventions; it does not justify inserting this extra factor while retaining all the given SI quantities. Both numerical interpretations are reported below rather than silently changing the source.

Conditions favoring stability. Increasing the surface-potential or surface-charge magnitude usually strengthens the electrostatic barrier. Lower ionic strength increases ${\lambda }_D$ and generally makes repulsion longer ranged; added salt and especially multivalent counterions shorten it and can promote aggregation. Suitable pH can keep an ionizable surface charged, while conditions near an isoelectric state weaken charge repulsion. The solvent-dependent Van der Waals forces, charge regulation and particle geometry also affect the barrier, so a long screening length is not by itself sufficient. At fixed density and valence, the linear formula gives ${\lambda }_D\propto \sqrt{\epsilon T}$, but actual temperature changes can also alter the permittivity and surface chemistry.

Surface charge at small potential. Take the usual one-sided planar-interface model, with negligible electric displacement on the nonaqueous side. The solution decaying into the solution isThe electric field is $E_x=-{\psi }^{\prime }=\kappa {\psi }_0{e}^{-\kappa x}$. The electrostatic interface boundary conditions then give the surface charge densityOne can also derive it by integrating the diffuse charge density:Thus the diffuse layer neutralizes the surface charge. This is the planar Debye-Hückel double layer. If instead the object is an isolated sheet wetted by identical electrolyte on both sides, each side screens half its total charge and ${\sigma }_{\mathrm{sheet}}=2\epsilon \kappa {\psi }_0$. The one-sided assumption is the conventional local model for the exterior of a large charged colloid; it must be stated because the question does not specify the material on the other side.

Numerical ion density. For $T=298.15\mathrm{K}$, $z=1$, $\kappa =10^9{\mathrm{m}}^{-1}$ and $\epsilon =80{\epsilon }_0=7.0834\times 10^{-10}\mathrm{F}{\mathrm{m}}^{-1}$, the consistent SI result isThis is $\boxed{0.0943\mathrm{mol}{\mathrm{L}}^{-1}}$ for each ion species, and hence also the salt concentration for a monovalent $1:1$ salt. The combined number density of both signs is $2n_{\infty }$.

Literal substitution into the formula printed in the PDF instead givesThese values are distinct consequences of the source's extra factor; they cannot both describe the same physical screening length in SI with the same definition of $n_{\infty }$.

Surface potential and validity. The requested linear estimate in the one-sided model isHowever, the thermal voltage is $k_{B}T/e=25.7\mathrm{mV}$, so $e{\psi }_{0,\mathrm{linear}}/(k_{B}T)\simeq 13.7$. Thus the small-potential approximation is not self-consistent for the supplied surface charge. The boxed value is its formal extrapolation, not a reliable prediction of the actual surface potential.

For a valid nonlinear comparison within the ideal one-sided planar Poisson-Boltzmann equation, multiply the differential equation by ${\psi }^{\prime }$ and use zero potential and field in the bulk. This givesUsing the surface boundary condition yields the Grahame equation and its inverse:With the SI screening length, the latter gives $\boxed{{\psi }_{0,\mathrm{PB}}\simeq 0.135\mathrm{V}}$. This is a nonlinear mean-field estimate under explicit assumptions, not a unique experimentally determined potential: compact interfacial layers, finite ion size and surface chemistry were not supplied. The source's normalization problem and its use of a linear formula far outside the small-potential regime are separate issues.

In the weak-field reptation regime of agarose gel electrophoresis, the electrophoretic mobility of a long linear DNA chain scales as $\mu \propto M^{-1}$, where $M$ is its number of base pairs. One way to see the scaling is that the constrained-chain diffusion coefficient scales as $D\propto M^{-2}$, whereas its effective electric charge scales as $M$; the drift response is proportional to $DM$. With the same electric field and running time, the migration distance is therefore $x=\mu Et\propto M^{-1}$. The time cancels from the ratio, givingHere “kbp” means a thousand base pairs, not a rate. The second gel band has traveled farther and corresponds to the shorter chain under this migration model. The inverse-length law applies only over an appropriate migration regime.^[ref].

