Permittivity and Conductivity Measured Using a Novel Toroidal Split-Ring Resonator

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.