  The resonant frequencies of three-dimensional microwave cavities are
explicitly dependent on the dielectric constant of the material filling the
cavity, making them an ideal system for probing material properties. In
particular, dielectric constant measurements allow one to extract the helium
density through the Clausius-Mossotti relation. By filling a cylindrical
aluminum cavity with superfluid helium, we make precision measurements of the
dielectric constant of liquid $^4$He at saturated vapor pressure for range of
temperatures 30 -- 300 mK and at pressures of 0-25.0 bar at 30 mK, essentially
the zero temperature limit for the properties of $^4$He. After reviewing
previous measurements, we find systematic discrepancy between low and high
frequency determination of the dielectric constant in the zero-temperature
limit and moderate discrepancy with previously reported values of
pressure-dependent density. Our precision measurements suggest 3D microwave
cavities are a promising choice for refining previously measured values in
helium, with potential applications in metrology.
