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An ideal closed loop with negligible resistance can sustain a current that compensates changes in externally linked magnetic flux. In a fixed-geometry circuit model with constant self-inductance $L$, Faraday's law of induction gives $L\dot{I}+{\dot{\Phi }}_{\mathrm{ext}}=0$, hence ${\Phi }_{\mathrm{ext}}+LI$ remains constant.

The conserved flux includes the loop's self-field. Increasing external flux need not increase the total linked flux: the induced current supplies the compensating contribution. With finite resistance, the circuit equation instead contains $RI$, permitting decay and resistive heating.

For an actual superconductor, quantum phase constraints introduce fluxoid and flux quantization in a superconductor. Significant kinetic inductance requires keeping the carrier-motion contribution to the fluxoid; a phase slip can change the winding sector. The simple constant-flux model is therefore a useful idealization with stated circuit assumptions.^[ref]

Start from a gauge-invariant Ginzburg-Landau theory of superconductivity gradient energy $|(-i\hslash \nabla -q\mathbf{A})\psi {|}^2/(2m)$, with $\psi$ the coherent pair order parameter and $q$ its charge. Varying the free energy with respect to $\mathbf{A}$ gives the current density displayed in the source. Write $\psi =\sqrt{n_s}{e}^{i\theta }$ to obtainSingle-valuedness around a loop in superconducting material requires $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. Integrating the current relation therefore gives the exact fluxoid conditionFor a thick ring choose the contour sufficiently deep in the material that the current term is negligible. Then flux itself is approximately quantized, with positive quantumThe sign of the pair charge is absorbed into the orientation of $n$. In a thin ring with appreciable kinetic inductance, it is the fluxoid, not exactly the geometrical flux alone, that remains quantized. A state with nonzero phase-gradient current is a concrete counterexample to dropping the current integral without justification. The flux-only identity thus uses the ordinary thick-ring or negligible-kinetic-inductance limit in addition to phase single-valuedness.

Cool a thick superconducting ring through its transition in a controlled external flux, remove or change the applied field while keeping the ring superconducting, and measure the persistent trapped flux using a calibrated pickup coil and sensitive magnetometer. Repeating while varying the cooling flux selects successive winding sectors. The measured flux is close to steps separated by ${\Phi }_0$ when the kinetic-inductance correction is negligible. A SQUID-based flux readout can resolve that spacing; scanning local magnetic imaging offers another way to observe the trapped flux without identifying the ring current with an ordinary dissipative transport current.

The experiment tests the phase-winding/fluxoid relation in a. Self-inductance, London penetration depth and the size of the ring determine how nearly the measured geometric flux follows integer steps. A periodic response alone should not be used to infer an absolute trapped flux without calibration and the relevant inductive corrections.

TODO — source figure gap: the surviving ordinary-ring and two-junction-ring outlines and labels are readable, but page 9's figure form calls 52 absent raster objects, /R514 through /R565. The original archived parent has identical damage. An intact figure is needed to recover any missing image-specific details; those raster components have not been inferred or recreated. The observation method above answers the physical request without requiring the unavailable artwork; the TODO concerns verification of the input illustration rather than a missing part of the solution. The original circuit schematic used in c is an independent diagram of the stated model, not a claimed reconstruction of the unavailable artwork.

Let ${\delta }_1$ and ${\delta }_2$ be gauge-invariant phase drops across the two identical junctions, oriented between the same terminal phases. Their currents obey the DC Josephson effect: $I_1=I_C\sin {\delta }_1$ and $I_2=I_C\sin {\delta }_2$. For negligible loop self-inductance and no additional phase drops in the arms, fluxoid consistency fixes their difference modulo $2\pi$:Define the mean terminal phase $\delta =({\delta }_1+{\delta }_2)/2$ and $\varphi =\pi \Phi /{\Phi }_0$. Then ${\delta }_{1,2}=\delta \pm \varphi$ and the two parallel branch currents add:Consequently the maximum magnitude of the current is $2I_C|\cos \varphi |$, not a negative critical current when the cosine changes sign. The phase shifts by $\pi$ between such stable branches. Junction asymmetry or appreciable self-generated flux modifies the simplest interference pattern, so the stated idealization is required before treating $\Phi$ as an externally prescribed flux.

Rearrange the source's condensate-current expression and integrate around a closed path lying in the superconducting material:Single-valuedness of the superconducting wavefunction requires $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$, and Stokes theorem gives $\oint \mathbf{A}\cdot d\mathbf{l}=\Phi$. Thus the exact result is fluxoid and flux quantization in a superconductor:Reversing the winding orientation absorbs the sign of $q$. Since a Cooper pair has $|q|=2e$, the positive superconducting flux quantum isFor a thick ring, choose the path deep enough that the supercurrent is negligible; then the enclosed magnetic flux is $n{\Phi }_0$. This is the flux quantization usually intended in the question.

Qualification of the unqualified flux claim: a thin ring can have a nonzero kinetic contribution around every available path, so it is the fluxoid that is exactly quantized. For example, with zero applied flux and geometric and kinetic inductances $L_g=L_k$, a metastable winding-$1$ state has $(L_g+L_k)I={\Phi }_0$ but $\Phi =L_{g}I={\Phi }_0/2$. This concrete counterexample satisfies the given current law while its magnetic flux is not an integer multiple of ${\Phi }_0$.

Write the condensate order parameter as $\Psi =|\Psi |e^{i\theta }$. Away from vortex cores, single-valuedness around a closed loop gives $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. For neutral condensed objects of mass $M$, ${\mathbf{v}}_s=\hslash \nabla \theta /M$, hence the quantized circulationHere $M$ is the atomic mass for a simple bosonic atomic condensate, or the pair mass for a paired-fermion condensate. A nonzero winding can surround a vortex core where the order-parameter amplitude falls to zero. The result concerns collective condensate phase winding, not separate orbit quantization of every constituent particle.

For charge $q$ per condensed object, the gauge-invariant relation is $M{\mathbf{v}}_s=\hslash \nabla \theta -q\mathbf{A}$. Integrating givesThis is fluxoid and flux quantization in a superconductor. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude isThe orientation fixes its sign. A loop taken deep in a thick superconductor has negligible superflow and therefore encloses an integer flux quantum. In a thin loop or one with appreciable screening current, it is the fluxoid, including the circulation term, that is quantized; the magnetic flux alone need not be exactly an integer multiple.

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.