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The Pauli exclusion principle says that each complete one-particle state can have fermionic occupation number zero or one. “Complete” includes spin and any other internal labels, as well as the spatial wavefunction. Thus two electrons can share a spatial orbital with opposite spin projections: these are two different spin orbitals. Equal energy alone does not mean equal quantum state, because degenerate energy levels contain several independent one-particle states.

This distinction explains how an electron shell can accommodate several electrons while exclusion still allows only one per complete one-particle state. It also makes a finite model of fermionic state counting surprisingly simple: two spatial orbitals with two spin choices each give four complete one-particle states, and exactly six two-electron basis states. The following diagram enumerates them; its boxes represent occupation numbers, not locations in space.

Pauli's Exclusion principle and quantum mechanics by Wolfgang Pauli is his retrospective account, delivered on 13 December 1946 for the 1945 Nobel Prize. On printed pp. 29–30, he describes proposing an extra two-valued electron property in autumn 1924 and formulating exclusion in 1925. The crucial clue linked counts of resolved energy eigenstates in alkali spectra under a magnetic field with the populations of closed electron shells. Resolving every label turned a complicated shell capacity into one electron per complete one-particle state. This historical route preceded the modern spin interpretation and the antisymmetric wavefunction formulation; it was not originally a deduction from a Slater determinant.

A Penning trap uses a static electric quadrupole field for axial confinement and a static axial magnetic field for transverse confinement. The Lorentz force bends a moving electric charge's trajectory without doing work. An RF probe may interrogate a trapped particle, but an RF confining voltage is unnecessary in this ideal model. This makes the Penning trap useful for relating cyclotron frequencies and spin precession to a common magnetic field.

Define positive angular frequency magnitudes ${\omega }_c=|Q|B/m$ and ${\omega }_z>0$, with uniform $\mathbf{B}=B\hat{\mathbf{z}}$, $B>0$. Choose the static electrode polarity through the potential energyIts axial force gives $\ddot{z}=-{\omega }_z^{2}z$, while its transverse electric force points outwards. For $Q>0$, let $w=x+iy$. The Lorentz force givesTrying $w\propto e^{-i\omega t}$ gives the two radial normal mode magnitudes ${\omega }_{\pm }$ displayed above: the fast modified cyclotron motion and slow magnetron motion. For $Q<0$ the rotation directions reverse. Bounded confinement in this ideal model requires ${\omega }_c^2>2{\omega }_z^2>0$; the repeated-root boundary needs exclusion because it allows a linearly growing solution. At large ${\omega }_c/{\omega }_z$, ${\omega }_{-}\simeq {\omega }_z^2/(2{\omega }_c)$ and ${\omega }_{+}\simeq {\omega }_c-{\omega }_{-}$.

The ideal identitiesfollow directly from that quadratic. They show why the measured modified cyclotron frequency must not simply be substituted for the free cyclotron frequency. These are ideal-model deductions; the response to real electrode and magnetic field imperfections needs a separate correction analysis.

At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

A permanent magnet retains magnetization after the applied magnetizing magnetic field is removed. Its working magnetic field persists without a driving electric current in a winding.

For a one-dimensional velocity perpendicular to a transverse magnetic field, $c_f^2=\gamma p/\rho +B^2/({\mu }_0\rho )$. The primitive-variable characteristic matrix has two advective eigenvalues and the two speeds $u\pm c_f$. Integrating the fast eigenvectors keeps $p/{\rho }^{\gamma }$ and $B/\rho$ constant. The state parameter obeys ${\rho }_t+(u\pm c_f){\rho }_x=0$, so the characteristic speed satisfies the Inviscid Burgers equation.

For a planar magnetohydrodynamic shock with normal velocity and tangential magnetic field, mass conservation gives $\rho u=m$ and the ideal magnetohydrodynamic induction equation gives $uB$ constant across the front. Hence $B/\rho$ is unchanged, expressing magnetic flux freezing. Normal momentum flux is $\rho u^2+p+B^2/(2{\mu }_0)$, and energy flux per unit mass flux is $u^2/2+\gamma p/[(\gamma -1)\rho ]+B^2/({\mu }_0\rho )$, apart from a smooth common gravitational potential. The amplified tangential field contributes both magnetic pressure and magnetic energy transport.

The equal normal stresses in the two directions perpendicular to the local magnetic field for a gyrotropic phase-space distribution function. The factor one half averages the two perpendicular velocity components. With first adiabatic invariant $\mu =v_{\perp }^2/(2B)$, the integrand is $m\mu Bf$.

The persistent current in a superconducting loop experiment prepares a closed cold path, changes the externally linked magnetic flux, and follows the resulting superconducting persistent current through its magnetic field. The essential measurement starts after the inducing magnetic field has stopped changing.

