The level of liquid helium (temperature \(\approx\) 4 K ) in its storage tank
can be monitored using a vertically aligned niobium-titanium (NbTi) wire,
whose length \(\ell\) spans the height of the tank. In this level-sensing setup,
an electronic circuit maintains a constant electrical current \(I\) at all times
in the NbTi wire and a voltmeter monitors the voltage difference \(V\) across
this wire. Since the superconducting critical temperature for NbTi is 10 K,
the portion of the wire immersed in the liquid helium is in the
superconducting state, while the portion above the liquid (in helium vapor
with temperature above 10 K) is in the normal state. Define \(f = x/\ell\) to be
the fraction of the tank filled with liquid helium (Fig. 18-38) and \(V_0\) to
be the value of V when the tank is empty (\(f = 0\)). Determine the relation
between \(f\) and \(V\) (in terms of \(V_0\)).