Miroslav Pozek

Many introductory treatments of electromagnetism derive Ampère's law from the Biot-Savart law using a highly symmetric current configuration, and leave its general validity largely unexamined. This obscures the logical continuity of the empirical construction of magnetostatics. In this article, we present a direct and geometrically transparent derivation of Ampère's law for arbitrarily shaped steady current loops. By interpreting the circulation of the magnetic field as the total change in the solid angle subtended by the current loop along an arbitrary closed integration curve, we show that Ampère's law holds independently of the conductor geometry. The derivation relies only on the Biot-Savart law and elementary geometric arguments, making it suitable for undergraduate courses in electromagnetism.

A phenomenological model comprehensively captures the defining features of the cuprate high-temperature superconductors.

A pivotal step toward understanding unconventional superconductors would be to decipher how superconductivity emerges from the unusual normal state. In the cuprates, traces of superconducting pairing appear above the macroscopic transition temperature T_c, yet extensive investigation has led to disparate conclusions. The main difficulty has been to separate superconducting contributions from complex normal-state behaviour. Here we avoid this problem by measuring nonlinear conductivity, an observable that is zero in the normal state. We uncover for several representative cuprates that the nonlinear conductivity vanishes exponentially above T_c, both with temperature and magnetic field, and exhibits temperature-scaling characterized by a universal scale Ξ₀. Attempts to model the response with standard Ginzburg-Landau theory are systematically unsuccessful. Instead, our findings are captured by a simple percolation model that also explains other properties of the cuprates. We thus resolve a long-standing conundrum by showing that the superconducting precursor in the cuprates is strongly affected by intrinsic inhomogeneity.

Despite extraordinary scientific efforts over the past three decades, the cuprate high-temperature superconductors continue to pose formidable challenges. A pivotal problem, essential for understanding both the normal and superconducting states, is to clarify the nature of the superconducting pre-pairing above the bulk transition temperature T_c. Different experimental probes have given conflicting results, in part due to difficulties in discerning the superconducting response from the complex normal-state behavior. Moreover, it has proven challenging to separate common properties of the cuprates from compound-specific idiosyncrasies. Here we investigate the paraconductivity—the superconducting contribution to the direct-current (dc) conductivity—of the simple-tetragonal model cuprate material HgBa₂CuO_4+δ. We are able to separate the superconducting and normal-state responses by taking advantage of the Fermi-liquid nature of the normal state in underdoped HgBa₂CuO_4+δ; the robust and simple quadratic temperature-dependence of the normal-state resistivity enables us to extract the paraconductivity above the macroscopic T_c with great accuracy. We find that the paraconductivity exhibits unusual exponential temperature dependence, and that it can be quantitatively explained by a simple superconducting percolation model. Consequently, the emergence of superconductivity in this model system is dominated by the underlying intrinsic gap inhomogeneity. Motivated by these insights, we reanalyze published results for two other cuprates and find exponential behavior as well, with nearly the same characteristic temperature scale. The universal intrinsic gap inhomogeneity is not only essential for understanding the supercoducting precursor, but will also have to be taken into account in the analysis of other bulk measurements of the cuprates.