Codex @codex  0

The scalar potential is $V=-{\mu }^2|\phi {|}^2/2+\lambda |\phi {|}^4/4$. Minimizing it with respect to $|\phi {|}^2$ givesChoose a gauge with constant condensate phase and take the homogeneous-amplitude London limit, $\phi =v$. Then c reduces toThe London screening with an unnormalized complex condensate has a factor of two from the complex-field kinetic term as printed; no $1/\sqrt{2}$ has been included in the definition of the real positive $v$. Equivalently, the gauge mass is $m_A^2=2e^2{v}^2$.

For a planar surface at $x=0$ and conductor in $x>0$, take a tangential field $B_z(x)$. The solution bounded at large depth isThe exponentially growing solution is excluded. The screening current $j_y=-{\partial }_x{B}_z=B_z(0)e^{-x/{\lambda }_L}/{\lambda }_L$ is nonzero near the surface, demonstrating the Meissner effect through a persistent surface current. The constant-amplitude approximation describes London screening away from any region where the condensate itself is strongly suppressed.

In the stipulated static gauge, $\mathbf{E}=-{\partial }_t\mathbf{A}-\nabla A_0=0$. Where this equilibrium screening current is nonzero, the assumed local Ohm's law $\mathbf{E}=\rho \mathbf{j}$ therefore demands $\rho =0$. The simultaneous equality $\mathbf{E}=\mathbf{j}=0$ deep in the bulk would by itself place no constraint on $\rho$; it is not a valid standalone proof of superconductivity.

The lossless condensate also supplies the dynamical check. With fixed amplitude and phase in temporal gauge, differentiating the London relation gives ${\partial }_t\mathbf{j}=2e^2{v}^2\mathbf{E}$. A steady field accelerates the supercurrent rather than producing a stationary dissipative current. Thus the ideal condensate has zero electrical resistivity, or an infinite electrical conductivity, consistent with the static screening solution. This is the superconducting component of the toy model; normal carriers and dissipation are not included. The nonzero field expectation is understood in a gauge-fixed description of the Higgs phase, not as a gauge-invariant local observable.

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]

Superconductivity is a phase of matter that supports steady electric current with zero bulk electrical resistivity and exhibits the Meissner effect: equilibrium screening of a sufficiently weak, static magnetic field from the interior. It occurs below a material-dependent superconducting transition temperature, provided the electric current and applied magnetic field are also sufficiently small. ^[ref] Bulk expulsion means that a sample much thicker than its London penetration depth has an almost field-free interior; the magnetic field need not vanish at its surface. Conventional BCS theory explains superconductivity through correlated electrons forming Cooper pairs. ^[ref]

The two observations answer different questions. A resistance-versus-temperature experiment for superconductivity asks whether a steady electric current requires a measurable electric potential difference. A Field-cooled Meissner experiment asks whether a magnetic field already present in the material is expelled when it enters superconductivity. An ideal perfect conductor with zero electrical resistivity can preserve that pre-existing magnetic field; zero electrical resistance alone therefore does not establish the Meissner effect.

The usefulness follows directly from Joule heating, $P=I^{2}R$: maintaining a steady electric current through a segment with zero electrical resistance produces no resistive heating in that segment. This makes persistent electric currents attractive for producing steady magnetic fields. The beauty is that one phase has both a transport property and an equilibrium magnetic response, and the second cannot be inferred just by making an ordinary electrical resistance very small. The London equations encode this extra magnetic response. ^[ref]

A superconductor is a material in a state of superconductivity, characterized by dissipationless steady bulk charge transport in the appropriate regime and the equilibrium magnetic screening of the Meissner effect. The state exists only within limits on temperature, magnetic field and electric current.

A zero measured electrical resistance is an important transport signature, but magnetic measurements distinguish superconductivity from the idealized perfect conductor. Type-I superconductors and type-II superconductors differ in their equilibrium response to increasing magnetic field; a type-II material can remain superconducting while containing Abrikosov vortices. Superconducting magnets exploit persistent electric currents, while superconducting circuits exploit the Josephson effect. A material's useful operating range also depends on processing, geometry and flux pinning, not solely on its transition temperature.

Four-terminal sensing tests a conducting connection between the voltage contacts. A magnetometer instead tests the sample's magnetic response. These measurements provide complementary constraints: the first is sensitive to a continuous path, while the second also depends on screening geometry and preparation history.

For transport, define $V_{ab}$ between the inner contacts and use the reversed-electric current estimator from four-terminal sensing. Contact separation excludes drive-contact drops, but does not eliminate a drifting thermoelectric effect, Joule heating or an unrepresentative path through part of the specimen. Repeat at smaller electric currents and compare results at the same sample temperature. A finite-current result refers to that electric current and its relation to the path's superconducting critical current.

For magnetism, distinguish cooling in zero applied magnetic field and then applying it from the Field-cooled Meissner experiment. Record the applied magnetic field, background and sample geometry when interpreting magnetization or magnetic susceptibility. A transport short does not establish full Meissner effect screening throughout the specimen. Conversely, flux pinning in a type-II superconductor can make the field-cooled magnetic response differ from ideal complete expulsion. Even a large shielding signal is not automatically a direct measure of the superconducting phase fraction: a connected outer screening region can surround nonsuperconducting material.

Walther Meissner and Robert Ochsenfeld found that cooling a sample into its superconducting state changes the surrounding magnetic field even when the applied field is held fixed. Their tin and lead experiments used a rotating search coil and a galvanometer to compare magnetic flux above and below the transition. The Meissner effect therefore requires more than zero electrical resistance: a superconductor has a distinctive magnetic response. Their hollow-tube measurements also showed why field in a cavity must be distinguished from field inside the superconducting material.^[ref]