Codex @codex  0

A researcher whose team developed an optical method for measuring magnetostriction in pulsed magnetic fields. A fiber Bragg grating attached to a small specimen changes its reflected wavelength as the specimen strains. Fast optical readout tracks tiny length changes during the short field pulse, including at cryogenic temperatures. The method connects mechanical deformation with the magnetic response of a material without requiring electrical strain readout beside the specimen.^[ref]

For inviscid cylindrical rotation without a magnetic field, an axisymmetric radial interchange conserves each parcel's specific angular momentum $l=r^2\Omega$. Its restoring coefficient is ${\kappa }^2=r^{-3}d(l^2)/dr$. Increasing $l^2$ gives centrifugal stability; decreasing $l^2$ gives an exponentially growing interchange mode when radial displacement is allowed by the boundaries. A vanishing derivative is marginal, not a strictly restoring condition.

// Target: astrophysics.bigb

Take a straight hollow rectangular electromagnetic waveguide with vacuum interior $0<x<a$, $0<y<b$ and axis $z$, where $a>b>0$. Its four walls are ideal perfect conductors. The time-dependent Maxwell equations require zero tangential electric field and zero normal oscillating magnetic field at these walls. Real metal has finite electrical conductivity, so real guides lose electromagnetic energy by Joule heating; the perfect-wall model isolates confinement and wave dispersion from that loss.

The simplest propagating transverse electric mode of a rectangular waveguide is ${\mathrm{TE}}_{10}$. “Transverse electric” means $E_z=0$, not that the magnetic field is also wholly transverse. With real peak scale $E_0$, its complex amplitudes, including the common axial factor, can be writtenThe physical fields are the real parts after multiplication by $e^{-i\omega t}$. In particular, $E_y$ is zero at $x=0,a$, maximal at $x=a/2$, and independent of $y$. At $y=0,b$ it is normal to the metal, so it need not vanish there. The ${\mathrm{TE}}_{10}$ electric field crosses the narrow gap, while its strength varies across the broad dimension.

For an original example choose $a=30\mathrm{mm}$, $b=12\mathrm{mm}$ and $f=1.25f_c$, where $f_c=c/(2a)$. With $c=299792458m/s$, this givesHere ${\lambda }_0=c/f$ is vacuum wavelength and ${\lambda }_g=2\pi /\beta$ is guided wavelength. The next cutoff is ${\mathrm{TE}}_{20}$ at $2f_c$; ${\mathrm{TE}}_{01}$ starts at $(a/b)f_c=2.5f_c$. Thus only ${\mathrm{TE}}_{10}$ can propagate at the chosen frequency, although a source or discontinuity can additionally excite local evanescent waves. These are calculated values, not specifications or measurements of a commercial guide.

For a charged particle in a uniform magnetic field $\mathbf{B}=B\hat{z}$ with no electric field, the relativistic Lorentz force preserves energy, so the Lorentz factor $\gamma$ is constant. Decompose the velocity into $v_{\parallel }$ and ${\mathbf{v}}_{\perp }$. The parallel component remains constant; the perpendicular component rotates. The generic orbit is a helix, becoming a circle only when $v_{\parallel }=0$.

Writing proper time as $\tau$ and spatial four-velocity components as $u_x,u_y$, the force equation givesThus the signed cyclotron parameter is ${\omega }_{\tau }=qB/m$; the counterclockwise phase of $u_x+i{u}_y$ changes at $-{\omega }_{\tau }$. The magnitude of the coordinate-time angular frequency is $|qB|/(\gamma m)$.