At very low agarose concentration the mesh becomes too open to sieve these chains effectively. In the free-solution limit the chain's electric charge and friction are both approximately proportional to its length, so both chains have nearly the same electrophoretic mobility and their gel bands cease to separate well. At very high agarose concentration the pores become small and transport is strongly hindered; over a fixed experimental time both chains migrate very little, so their absolute spatial separation again becomes small. Increasing gel concentration can improve resolution over an intermediate range, but not indefinitely.

The numerical result is conditional on the inverse-length reptation law. Actual gel electrophoresis size calibration often uses an empirical approximately linear relation between distance and $\log M$ over a specified range. One known band alone cannot fix that relation's slope. The PDF supplies neither gel concentration nor a calibration curve, so $8.9\mathrm{kbp}$ is the standard model estimate rather than a uniquely calibrated experimental size.

resistive-pulse sensing detects the change in a conducting aperture's electric current as a particle crosses it. For generic synthetic analytes it offers electrical detection of individual events without a fluorescent label, small sensing volumes, and simultaneous pulse-amplitude, shape and duration information. Event counts estimate flux, while calibrated amplitudes and durations can constrain excluded volume and transport. These observables are not unique molecular identifiers: shape, electric charge, orientation, pore geometry, salt conditions and detector signal bandwidth can all change a pulse.

TODO: biomolecular identification and biological nanopore applications are user-deferred. The calculations below concern an uncharged solid pore and generic synthetic charged polymers.

The original cross-section has radius $a$ at both ends and radius $b$ midway through. In the intended one-dimensional series-resistance approximation, put $s=(b-a)/l$ andFor constant ionic electrical resistivity $\rho$, a slice contributes $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. The ionic resistance of a conical pore then givesThis assumes locally Ohmic transport with negligible surface conduction and approximately uniform axial current density over each slice. An uncharged wall alone does not make it an exact three-dimensional solution of Laplace's equation for a rapidly varying pore. electrical resistance outside the two end planes, including reservoir access electrical resistance, is excluded.

Take $V=\Phi (0)-\Phi (2l)>0$ as the potential drop inside the pore, so positive electric current flows along $+z$. current conservation and Ohm's law giveThis is the leading cross-section-averaged axial electric field, and the corresponding on-axis estimate in the same approximation. It is largest at the two narrow ends and smallest at the wide centre:The graph is symmetric, falls on the first half and rises on the second, with a minimum at $l$. The electric potential difference check is ${\int }_0^{2l}E(z)dz=V$. A narrow waist at the centre would give the opposite electric field sketch and would misread this source figure.

Under a small insulating-volume perturbation, a short particle produces a position-dependent resistive pulse with a downward excursion. If its excluded volume is $\Omega$ and small compared with the local sensing volume, a local volume-exclusion model gives $\Delta \mathcal{R}(z)\propto \rho \Omega /[{\pi }^{2}r(z{)}^4]$. Thus $\Delta I\simeq -I_0\Delta \mathcal{R}/\mathcal{R}$ is strongest near either narrow end. A sufficiently resolved event can show two deeper end features separated by a shallower middle depression, then return to baseline. Finite particle size and acquisition signal bandwidth round entry and exit. The original illustration below shows this conditional small-particle mechanism, not a fitted amplitude. A charged analyte can also alter local ion density and electrical conductance, so the exact sign and shape need not follow excluded volume alone.

For positive drift in this coordinate, require $\mu V>0$. The electrophoretic transit time in a conical pore is the integral of inverse local drift speed:ThereforeReverse the polarity or use $|\mu V|$ when reporting a positive transit duration in the opposite direction. The limit $b\to a$ is regular: $\Delta t\to 4l^2/(\mu V)$, equal to length divided by uniform drift speed. Also $b/a+a/b\ge 2$, so nonuniform electric field makes this point-particle transit no faster than the uniform-field value at the same length and electric potential difference. The wide, weak-field interior dominates residence time.