Heike Kamerlingh Onnes's 1914 Communication 140c describes a lead coil closed by fused lead connections. Cooling in a magnetic field and removing the inducing magnet prepared the circulation. A magnetic compass sensed it without contacts; a separate copper coil, cooled in liquid air, supplied a measured balancing magnetic field. Reversing orientation and compensation reduced offsets. ^[ref]

The liquid-helium cryostat maintained low temperature, but refilling temporarily warmed the apparatus and changed the inferred electric current. A later run used a smaller initial electric current. Apparatus deformation and needle-zero drift limited comparison; cutting the closure still left a smaller magnetic signal, warning against assigning every signal to circulation around the whole loop. The report consequently gave an upper bound under a residual Ohm's law hypothesis. ^[ref]

For interpreting such a measurement, write a locally calibrated magnetic signal asHere $k$ is the coupling and gain, while $b$ contains the background magnetic field and readout offset. Only after establishing stable $k$ and correcting $b$ does a ratio of signals determine a ratio of electric currents. A gradual change in geometry, a drifting background magnetic field, or a change of temperature can imitate or conceal a decay. A pickup coil connected only to a voltage meter obeys Faraday's law of induction and detects a changing linked magnetic flux; its zero steady output cannot demonstrate that the underlying static magnetic field has disappeared. A magnetic compass can respond to that static magnetic field.

Opening the path, reversing the preparation, and repeating with smaller initial electric current are useful controls for separating whole-loop circulation, magnetic background and a superconducting critical current. Keep the temperature and external magnetic field stable during the interval used to infer electrical resistance. A sudden reduction near a superconducting critical current is not evidence for the constant-$R$ exponential decay model at smaller electric current. Persistence also does not replace the Field-cooled Meissner experiment: transport around a hole and magnetic expulsion from the material test different properties.

A superconducting persistent current is an electric current circulating in a closed path with superconductivity after the external driving electromotive force has ceased. Its associated magnetic field makes the circulation observable without inserting a measuring instrument into the path. This article concerns macroscopic circulation in superconductivity. A finite experiment establishes persistence over its observation window with a stated measurement error, rather than directly observing an infinite lifetime.

The ideal circuit connection is flux conservation in a superconducting loop. In a fixed loop with self-inductance $L$, a single circulating electric current $I$, and externally imposed linked magnetic flux ${\Phi }_{\mathrm{ext}}$, the lumped circuit approximation givesWhen the electrical resistance $R$ vanishes, integrating gives $LI+{\Phi }_{\mathrm{ext}}=\text{constant}$. Removing the external linked magnetic flux therefore changes the circulating electric current; holding it fixed afterwards preserves that electric current. This is a circuit approximation with a specified self-inductance, not a derivation of the microscopic state of superconductivity. For thin paths where the carrier contribution matters, fluxoid and flux quantization in a superconductor supplies the more careful description.

For constant $L>0$, constant effective electrical resistance $R\ge 0$, and no changing external linked magnetic flux, the RL circuit instead predictsThis exponential decay follows by separating $dI/I=-(R/L)dt$. A long relaxation time tests extremely small electrical resistance without trying to resolve an equally tiny steady electric potential difference. The stored magnetic energy is $U=L{I}^2/2$, so the same model gives $dU/dt=-R{I}^2$, exactly the Joule heating loss. The finite energy stored in the loop supplies the magnetic readout; the circulating electric current is not a continuously powered source of useful work.

Differential circulation of a weak magnetic field winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order $U/L^2$, its radial wavenumber grows as $k_r\sim Ut/L^2$. Resistive damping becomes substantial when $\eta {\int }_0^t{k}_r^{2}dt\sim 1$, giving $t\sim (L^4/(\eta U^2){)}^{1/3}=(L/U){\mathrm{Rm}}^{1/3}$. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite magnetic diffusivity then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.

A physicist.^[ref]

The Zeeman effect connects a laboratory magnetic field to the frequencies and wave polarization of light emitted by chemical atoms. Its experimental history makes the viewing direction and spectral resolving power essential to the inference.

In the standard two-dimensional setting, with planar velocity and magnetic field independent of the third coordinate and homogeneous energy-closed boundary conditions, the in-plane magnetic potential is an advected diffusing scalar. Its quadratic integral decreases, preventing regeneration of a dynamo field. A third field component independent of that coordinate also obeys a passive advection-diffusion equation. A flow depending on two coordinates but possessing a third velocity component is not covered by this strictly planar formulation.

Take a homogeneous material exhibiting superconductivity in the half-space $z\ge 0$, bounded by vacuum at $z=0$. Fix the temperature below the superconducting transition temperature, take a constant positive London penetration depth ${\lambda }_L$, and assume a local, linear London equation description in a Meissner state. The applied magnetic field is weak enough that it neither nucleates a quantum vortex nor appreciably suppresses the material's superconductivity. Assume translation invariance in $x,y$, no imposed transport electric current, and a tangential boundary magnetic field $\mathbf{B}(0)=B_0\hat{\mathbf{x}}$. Here $B_0$ is the local surface magnetic flux density, equal to the imposed vacuum magnetic field in this infinite planar geometry; it is not automatically the far-away applied value for a finite object.