The orbit radius depends on transverse momentum,not on the total speed unless the motion is perpendicular to the field. At fixed $p_{\perp }$, adding parallel momentum leaves $R$ unchanged but increases $\gamma$ and the coordinate-time orbit period. These formulas neglect radiation reaction and collisions.^[ref]

At larger kinetic energy, special relativity changes the relation between momentum and velocity. With $m$ the invariant mass, the Lorentz factor is $\gamma =1+K/(m{c}^2)$, and a relativistic cyclotron orbit in a fixed uniform magnetic field satisfiesThe cyclotron frequency decreases as the kinetic energy increases. If the drive stays at the classical value ${\omega }_0=|q|B/m$, the half-turn time is $\pi \gamma /{\omega }_0$. During it the drive advances through $\pi \gamma$ rather than $\pi$ radians. Its excess phase per half-turn is thereforeEven a small fractional frequency mismatch accumulates across many turns. A particle eventually reaches a weak or decelerating gap phase, so the classical fixed-field, fixed-frequency design has a practical kinetic energy limit. The limit depends on the kick size and acceptable accumulated phase error, rather than one universal cutoff energy. Changing drive frequency or shaping the magnetic field can restore synchronism; the argument does not forbid relativistic circular accelerators.

For the illustrative proton at $1.00\mathrm{MeV}$, $\gamma -1\simeq 0.00106579$: the frequency is lower by about $0.1065\%$, and the next half-turn adds about $0.19184^{\circ }$ of drive phase excess. To see cumulative sensitivity, prescribe an ideal ramp $K_j=jϵ$ with $ϵ=10\mathrm{keV}$ and sum the errors of the first $N$ coasts after kicks:This gives ${\Phi }_{100}\simeq 9.688^{\circ }$ after the coast following kick $100$; the arrival at kick $100$ instead has ${\Phi }_{99}\simeq 9.496^{\circ }$. The prescribed ramp diagnoses detuning but does not feed the resulting phase error back into the kicks. A self-consistent thin-gap model uses the cyclotron gap energy gain with $\cos {\varphi }_n$, then advances the phase with the new kinetic energy during the following coast.

For $\mathcal{E}=\gamma m{c}^2$, the temporal component of the covariant Lorentz-force equation givesThe magnetic field does no work because $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.

Relativistic electromagnetism describes electric fields and magnetic fields as observer-dependent components of one electromagnetic field tensor. A Lorentz transformation changes their components together while preserving the physical predictions of the Lorentz force.

For example, a charged particle can experience a magnetic force in a frame where it moves through a magnetic field. In its instantaneous rest frame, the transformed fields can produce an electric force instead. The two descriptions refer to the same interaction, with forces and time intervals transformed consistently. The tensor formulation also packages the Maxwell equations so their covariance is explicit.

A spatial image after a chosen dark interval is a different readout from continuously recording arrivals at a fixed probe. Here is an original idealised fluorescence imaging model: release a dilute cloud, let it move freely, then apply a short uniform probe pulse. Neglect collisions, residual forces, attenuation and motion during the pulse. Use each image's cloud centre to remove mean drift. Uniform gravity shifts that centre without changing the centred variance.

For each atom let $x(t)=x_0+v_{0}t$ along a horizontal imaging axis. Expanding the square about the expected values gives exactlyNo Gaussian distribution is needed for this variance identity. A jointly Gaussian distribution of initial positions and velocities does make the later density Gaussian. For a separable three-dimensional cloud viewed along $z$, integration along the viewing direction gives the column densityHere $N$ is atom number. Integrating this image along $y$ leaves a one-dimensional Gaussian distribution with standard deviation ${\sigma }_x(t)$. This is a projected density measurement, not a three-dimensional radius or a direct velocity histogram. Under the uniform, optically thin probe assumptions, expected fluorescence is proportional to this column density. A known, independent point spread function with finite variance adds its own variance to the measured width; probe duration and intensity gradients need separate treatment.

For a Maxwell-Boltzmann distribution, $T_x=m{\sigma }_v^2/k_B$, where $m$ is atomic mass and $k_B$ the Boltzmann constant. For a nonthermal distribution this defines a directional second-moment temperature, which need not describe every feature of the distribution. A covariance between position and velocity can initially narrow the cloud even though ${\sigma }_v^2$ is positive. The physical covariance matrix requires $C_{xv}^2\le {\sigma }_{x0}^2{\sigma }_v^2$.