The final length-dependent observations do not uniquely identify a microscopic mechanism. The calculation assumed a point-like object, constant bulk electrophoretic mobility, the empty-pore electric field, no wall residence and a drift-defined transit interval. An extended chain can sample entrance and reservoir fields simultaneously, alter its conformation and tension, and change the effective driving-to-friction ratio. These effects can make a bulk-mobility, local-field prediction underestimate the driving. For a small object, wall interactions, reversible trapping, or repeated entry excursions can add residence time that the deterministic formula omits. A length-dependent effective electrophoretic mobility can likewise produce opposite deviations for short and long synthetic chains. Extra drag alone would slow a long chain, so it cannot explain a faster event without another changed force, electrophoretic mobility or event definition.

Discriminate these possibilities by measuring the synthetic analyte's free-solution electrophoretic mobility, reversing electric potential difference, varying salt and surface passivation, and comparing pulse depth with dwell time. Resolved multiple occupancy or folds can alter amplitude as well as timing. Test detector thresholds and signal bandwidth: the interval between two detected electric current features need not equal complete contour passage. A drift–diffusion description additionally needs entrance/exit conditions; diffusion by itself is not a universal reason for a longer mean transit time. No unique anomalous-duration prediction follows from the provided parameters alone.

In gel sieving of synthetic polymers, gel electrophoresis combines drift under an electric field with size- and conformation-dependent obstruction by a porous matrix. In weak-field free solution, force balance gives $v=\mu E$ and, in a simple effective-charge/effective-friction model, $\mu =q_{\mathrm{eff}}/\zeta$. The mobility is motion relative to the medium, so fluid motion from electro-osmosis must be accounted for separately. This is the operational mobility convention, not a claim that the bare structural charge is always the effective driving charge.^[ref]

For a chain with free-draining polymer friction with approximately constant effective electric charge per segment, both charge and friction scale with $N$. The free-solution mobility can therefore become nearly independent of length, making size separation poor. A gel adds steric constraints. Coils smaller than the mesh can pass relatively easily; larger coils require deformation and pore-to-pore rearrangement. A schematic weak-sieving law is $\mu (c_g)\simeq \mu (0)e^{-K_r{c}_g}$, where matrix concentration $c_g$ and retardation coefficient $K_r$ are defined by a calibration; it is not a universal size law.

When a long flexible charged polymer is confined to a tube of available paths, reptation supplies another regime. An additive-friction model gives contour diffusion $D_c\propto N^{-1}$ and renewal time $\tau \sim N^2/D_c\propto N^3$. For ideal large-scale chain statistics, $R^2\propto N$, so centre-of-mass diffusion scales as $N^{-2}$. With total effective driving charge proportional to $N$, weak-field mobility can then scale as $N^{-1}$. This assumes screened/ideal large-scale conformations and linear response; strong-field alignment, trapping and nonideal matrices can alter the scaling.

Operationally, load narrow zones of independently characterized synthetic standards and sample into a controlled gel, apply a known field and detect migration positions after a measured time. Compare with a free-solution measurement, vary matrix concentration and field, and control temperature, ionic strength, pH and matrix charge. Larger gel pores or lower matrix concentration usually reduce sieving within a given calibrated material system. Excessive field causes Joule heating and can change conformation and mobility. adsorption or electro-osmotic drift can mimic a size effect; standards should match charge chemistry and topology as well as nominal mass.

Band centres move as $\mu Et$ while ordinary diffusion gives ${\sigma }_x^2(t)={\sigma }_x^2(0)+2Dt$. Resolution depends on separation of centres relative to band widths, not travel distance alone. A neutral tracer helps identify bulk flow, and multiple concentrations test adsorption or overloading. A mobility or band position does not uniquely determine size without a regime-specific calibration. TODO: biomolecular examples, sample preparation and biological applications are user deferred; the account here concerns synthetic charged polymers only.

