= Solution
<Ginzburg-Landau theory of superconductivity> describes the superconducting phase by a complex <order parameter> and a gauge-invariant free-energy functional. A common normalization is
$$
F=\int d^3x\left[f_n+\alpha|\Psi|^2+\frac\beta2|\Psi|^4
+\frac1{2m_*}|(-i\hbar\nabla-q\mathbf A)\Psi|^2+\frac{B^2}{2\mu_0}\right],
$$
with $\beta>0$, pair charge $q=-2e$ and $\alpha\simeq a(T-T_c)$. In zero field the uniform minimum has $|\Psi|^2=-\alpha/\beta$ below $T_c$. Variation gives the order-parameter equation and supercurrent. The characteristic lengths are
$$
\xi=\sqrt{\frac{\hbar^2}{2m_*|\alpha|}},\qquad
\lambda_L=\sqrt{\frac{m_*}{\mu_0q^2|\Psi|^2}},\qquad
\kappa_{\rm GL}=\lambda_L/\xi.
$$
The Meissner response screens static <magnetic fields> over $\lambda_L$. In the simple isotropic theory, $\kappa_{\rm GL}<1/\sqrt2$ gives type-I behavior; $\kappa_{\rm GL}>1/\sqrt2$ allows type-II vortices and a mixed state between lower and upper critical fields. Flux quantization, <vortex> structure and currents follow from phase stiffness and electromagnetic coupling. This is a phenomenological expansion most directly controlled near $T_c$; static coefficients alone do not determine every dynamical mode or dissipation rate.
<BCS theory> supplies a microscopic pairing description. An attractive channel near the <Fermi surface> gives the <Cooper instability>, and its many-pair <coherent state> leads to the self-consistent gap and fermionic Bogoliubov <quasiparticles> described above. For an isotropic weak-coupling model with constant single-spin <density of states> $N_s(0)$, attraction $V>0$ and cutoff $\hbar\omega_D$,
$$
\Delta(0)\simeq2\hbar\omega_D e^{-1/[N_s(0)V]},\qquad
k_BT_c\simeq1.14\hbar\omega_D e^{-1/[N_s(0)V]}.
$$
It predicts $2\Delta(0)/(k_BT_c)\simeq3.53$ in that specified model. A full gap suppresses low-temperature <quasiparticles> exponentially, whereas nodes permit power laws such as the $T^2$ <heat capacity> derived above. The weak-coupling ratio is not a universal value for all superconductors. Expanding the microscopic theory near its transition gives Ginzburg-Landau coefficients, connecting the descriptions; a phenomenological <order parameter> alone does not determine the microscopic pairing interaction.
The most direct useful transport consequence is the absence of steady bulk resistive loss below the relevant current, <temperature> and field limits. <Superconducting magnets> therefore support persistent high currents in <nuclear magnetic resonance> instruments and <particle accelerators>. Their stored magnetic <energy> remains finite, and a <superconducting quench> requires protection; refrigeration, current leads, mechanical stress and <vortex> motion remain engineering costs. Zero dc resistance does not eliminate all ac loss or justify unlimited current.
<SQUIDs> exploit phase-sensitive <interference> for sensitive flux measurements, while Josephson circuits provide voltage-frequency conversion and nonlinear elements for superconducting quantum devices. These applications use coherence, not just low resistance. Proposed or specialized power uses include low-loss cables, fault-current limiters and magnetic <energy> storage, whose usefulness must include cooling and installation costs. Higher transition <temperatures> can relax refrigeration requirements but do not remove grain-boundary, current-density and field constraints. Neither BCS nor Ginzburg-Landau theory alone supplies a complete device design or guarantees that a particular material is economical.
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