Choose ${\sigma }_{x0}=0.60\mathrm{mm}$, ${\sigma }_v=0.060\mathrm{m}{\mathrm{s}}^{-1}$ and $C_{xv}=-0.018\mathrm{m}{\mathrm{m}}^2\mathrm{m}{\mathrm{s}}^{-1}$, giving correlation coefficient $-1/2$. With $t$ in milliseconds, the numerical variance in square millimetres is $0.36-0.036t+0.0036t^2$. Its minimum occurs at $5\mathrm{ms}$, with standard deviation $\sqrt{0.27}\mathrm{mm}$. These chosen values satisfy the covariance matrix constraint; early contraction is compatible with free motion.

Generate thirteen synthetic variance observations at equally spaced times from $0$ to $30\mathrm{ms}$, adding independent Gaussian noise of standard deviation $0.012\mathrm{m}{\mathrm{m}}^2$. Linear least squares fits all three coefficients in $a+bt+c{t}^2$. The example yields $a\simeq 0.37026\mathrm{m}{\mathrm{m}}^2$, $b\simeq -0.036413\mathrm{m}{\mathrm{m}}^2\mathrm{m}{\mathrm{s}}^{-1}$ and $c\simeq 0.0035820\mathrm{m}{\mathrm{m}}^2\mathrm{m}{\mathrm{s}}^{-2}$. With the chosen approximate sodium mass $m=3.82\times 10^{-26}\mathrm{kg}$, the true $T_x$ is $9.96\mu \mathrm{K}$ and the fitted value is $9.91\mu \mathrm{K}$. This is a fit to generated width data, not historical observations.

If the same data are fitted to $a+c{t}^2$, forcing $C_{xv}=0$, the inferred $T_x$ falls to $6.79\mu \mathrm{K}$. The omitted term is partly absorbed into both fitted coefficients. In noiseless linear least squares with a free intercept, its precise effect on the quadratic coefficient iswhere the subscript denotes equal-weight moments over the chosen observation times, using one consistent unit system. Since the ratio is positive for distinct nonnegative times, negative $C_{xv}$ biases this estimate downwards. An incorrect release-time origin can also generate a linear term, so fitting it does not by itself identify a physical initial covariance. Timing calibration, spatial resolution, residual magnetic fields and probe-induced recoil energy remain separate controls. Neither a cold-looking image nor a single long expansion time identifies all three coefficients.

Remanent magnetization is magnetization retained after the applied magnetic field has been removed. Magnetic minerals cooling in an ambient field can preserve information about that field in a planetary crust. Such a record may outlive the planetary dynamo which originally produced it. Crustal remanence therefore differs from a present field maintained by conducting-fluid motion.

A rectangular loop entering a uniform magnetic field while only one horizontal segment is immersed has motional electromotive force $Bav$. Neglecting self-inductance, its electric current is $Bav/R$ and its upward magnetic force is $(Ba{)}^{2}v/R$. The equation $m\dot{v}=mg-(Ba{)}^{2}v/R$ gives exponential relaxation toward $mgR/(Ba{)}^2$. This entry model ends when both horizontal segments enter the field; then their net magnetic force and the rate of change of external magnetic flux vanish.

For a magnetic vector potential $\mathbf{A}$, a solenoidal magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$ and constant magnetic diffusivity ${\eta }_m$, write ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-{\eta }_m\nabla \times \mathbf{B}-\nabla \varphi$. In a fixed closed volume with zero helicity surface flux, integration by parts makes the two differentiated factors in $H={\int }_V\mathbf{A}\cdot \mathbf{B}dV$ equal; the scalar-potential boundary term vanishes when $\mathbf{B}\cdot \mathbf{n}=0$. One sufficient boundary gauge is $\mathbf{n}\times \mathbf{A}=0$; another standard setting is integration over all space with sufficient decay. Orthogonality of $\mathbf{u}\times \mathbf{B}$ to $\mathbf{B}$ gives the displayed identity. Unlike the resistive magnetic energy loss, $\dot{H}$ is not sign definite. The zero normal magnetic field also makes $H$ gauge invariance under single-valued smooth gauge transformations.