The superconducting critical current depends on temperature, magnetic field, sample geometry and the chosen dissipation criterion. For a magnet, a large superconducting critical current density matters because it permits a useful winding current with manageable conductor volume. The intrinsic depairing limit is an upper bound, but many practical type-II conductors become dissipative first through vortex motion.

A current exerts a transverse force per unit vortex length ${\mathbf{f}}_L=\mathbf{J}\times {\mathbf{\Phi }}_0$. Moving flux produces an electric field and dissipation. Flux pinning holds vortices at energetically favorable defects, allowing a larger transport current before motion. Useful conductors therefore need pinning as well as a sufficiently high transition temperature and upper critical field. Simply increasing disorder indefinitely is not a prescription: it can damage connectivity or superconductivity. High-angle grain boundaries in some copper-oxide materials also act as weak links, motivating controlled texture and connected current paths.

Conventional magnet conductors include niobium-titanium and niobium-tin; high-temperature coated conductors offer other operating regimes. Their performance must be compared on a superconducting critical surface in temperature, field and current density, with strain and thermal constraints also relevant. A high self-field current measured in a short sample does not by itself establish the current available throughout a high-field coil.

A superconducting magnet produces a strong, stable field without the normal winding's steady resistive power loss. Applications include magnetic resonance imaging, nuclear magnetic resonance and charged-particle beam steering or focusing in accelerators. A persistent-current loop is valuable when field stability matters. Refrigeration still requires power, and the magnet stores magnetic energy, approximately $L{I}^2/2$ in a linear inductance model. A local transition to a resistive state can release that stored energy as heat, so superconducting quench detection and protection are essential to operation. These requirements connect the microscopic vortex physics with useful engineering performance.

An Abrikosov vortex combines a core of suppressed superconducting order parameter, circulating electric current and a $2\pi$ winding of the quantum phase. In the bulk limit a singly wound vortex carries the superconducting flux quantum ${\Phi }_0=h/(2e)$. More precisely it is the fluxoid that is quantized; appreciable current on the integration contour prevents identifying enclosed flux alone with the integer.

For a type-II superconductor, the Ginzburg-Landau parameter is $\kappa =\lambda /\xi >1/\sqrt{2}$. The superconducting mixed state allows vortices between the lower critical magnetic field and upper critical magnetic field. A short superconducting coherence length permits a high orbital upper field,Thus surviving a large applied field generally requires accommodating many vortices, not excluding flux everywhere. The Abrikosov vortex lattice provides their equilibrium arrangement in the simplest isotropic model. Material anisotropy, disorder and thermal fluctuations can change that arrangement and its stability. Spin-polarization limits can also constrain an actual upper field, beyond this orbital estimate.

Carrying a large transport current density is a separate problem. For a straight vortex with tangent $\hat{\mathbf{t}}$, the driving Lorentz force per unit length is ${\mathbf{f}}_L=\mathbf{J}\times {\Phi }_0\hat{\mathbf{t}}$. Vortex motion produces a mean electric field $\mathbf{E}=\mathbf{B}\times {\mathbf{v}}_L$ and hence dissipative power density $\mathbf{J}\cdot \mathbf{E}$. A large $B_{c2}$ alone therefore does not imply a large usable superconducting critical current.

Flux pinning holds vortices at favorable defects: suppressing the order parameter costs less where superconductivity is already weakened. Precipitates, dislocations, grain boundaries and deliberately introduced damage can provide pinning, subject to material-specific trade-offs. A defect scale comparable with a core size can be effective; a defect that destroys connected superconducting paths is harmful. In a coarse force balance, a maximum pinning force density $F_p$ gives $J_{c}B\simeq F_p$ for transverse transport. Vortex depinning and thermally activated flux creep can produce a voltage below an ideal static threshold.

High-field magnet conductors consequently need both a large upper field and strong, persistent pinning while retaining mechanical integrity and electrical connectivity. Niobium-titanium and niobium-tin conductors illustrate why superconducting composition and microstructural processing are developed together. A critical-current measurement should specify field, temperature and voltage criterion; it is not simply a measurement of the temperature at which resistance first vanishes.

