= Solution
A <Josephson junction> weakly couples two superconducting condensates. Its coupling <energy> is $-E_J\cos\varphi$, with a gauge-invariant phase difference across the junction. Choose current and voltage orientations consistently so that
$$
\boxed{I=I_c\sin\varphi,\qquad E_J=\frac{\hbar I_c}{2e},\qquad
\dot\varphi=\frac{2eV}{\hbar}.}
$$
The <DC Josephson effect> permits a stationary zero-voltage supercurrent up to $I_c$. A constant nonzero voltage produces the <AC Josephson effect>, with <frequency> $f=2e|V|/h$. Resistive and capacitive currents are additional circuit contributions, not part of the ideal sinusoidal supercurrent alone.
In a symmetric <DC SQUID> with negligible loop <self-inductance>, the two junction phases differ by $2\pi\Phi/\Phi_0$. Adding their currents and maximizing over the common phase gives
$$
\boxed{I_{c,\rm SQUID}(\Phi)=2I_c\left|\cos\frac{\pi\Phi}{\Phi_0}\right|.}
$$
Flux changes therefore modulate the critical current or a suitably biased output voltage. This converts phase <interference> into sensitive flux measurement. Finite loop <inductance>, unequal junctions, noise and readout conditions alter the simple modulation; the loop flux then includes screening currents. A <SQUID> is useful in magnetometry, but achieving sensitivity requires both a suitable circuit and controlled environmental noise.
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