For a bounded, decaying interior solution, the London equations, magnetostatic Ampère law and Gauss's law for magnetism giveThe solution isAt one London penetration depth the magnetic field is $e^{-1}\simeq 0.368$ of its surface value; at five it is $e^{-5}\simeq 0.00674$. There is no finite depth where this exponential is exactly zero. The differential equation excludes the growing exponential through the deep-interior boundary condition. A constant normal component cannot decay in this geometry: Gauss's law for magnetism would require ${\partial }_z{B}_z=0$. A perpendicular applied magnetic field on a finite plate therefore requires a different spatial problem.^[ref]

A planetary dynamo maintains an intrinsic magnetic field by motion of an electrically conducting fluid. The resistive induction equation balances stretching and transport against magnetic diffusion. A sufficiently large magnetic Reynolds number is necessary but does not ensure that the flow regenerates every required field component. Thermal or compositional convection supplies mechanical power; rotation organizes the flow through the Coriolis force, and the Lorentz force density eventually modifies the flow and limits growth. A fluid core can fail to operate a dynamo if it is stably stratified or inadequately driven.

The inertial correction to the electric drift ${\mathbf{u}}_E=c\mathbf{E}\times \mathbf{B}/B^2$ in a slowly varying magnetic field. In the simplest uniform-field electrostatic case it is ${\mathbf{v}}_{\mathrm{pol}}=m_s{c}^2{\partial }_t{\mathbf{E}}_{\perp }/(q_s{B}^2)$. Its species-dependent current density supplies the perpendicular polarization response needed in many slow plasma instabilities.

Viewed perpendicular to the magnetic field, the normal Zeeman effect has a $\pi$ component with linear polarization parallel to the magnetic field and two $\sigma$ components polarized perpendicular to it. Along the magnetic field, $\pi$ vanishes and the $\sigma$ pair has opposite circular polarizations. Pieter Zeeman described these geometry-dependent patterns in his lecture.^[ref]

The geometry can be derived from electric dipole radiation. Let the observation unit vector be $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, with $\theta$ measured from the magnetic field. Radiation retains only the transverse part of the oscillating electric dipole moment $\mathbf{d}$:For $\pi$, take $\mathbf{d}\parallel \hat{\mathbf{z}}$; its angular light intensity factor is ${\sin }^2\theta$, so it vanishes on axis. For ${\sigma }^{\pm }$, use normalized complex dipoles $(\hat{\mathbf{x}}\pm i\hat{\mathbf{y}})/\sqrt{2}$. Their angular factors are $(1+{\cos }^2\theta )/2$. Projection gives circles on axis, lines at $\theta =\pi /2$, and generally elliptical polarization between. These factors assume equal dipole magnitudes; actual line light intensities also depend on initial quantum state populations, transition strengths and collection geometry. No absolute handedness name is needed: it depends on the viewing and phase conventions.

A transverse polarizer can separate $\pi$ from $\sigma$ even before the spectral line components are fully resolved. Along the magnetic field, a quarter-wave plate followed by a polarizer converts the two opposite circular polarizations into distinguishable linear settings. This explains why polarization analysis carried information beyond a small increase in line width.

For the electronic ${}^2{S}_{1/2}\to {}^2{P}_{1/2}$ transition, neglect hyperfine structure and use weak-field LS coupling. The Landé g-factors are $g_g=2$ and $g_e=2/3$, so a transition from $m_g$ to $m_e$ shifts by $({\mu }_{B}B/h)(g_e{m}_e-g_g{m}_g)$.

With propagation perpendicular to the magnetic field, linear polarization parallel to the field drives the $\pi$ electric-dipole selection rule $\Delta m=0$. The two shifts are $\pm 2{\mu }_{B}B/(3h)$. Perpendicular linear polarization is an equal-amplitude combination of the two field-axis circular components, driving ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$ and shifts $\pm 4{\mu }_{B}B/(3h)$. Each geometry therefore selects a doublet, with twice the separation in the perpendicular-polarization case. Equal line heights within a pair require equal lower-state populations and an unsaturating probe; line positions do not require that extra population assumption.

For an isolated solenoidal magnetic field, zero flux through every sphere removes the degree-zero spherical harmonic of $P=\mathbf{B}\cdot \mathbf{x}$. In a poloidal-toroidal decomposition, let ${\mathcal{L}}^2=-{\Delta }_{S^2}$. The spherical poloidal component obeysIf $P={\sum }_{l\ge 1,m}{p}_{lm}(r)Y_{lm}$, put $D_{lm}={\int }_0^{\infty }[r^2|p_{lm}^{\prime }{|}^2+l(l+1)|p_{lm}{|}^2]dr$. The two sides of the bound are respectively $\sum D_{lm}/[l(l+1)]$ and $\frac{1}{2}\sum D_{lm}$. Since $l(l+1)\ge 2$, the bound follows, with equality for degree-one radial scalars. Together with the radial-flow dynamo energy bound it gives $(q/\eta {)}^2\ge 2E_P/E_{\mathrm{total}}$ for a marginal or growing dynamo field.

// Target: fluid-mechanics.bigb

A poloidal magnetic field lies in meridional planes through the symmetry axis, in contrast to a toroidal magnetic field. For an axisymmetric vector field, it is represented by a poloidal magnetic flux function as ${\mathbf{B}}_p=R^{-1}\nabla \Psi \times {\mathbf{e}}_{\varphi }$.