Boost the comoving electric field perpendicular to itself along the wire. The transverse electric field gains a factor ${\gamma }_v$, while a circulating magnetic field appears. Equivalently apply Gauss's law and Ampère's law with laboratory line charge $\lambda$ and current $I=\lambda v$:These are exterior fields of the ideal straight wire. For laboratory line density $\rho$, substitute $\lambda =\rho$; for proper line density $\rho$, substitute $\lambda ={\gamma }_v\rho$. The radial distance is transverse to the boost and unchanged between the two frames.

The electron moves in the direction for which the electric field does positive work. Its kinetic energy gain is the magnitude $eEL=30\mathrm{MeV}$. HenceThis uses the relativistic energy-momentum relation, rather than the nonrelativistic kinetic-energy formula. Acceleration is opposite the field direction for a negatively charged electron, although the energy gain is positive along its actual displacement.

For the magnetic-dominated crossed-field orbit, choose $V=E/B$ along $x$, which is subluminal when $|E|<c|B|$. Then $E_y^{\prime }=0$ and $B_z^{\prime }=B/\Gamma$. The initial velocity in this frame is $(-V,0,0)$, and the constant particle Lorentz factor is ${\gamma }^{\prime }=\Gamma$ because a magnetic field does no work.

Define the signed relativistic cyclotron frequency $\Omega =q{B}_z^{\prime }/(\Gamma m_e)$, retaining $q=-e$. With $t^{\prime }$ as parameter, the exact primed trajectory isIt is a circle of radius $|V/\Omega |$ with centre $(0,V/\Omega )$; its initial velocity is $(-V,0,0)$ as required. Transforming back givesIn the laboratory this is a stretched cycloidal trajectory with cusps where the electron is instantaneously at rest. Its average drift is $⟨v_x⟩=V=E/B$, along $\mathbf{E}\times \mathbf{B}$ for the positive field orientation. Its energy oscillates as the electric field exchanges energy with it; it does not accelerate without bound. If $E=0$, the initially stationary electron remains stationary. At $|E|=c|B|$, neither of these finite boosts exists; that separate null-field limit is not one of the two strict cases requested.

Use a neutral, infinitely long wire on the $x$ axis. Denote the total current by $I=\int j_{x}d{A}_{\perp }$, and write $r^2=y^2+z^2$. Outside the wire the magnetic field and electric field areThese follow from cylindrical symmetry and Ampère's law. The zero temporal component of the given total four-current specifies lab neutrality; the $\rho$ in $j_x=\rho v_0$ must refer to a carrier density, not that zero net charge density.

TODO: normalization from a literal volume current density requires the unspecified wire cross-section. The source does not supply its area or profile. The exterior field family below is fully determined in terms of $I$; it cannot be reduced to $j_x$ alone without that missing datum. If $j_x$ was intended as a cross-section-integrated line current instead, set $I=j_x$ and interpret its accompanying charge density as charge per length. No inside-wire profile is invented.

For a boost $u=v_0$ along positive $y$, let $\Gamma =(1-u^2/c^2{)}^{-1/2}$. Apply the Lorentz transformation of electromagnetic fields at corresponding events:Here the subscript labels the perpendicular-to-wire boost, not the direction of the transformed vector. In these formulas evaluate the original field at $y=\Gamma (y^{\prime }+u{t}^{\prime })$, $z=z^{\prime }$ and $x=x^{\prime }$. The wire moves in this frame, so the fields are time-dependent. The local current transforms to $J^{\prime 0}=0$, $J^{\prime x}=j_x$, $J^{\prime y}=J^{\prime z}=0$. A nonzero transformed electric field is possible even with zero net charge density: it is not a purely electrostatic field.

For a boost $u=v_0$ along positive $x$, the transverse coordinates are unchanged. The transformed fields areThe transformed line charge and current are ${\lambda }^{\prime }=-\Gamma uI/c^2$ and $I^{\prime }=\Gamma I$. These reproduce the radial electric field through Gauss's law. A boost comoving with the electrons removes their individual current contribution, but the positive background then moves and the total current is not zero.

