A Josephson junction consists of two superconductors separated by a non-superconducting layer, typically an insulator that is thinner than the Josephson penetration depth. We will explore this structure using the magnetic vector potential to describe the electromagnetic fields of a superconducting transmission line. However, we will revisit the beginnings of the Josephson junction and the conventional formulation techniques used to describe the electromagnetism of layered superconducting and Josephson structures. We will then formally derive the field equations, for a transverse magnetic to z mode, for a superconducting transmission line, and take an in depth look at what these electromagnetic field equations represent. We will then discuss the Sine-Gordon equation and its role in the description and solution of Josephson junctions. This equation governs the coupling between superconductors separated by an insulating barrier, and it is this coupling that Josephson predicted. We continue on this path by revisiting the same superconducting transmission line, but take into account tunneling through the barrier using the Josephson current in our solution method. Finally, we will investigate using the finite difference method in order to numerically solve for the electromagnetic fields in our superconducting transmission line.