Supercurrent density detailed

The supercurrent density, denoted as \mathbf{J}_s, is a fundamental concept in the theory of superconductivity. It describes the flow of supercurrent, which is the current that flows without resistance in a superconducting material. The supercurrent density is defined as the current per unit area flowing through a cross-section of the superconductor. The supercurrent density can be expressed in terms of the wave function \psi of the superconducting state as follows:

\begin{equation} \mathbf{J}_s = \frac{n_s e}{m} \mathbf{v}_s = \frac{n_s e}{m} \nabla \phi \end{equation}

\textbf{where:}

\begin{aligned} n_s &\text{ is the density of superconducting carriers,} \\ e &\text{ is the charge of the carriers,} \\ m &\text{ is the effective mass of the carriers,} \\ \mathbf{v}_s &\text{ is the velocity of the supercurrent,} \\ \phi &\text{ is the phase of the wave function.} \end{aligned}">\begin{aligned} n_s &\text{ is the density of superconducting carriers,} \\ e &\text{ is the charge of the carriers,} \\ m &\text{ is the effective mass of the carriers,} \\ \mathbf{v}_s &\text{ is the velocity of the supercurrent,} \\ \phi &\text{ is the phase of the wave function.} \end{aligned}

\text{The supercurrent density is also related to the electromagnetic field in the superconductor.}

\text{According to the London equations,}

\mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{A}

The symbol n is traditionally used to represent number density. // That is number of particles per unit volume. n should automatically translate to number of particles divided by volume. Details Here The history of supercurrent density is closely tied to the development of the theory of superconductivity. The phenomenon of superconductivity was first discovered by Heike Kamerlingh Onnes in 1911 when he observed that mercury exhibited zero electrical resistance at low temperatures. In the 1930s, the London brothers, Fritz and Heinz London, formulated the first theoretical framework for superconductivity, introducing the concept of supercurrent density and the London equations. Later, in 1957, John Bardeen, Leon Cooper, and Robert Schrieffer developed the BCS theory, which provided a microscopic explanation of superconductivity and further elaborated on the nature of supercurrent density. The understanding of supercurrent density has evolved over the years, leading to significant advancements in the field of superconductivity, including the discovery of high-temperature superconductors and the development of applications such as superconducting magnets and quantum computing.