For two identical wires, their lab interaction is attractive, with magnitude ${\mu }_0{I}^2/(2\pi d)$ per length. The total four-current is spacelike, $J^2=-j_x^2<0$, so no inertial frame makes it a pure charge density. Also $B^2-E^2/c^2=B^2>0$ forbids a frame in which the exterior field is purely electric. For an arbitrary boost along the wires, the electrical part is repulsive and the magnetic part attractive:The latter always dominates, and their sum remains $-{\mu }_0{I}^2/(2\pi d)$.

Even a more general boost cannot make the entire-wire force electrical alone. At the second wire choose $y=d,z=0$, so $\mathbf{B}=B{\mathbf{e}}_z$ and $\mathbf{j}=j{\mathbf{e}}_x$ with $jB>0$. The physical Lorentz force density is $f=(0,0,-jB,0)$. If the boosted magnetic force vanished, ${\mathbf{j}}^{\prime }\times {\mathbf{B}}^{\prime }=0$, then ${\mathbf{E}}^{\prime }\cdot {\mathbf{B}}^{\prime }=0$ would also give ${\mathbf{j}}^{\prime }\cdot {\mathbf{E}}^{\prime }=0$. Thus $f^{\prime 0}=0$, which from the boost of $f$ requires $V_y=0$. But for such a boost $f^{\prime y}=-jB<0$, whereas ${\rho }^{\prime }{E}_y^{\prime }={\Gamma }_V^2{V}_x^{2}jB/c^2\ge 0$. The electrical term alone cannot equal the actual force. Therefore there is no such frame for both complete wires, although the force on electrons considered separately can be electrical in their rest frame.

The upper-left panel shows the three causal structure regions, with null boundaries included in the appropriate causal future and causal past. For the charged trajectory, use part b with $a=q{E}_x/m$, choosing the $x$ direction so that $a>0$. The electric field gives the constant three-force $q{E}_x$ and hence the shifted hyperbolaIt starts with a vertical tangent at the origin, stays strictly inside the future light cone, and asymptotes to $x=ct-c^2/a$. The other panels implement the coordinate and topology changes used below; they are original spacetime diagrams, not copies of source artwork.

The scalar contraction of the electromagnetic field tensor isWith orientation ${ϵ}^{0123}=+1$, define the dual electromagnetic field tensor by ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. Direct substitution givesThe first is a Lorentz scalar; the second is a pseudoscalar, invariant under proper Lorentz transformations such as boosts and rotations. Under spatial inversion $\mathbf{E}$ changes sign while $\mathbf{B}$ does not, so $\mathbf{E}\cdot \mathbf{B}$ reverses sign. For example, $\mathbf{E}=\mathbf{B}=\hat{\mathbf{z}}$ has product $+1$ before inversion and $-1$ afterwards. Thus the unrestricted statement including improper transformations needs this qualification.

For the spacetime integral, assume a globally regular electromagnetic four-potential on ordinary Minkowski space and sufficient boundary decay. Using $\nabla \cdot \mathbf{B}=0$ and ${\partial }_t\mathbf{B}=-\nabla \times \mathbf{E}$,ConsequentlyThe magnetic helicity boundary term controls the integral. If its surface flux vanishes and the magnetic helicity $H_B(t)=\int \mathbf{A}\cdot \mathbf{B}{d}^{3}x$ vanishes at both temporal ends, thenCompactly supported smooth potentials are one sufficient version of the intended falloff assumption. The local identity is exact, but vanishing fields at spatial infinity alone is insufficient. For a concrete counterexample, take $\varphi =0$ and, in dimensionless coordinates,where $a$ changes smoothly from zero to one and $\dot{a}$ has compact temporal support. Both electric fields and magnetic fields vanish rapidly at spatial infinity, butThe counterexample leaves a final magnetic field and therefore fails the extra temporal boundary condition. It shows exactly why the printed integral cannot follow from a merely spatial reading of “at infinity”; vanishing of the full boundary term is the needed hypothesis.

