  We describe and demonstrate the use of a novel toroidal split-ring resonator
to make accurate and precise measurements of the permittivity of liquids and
gases. We first analytically show how the resonance frequency and quality
factor of the resonator are related to the complex permittivity of the material
filling its gap. We then use the resonator to experimentally determine the
permittivity of a number of different materials. First, the compact and high-Q
resonator is used to measure both the real and imaginary parts of the complex
permittivity of methyl alcohol at 185 MHz. Second, the resonator was placed
inside a vacuum-tight Dewar. We measured the resonance frequency with the
resonator suspended in vacuum and then immersed in an atmosphere of air. From
these data, the dielectric constant of air was accurately determined. Next, the
resonator was submerged in liquid nitrogen and the boiling temperature of the
nitrogen bath was manipulated by regulating its vapor pressure. This system
allowed for a precise measurement of the dielectric constant of liquid nitrogen
over a temperature range of 64 to 77 K. Finally, we monitored the quality
factor of the copper resonator as its temperature drifted from 80 K to room
temperature. From these data, we extracted the linear temperature dependence of
copper's resistivity.