Resistive magnetohydrodynamics retains finite magnetic diffusivity in the coupled evolution of a conducting fluid and its magnetic field. The resistive induction equation allows field diffusion relative to the fluid, while the magnetohydrodynamic momentum equation includes the Lorentz force density. It contrasts with the zero-diffusivity limit of ideal magnetohydrodynamics.

By National Institute of Standards and Technology.

Official text: www.nist.gov/pml/nanoscale-device-characterization-division/popular-links/hall-effect/resistivity-and-hall

Institutional measurement procedures covering contact geometry, reversal of electric current and magnetic field, and extraction of sheet carrier density and Hall mobility. Bulk density requires the conducting thickness.

The right-hand rule fixes the orientation of a cross product and relates the circulation direction of an electric current to the direction of its magnetic field.

An RLC circuit combines electrical resistance $R$, inductance $L$, and capacitance $C$. The capacitor stores energy in its electric field; the inductor stores energy in its magnetic field. Their exchange can produce an oscillation, while Joule heating removes energy. A series RLC circuit gives a particularly clear example: changing $R$ turns ringing into a direct return to equilibrium, and changing $L$ or $C$ tunes a resonance.

These circuits connect two useful viewpoints. In time, they describe how stored energy is released after switching. In frequency, they select a range of signals: the resonant series RLC response explains radio tuning and simple filters. The same mathematics describes a damped harmonic oscillator, so insight about mechanical ringing transfers directly to an electrical circuit. ^[ref]; ^[ref].

An experimental physicist who, with Walther Meissner, discovered that cooling a material into superconductivity can reorganize an already present magnetic field. Their 1933 measurements on tin and lead cylinders showed the field being expelled from the superconducting material. This Meissner effect supplies information beyond zero electrical resistance: merely preserving an existing field, as an ideal conductor would, does not explain the observed change on cooling.^[ref]

With aligned uniform magnetic field and rotation, pressure projection couples the two incompressible transverse wave polarization states at rate ${\omega }_I\sim 2\Omega |k_{\parallel }|/k_{\perp }$. Each nonrotating Alfvén wave has rate ${\omega }_A=|k_{\parallel }|v_A$. Their ratio is independent of the anisotropy. Leading finite-timescale decoupling therefore needs $k_{\perp }{v}_A\gg 2\Omega$, with rotational coupling also slow relative to any retained nonlinear evolution. At finite amplitude the usual anisotropic reduction still permits passive advection of pseudo-Alfvénic fluctuations by the Alfvénic subsystem; removing rotational mixing does not eliminate that one-way interaction. Because the two Alfvén polarizations are degenerate, weak coupling can still mix them over the longer time ${\omega }_I^{-1}$.

A rotational discontinuity changes the tangential direction of the magnetic field and velocity while leaving density, pressure, and normal components continuous. The normal flow is Alfvénic.

For the high-frequency Faraday effect in a cold plasma, the rotation measure is the coefficient in $\Delta \varphi ={\lambda }^2\mathrm{RM}$. It is a density-weighted line integral of the line-of-sight magnetic field; the coefficient $a_0$ depends on the units. A sign convention for the line of sight fixes its sign. Correlations across an extended source can constrain a foreground magnetic energy spectrum, subject to statistical assumptions.

A qubit encoded in the two $m_F=0$ ground state hyperfine levels of rubidium-87. The electronic $J=1/2$ and nuclear spin $I=3/2$ combine to $F=1,2$. At zero magnetic field both levels have vanishing linear Zeeman shift, suppressing first-order magnetic dephasing; the quadratic Zeeman shift of a clock transition remains. Coherent microwave magnetic dipole transitions or stimulated Raman transitions implement single-qubit rotation gates. These are coupled electron-nucleus states, not bare nuclear-spin states.