Let $e>0$ be the magnitude of the electron's electric charge, $m=m_e$, $F=eE>0$, and $\ell =m{c}^2/F$. The Lorentz force is $d{p}_x/dt=qE$, so the electron and Positron acquire opposite momenta:Using the relativistic energy-momentum relation, $u_x=p{c}^2/\mathcal{E}$, and integrating from rest gives the world linesFor $t<0$, the world lines are vertical at $x=0,d$ on a spacetime diagram. They join smoothly to the accelerated branches with zero initial velocity. At small $t$, $x_e\simeq -F{t}^2/(2m)$ and $x_p-d\simeq F{t}^2/(2m)$; at large $t$, their velocities approach $-c,+c$, respectively. Each accelerated world line is a branch of a hyperbola, since $(x_e-\ell {)}^2-c^2{t}^2={\ell }^2$ and $(x_p-d+\ell {)}^2-c^2{t}^2={\ell }^2$. The electric field separates the opposite electric charges in this orientation. The asymptotes are lightlike, while every finite-time tangent is timelike.

Write $\Gamma =(1-v^2/c^2{)}^{-1/2}$ for the Lorentz factor of the inertial frame $S^{\prime }$, distinguishing it from the particles' changing Lorentz factors. Apply the Lorentz transformationto each world line in a. Both particles initially have velocity $-v$ in $S^{\prime }$. The switching events have coordinatesThus the Positron starts accelerating earlier in $S^{\prime }$. Its velocity first becomes less negative, vanishes when its laboratory velocity reaches $v$, and then becomes positive. The electron continues at $-v$ until $t^{\prime }=0$, then accelerates toward $-c$. The relativistic velocity-addition formula makes this precise:The electric field parallel to the Lorentz boost has unchanged magnitude, but its switching surface is $t^{\prime }+v{x}^{\prime }/c^2=0$, rather than $t^{\prime }=0$ everywhere.

In the laboratory, the total three-momentum is zero for all $t$: the two Lorentz forces cancel at equal laboratory times. In $S^{\prime }$, each active force is still $qE$, since this is one-dimensional motion with a parallel electric field. Define $t_p^{\prime }=-\Gamma vd/c^2$. For $v>0$ the total three-momentum, evaluated at equal $t^{\prime }$, isThis follows by integrating $d{p}_p^{\prime }/d{t}^{\prime }=F$ from its switching event and $d{p}_e^{\prime }/d{t}^{\prime }=-F$ from its later switching event. During the interval between them only the Positron feels the prescribed electric field, so momentum conservation for the particle pair fails. Once both forces act, the total three-momentum is constant again.

The equal-time momentum of an externally driven particle pair illustrates a distinction. The sum of particle four-momenta at a fixed pair of events is always a four-vector. However, a fresh sum at equal time in each inertial frame generally uses different events. Here transforming an equal-$t$ laboratory sum would give $-2\Gamma v\sqrt{m^2{c}^4+F^2{c}^2{t}^2}/c^2$, which varies with $t$; this differs from the equal-$t^{\prime }$ answer above. For example, with $c=m=F=d=1$ and $v=0.6$, the late-time equal-$t^{\prime }$ sum is $-0.75$, whereas transforming the equal-$t=0$ sum gives $-1.5$. This is a concrete failure of treating the pair's equal-time sums as components of a single frame-independent four-vector.

A closed system's total four-momentum, including the electromagnetic field and the apparatus supplying it, is conserved and independent of the choice of spacelike integration surface under the usual isolated-system boundary conditions. The particles alone exchange energy and momentum with an external system, so neither that independence nor universal particle-only momentum conservation is guaranteed.

