The measurement of electrical resistance in terms of a mutual inductance and a period
H Redmayne Nettleton, F H Llewellyn · Proceedings of the Physical Society · 1932
It is shown that if the ratio α/ a of the radii of the concentric circles forming a simple inductometer is 0.506078, the mutual inductance is so accurately proportional to the angle through which the turning coil is displaced that the rising deviation from a linear law is less than 5 parts in a million at 7° of deflection. If the outer circle is replaced by two twin circles which are separated symmetrically with regard to the turning coil, the ratio α/ a must be increased to bring about a similar approach to linearity. If a heavy dynamometer is constructed on this principle, then, over the range indicated, the deflecting couple is strictly proportional to the current-products. Moreover if such a dynamometer be damped electrically, the damping-couple is accurately proportional to the velocity and, over a range of oscillation of 14° we have a remarkably close approach to true damped simple harmonic motion. It is shown that the resistance of the oscillator circuit is very accurately given by the expression R = ( CM / c )/[λ T /(π 2 + λ 2 ) - λ 0 T 0 /(π 2 + λ 0 2 )] where λ is the logarithmic decrement and T the period on closed circuit of the oscillator when a current C traverses the field coils, while λ 0 and T 0 correspond to open circuit and M is the mutual inductance between the oscillating coil and the field coils when the former is in the position of steady deflection due to the passage through it of a steady current c , in conjunction with the current C through the field coils. This expression eliminates quantities difficult to measure in Weber's method. Experimental tests are described. Table of Legendre and other functions, which facilitate the calculation of mutual inductances of the nature here considered, are appended.