Editing Josephson effect (section)

== Josephson inductance ==
When the current and Josephson phase varies over time, the voltage drop across the junction will also vary accordingly; As shown in derivation below, the Josephson relations determine that this behavior can be modeled by a [[kinetic inductance]] named Josephson Inductance.<ref>{{cite arXiv |eprint=cond-mat/0411174 |first1=M. |last1=Devoret |first2=A. |last2=Wallraff |title=Superconducting Qubits: A Short Review |date=2004 |last3=Martinis |first3=J.}}</ref>

Rewrite the Josephson relations as:

:<math>
\begin{align}
\frac{\partial I}{\partial \varphi} &= I_c\cos\varphi,\\
\frac{\partial \varphi}{\partial t} &= \frac{2\pi}{\Phi_0}V.
\end{align}
</math>

Now, apply the [[chain rule]] to calculate the time derivative of the current:

:<math>
  \frac{\partial I}{\partial t} = \frac{\partial I}{\partial \varphi}\frac{\partial \varphi}{\partial t}=I_c\cos\varphi\cdot\frac{2\pi}{\Phi_0}V,
</math>

Rearrange the above result in the form of the [[current–voltage characteristic]] of an inductor:

:<math>
  V = \frac{\Phi_0}{2\pi I_c\cos\varphi} \frac{\partial I}{\partial t}=L(\varphi)\frac{\partial I}{\partial t}.
</math>

This gives the expression for the kinetic inductance as a function of the Josephson Phase:

:<math>
L(\varphi) = \frac{\Phi_0}{2\pi I_c\cos\varphi} = \frac{L_J}{\cos\varphi}.
</math>

Here, <math> L_J=L(0)=\frac{\Phi_0}{2\pi I_c} </math> is a characteristic parameter of the Josephson junction, named the Josephson Inductance.

Note that although the kinetic behavior of the Josephson junction is similar to that of an inductor, there is no associated magnetic field. This behaviour is derived from the kinetic energy of the charge carriers, instead of the energy in a magnetic field.