With the reversed electric field, reflection about $x=d/2$ exchanges the electron and Positron. They therefore meet there with equal and opposite momenta. Each travels $d/2$, and the work-energy theorem gives its kinetic energy:Their combined kinetic energy is $1.000\mathrm{MeV}$, while each total relativistic energy is $\mathcal{E}=0.511+0.500=1.011\mathrm{MeV}$. The laboratory is the center-of-momentum frame of the colliding pair.

Local four-momentum conservation in the electron-positron annihilation excludes one real photon: the incoming total four-momentum is timelike, with zero spatial component and positive energy, while a single photon's four-momentum is null and has $|\mathbf{p}|=\mathcal{E}/c>0$. Two photons can balance their momenta by travelling in opposite directions, soThe equal photon energies follow from opposite equal momenta, not merely from sharing the incoming energy arbitrarily. Here $hc\simeq 1.23984\mathrm{MeV}\mathrm{pm}$. This is the free local annihilation idealization implicit in the question; significant momentum transfer to the external apparatus would change the kinematic system being considered.

In the magnetic-dominated case the suitable Lorentz boost instead hasThe hypothesis gives $|w|<c$. The electromagnetic field invariants now have $E^2-c^2{B}^2<0$, allowing a purely magnetic inertial frame. The initial velocity is again $-w\hat{\mathbf{z}}$, perpendicular to ${\mathbf{B}}^{\prime }$. Since a magnetic field does no work, the electron's relativistic energy and speed in this inertial frame remain constant. Its trajectory is a circle in the $x^{\prime }{z}^{\prime }$ plane with Lorentz factor ${\gamma }_w$ and cyclotron frequency $\Omega =e{B}^{\prime }/({\gamma }_{w}m)$.

For $E,B>0$, set $\theta =\Omega t^{\prime }$ and $\rho =w/\Omega$. A parametrization with the correct initial velocity and Lorentz force isTransforming the world line back to the laboratory givesThe laboratory trajectory is a stretched cycloid, with cusps at $\theta =2\pi n$ where the electron is momentarily at rest. It first accelerates toward $-x$, then curves toward $+z$, returning to its original $x$ coordinate each cycle while advancing in $z$. Its mean drift is $\Delta z/\Delta t=w=E/B$, the E-cross-B drift. Its laboratory kinetic energy oscillates as the electric field does positive and then negative work; it does not grow indefinitely. The strict inequalities in the two branches exclude the null case $|E/B|=c$, where neither field can be removed by a finite Lorentz boost.

Assume $E,B>0$ for the directions and parametrizations; the boost formulas retain their signed meaning for reversed fields. A Lorentz boost with velocity $w\hat{\mathbf{z}}$ along $\mathbf{E}\times \mathbf{B}$ transforms the transverse electric field and magnetic field asIn the electric-dominated case chooseThe inequality ensures $|w|<c$, so this is an inertial frame. The field-removing boosts for perpendicular electric and magnetic fields are governed by the electromagnetic field invariants $\mathbf{E}\cdot \mathbf{B}=0$ and $E^2-c^2{B}^2>0$ explain why a purely electric inertial frame exists and a purely magnetic one does not.

The electron is not initially at rest in this convenient inertial frame: its initial velocity is $-w\hat{\mathbf{z}}$ and its initial three-momentum is $p_z^{\prime }=-{\gamma }_{w}mw$. Thereafter $p_x^{\prime }=-e{E}^{\prime }{t}^{\prime }$, while $p_z^{\prime }$ stays constant. The electric field accelerates it toward $-x$; as its relativistic energy increases, its negative transverse velocity decreases in magnitude. Defining $a=e{E}^{\prime }/m>0$ and $s=a\tau /c$ gives a useful parametric world line:Here $\tau$ is the proper time. These formulas follow by integrating $U^{\prime 0}={\gamma }_{w}c\cosh s$, $U^{\prime x}=-{\gamma }_{w}c\sinh s$, and constant $U^{\prime z}=-{\gamma }_{w}w$. Its projection onto the $t^{\prime },x^{\prime }$ plane is a hyperbola, but the spatial trajectory includes the initial transverse motion and is not a straight line.