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The linear carrier acceleration equation $\dot{\mathbf{j}}={ϵ}_0{\omega }_p^2\mathbf{E}$ and Faraday's law imply this conserved local quantity. Setting its initial value to zero gives the second London equation and London screening. An undamped response alone does not select that initial value, so it must not be used to infer expulsion of a pre-existing magnetic field for every magnetic history.^[ref]

To first order in the applied fields, the carrier acceleration gives $m\dot{\mathbf{v}}=q\mathbf{E}$. The current density $\mathbf{j}=nq\mathbf{v}$ consequently obeys the first London equationTaking its curl and using Faraday's law yieldsFor an initially field-free, current-free preparation, the integration constant is zero. Equivalently, impose the second London equation appropriate to the Meissner state, $\nabla \times \mathbf{j}=-{ϵ}_0{\omega }_p^2\mathbf{B}$. In the final static state, Ampère's law and $\nabla \cdot \mathbf{B}=0$ then giveFor a planar half-space $z>0$ with a tangential surface field, the bounded solution isThe plasma-frequency form of London penetration depth therefore shows that an applied field is exponentially small several penetration depths inside the material.

The magnetic-history constant in an undamped plasma explains the required qualification. Undamped carrier motion alone fixes a time-independent value of $\nabla \times \mathbf{j}+{ϵ}_0{\omega }_p^2\mathbf{B}$; it does not force that value to vanish. A perfect conductor can retain a pre-existing field. The additional London equation describes London screening independently of that arbitrary ideal-conductor history. The original London paper explicitly distinguishes its description from acceleration theory.^[ref]

Ginzburg-Landau theory of superconductivity organizes a continuous superconducting transition through a slowly varying complex order parameter. In the simplest isotropic single-component SI model,Here $\beta >0$, $m_{\ast }>0$, $\alpha$ changes sign at $T_c$, and $q=-2e$ fixes a charge/phase convention. The phenomenological normalization of $\psi$ determines the coefficient values; it should not be confused without qualification with a literal density of independently identifiable pairs.

Microscopic derivation of Ginzburg-Landau theory expands the BCS response near $T_c$ in small gap amplitude and slow spatial gradients. It identifies the complex order parameter with a scaled pair amplitude, supplies the coefficients from the electronic microscopic parameters, and explains the pair charge in the gauge coupling. The agreement is a controlled near-transition limit, not a statement that the truncated local functional remains exact at all temperatures. Lev Gor'kov established this connection explicitly in 1959.^[ref]

Below the transition, minimization gives $|{\psi }_0{|}^2=-\alpha /\beta$ and condensation-energy density ${\alpha }^2/(2\beta )$. A variation with respect to ${\psi }^{\ast }$ givesVariation with respect to $\mathbf{A}$ gives the supercurrentcoupled to the magnetostatic Ampère law. For nearly uniform amplitude this reduces to London screening and the Meissner effect. The theory supplies two different length scales:The amplitude can vary over the superconducting coherence length, whereas the magnetic field is screened over the London penetration depth.

The surface-energy distinction at $\kappa =1/\sqrt{2}$ explains why type-I and type-II behavior are possible within the same framework. In the type-II regime, quantized winding produces an Abrikosov vortex with a finite core and circulating current. Periodic solutions yield an Abrikosov vortex lattice. Linearization about the normal state gives $B_{c2}={\Phi }_0/(2\pi {\xi }^2)$, while core and field energetics determine the lower entry field. These predictions connect a spatial field theory to measurable vortex patterns and critical fields, beyond a spatially uniform BCS gap equation. Thin-film critical-field observations helped motivate the large-$\kappa$ regime; magnetic-decoration experiments subsequently made vortex positions visible.^[ref]

These phenomena also enable applications. Type-II conductors support high-field magnets, but flux pinning is needed to limit vortex-motion losses. Phase coherence and fluxoid quantization permit persistent-current loops and interference devices. A weak link supplies Josephson coupling energy, allowing SQUID magnetometry and nonlinear quantum circuits; a junction's detailed current–phase law requires a weak-link model, rather than following automatically from the spatially uniform bulk functional. Microstructure can raise a useful critical current without greatly changing the equilibrium upper field, illustrating the difference between thermodynamic stability and transport performance.

The simple functional has identifiable limits. Far below $T_c$, the small-gap expansion is uncontrolled; rapidly varying fields can require nonlocal microscopic electrodynamics. Very near a transition in a fluctuating low-dimensional or strongly anisotropic material, mean-field minimization can miss important thermal or quantum fluctuations. Unconventional or multiband pairing can require several order-parameter components and additional gradient couplings. Strong fields can introduce spin-polarization pair breaking omitted by the orbital model. Finally, the static functional predicts equilibrium extrema, not quasiparticle spectra, dissipation, vortex velocities or a unique amplitude-mode frequency without further dynamical input. These are reasons to choose an appropriate extension or a microscopic calculation, not reasons to discard the successful near-transition predictions.

Keep the index convention of b: $A^{\mu }=(A_0,\mathbf{A})$, so $A_{\mu }=(A_0,-\mathbf{A})$ and the spatial gauge covariant derivative is $\nabla +ie\mathbf{A}$. The physical three-current is the spatial part of $J^{\mu }$:Equivalently, writing $\phi =R{e}^{i\theta }$ gives $\mathbf{j}=-2e{R}^2(\nabla \theta +e\mathbf{A})$. Under the gauge transformation in a, $\theta \mapsto \theta +\alpha$ and $\mathbf{A}\mapsto \mathbf{A}-\nabla \alpha /e$, so this expression is manifestly invariant. Switching to a convention with the opposite spatial gauge sign reverses the phase-gradient term, not the screening term proportional to $e^2$.

For static fields the spatial Ampère-Maxwell equation is $\nabla \times \mathbf{B}=\mathbf{j}$, while $\nabla \cdot \mathbf{B}=0$. Taking its curl yields the general magnetic-field equationNo uniform-condensate approximation has yet been used. For a varying scalar amplitude the gradient terms must be kept; simply replacing $|\phi {|}^2$ by a constant before deriving the current would conceal their origin. The homogeneous-condensate limit supplies the London screening equation in d.

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.