= DC Josephson effect
{c}
{title2=$V=0,\quad |I_s|\leq I_c$}
At zero <electric potential difference>, $\dot\phi=0$ permits a nonzero constant <quantum phase> and hence a steady pair <electric current>. Under an imposed <electric current> $I_b$ with $|I_b|<I_c$, a static solution satisfies $\sin\phi=I_b/I_c$. The low-energy branch has $\cos\phi>0$. For example, a stipulated $I_b=I_c/2$ gives $\phi=\pi/6$ modulo $2\pi$; it does not require a voltage drop. In the ideal current-biased <energy> $-E_J\cos\phi-(\hbar I_b/2e)\phi$, the positive second <derivative> on that branch explains its local stability.
The <superconducting critical current> $I_c$ is a maximum available static pair <electric current>, not the inevitable <electric current> under a voltage bias. At $V=0$ the initial state and circuit select a phase and a current. Above $I_c$ this sinusoidal model has no static phase solution, but its two relations alone do not predict the complete resistive <current-voltage characteristic>, switching dynamics or hysteresis. Those require the circuit and dissipative channels.
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