On the Utilization of Magnetic Vector Potential for a Description of a Superconducting Transmission Line

We will present and examine an alternative approach to describe the behavior of superconducting transmission lines, and possibly weak link Josephson junction using the magnetic vector potential
$(bf A)$
. While the utilization of magnetic vector potential is known in this field since the inception, a device level formulation based on
$bf A$
has not been fully investigated. We will show that for device level formulation, the magnetic vector potential is a mathematically simpler quantity to deal with than the combination of magnetic flux density
$(bf B)$
and electric field intensity
$(bf E)$
and still contains all the electromagnetic information of the junction under construction. In addition, other benefits of this formalism arise when dealing with Lagrangian and Hamiltonian mechanics. In this paper, we present a detailed description of how
$bf A$
can be determined for an infinite weak link Josephson junction with planar geometry. Utilizing the magnetic vector potential formulation, both electric and magnetic fields are calculated, and we show that we obtain the same dispersion relation as other approaches have previously demonstrated. We then discuss the advantages of this formalism. In addition, as an application of the present approach, we solve the Josephson junction's nonstationary equation numerically to get a realization of the actual coupling that occurs across the junction.