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Different dielectric sensors for measuring the liquid water content of snow are compared and described in detail. The instruments make use of the significant difference in the dielectric properties of ice and liquid water at radio frequencies; they are operated with frequencies ranging from 1 MHz up to 1.3 GHz. Plate condensers in connection with ac bridges are used as sensors in the frequency range up to 100 MHz whereas open resonators are used in the GHz regime. Test measurements with the different sensors on homogeneous samples like dry sand and mixed and prepared snow showed the same results for the dielectric constant: the discrepancies are less than 1%. In the natural, inhomogeneous snow cover, the special properties of the different sensors appear. Snow wetness is calculated from the measured dielectric constant and the snow density using the model of Polder and van Santen. The comparative field measurements were made with Alpine snow in the Stubai Alps in Austria.

The best analytical formulae for the self-inductance of rectangular coils of circular cross section available in the literature were derived from formulae for the partial inductance of straight wires, which, in turn, are based on the well-known formula for the mutual inductance of parallel current filaments, and on the exact value of the geometric mean distance (GMD) for integrating the mutual inductance formula over the cross section of the wire. But in this way, only one term of the mutual inductance formula is integrated, whereas it contains also other terms. In the formulae found in the literature, these other terms are either completely neglected, or their integral is only coarsely approximated. We prove that these other terms can be accurately integrated by using the arithmetic mean distance (AMD) and the arithmetic mean square distance (AMSD) of the wire cross section. We present general formulae for the partial and mutual inductance of straight wires of any cross section and for any frequency based on the use of the GMD, AMD, and AMSD. Since partial inductance of single wires cannot be measured, the errors of the analytical approximations are computed with the help of exact computations of the six-dimensional integral defining induction. These are obtained by means of a coordinate transformation that reduces the six-dimensional integral to a three-dimensional one, which is then solved numerically. We give examples of an application of our analytical formulae to the calculation of the inductance of short-circuited two-wire lines. The new formulae show a substantial improvement in accuracy for short wires.

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