Brownian fluctuations in galvanometers and galvanometer amplifiers

Abstract
The Brownian fluctuations in moving-coil galvanometers and galvanometer amplifiers have been investigated theoretically and experimentally. Previous work is summarized and a simple treatment given of the fluctuations of a galvanometer with negligible inductance in its circuit; here, as elsewhere in the paper, the correlation function of the random force is used, not the frequency spectrum. Both molecular bombardment of the suspended mirror and Johnson noise in the circuit resistance are considered, and a comment is made on why the coexistence of these two effects does not increase the r.m.s. deflexion above its equipartition value. It is further shown by an exact random force calculation that the presence of an appreciable inductance in the galvanometer circuit does not change the r.m.s. values of deflexion and angular velocity. The same result is obtained by a statistical mechanical argument based on the assumptions already implicit in the application of the equipartition principle to, say, a suspended mirror. Expressions are obtained for the magnitude and correlation function of the Brownian fluctuations of a galvanometer amplifier with two galvanometers of arbitrary periods and damping, these expressions are in terms of the periods and damping constants of the two galvanometers, the mechanical damping of the primary galvanometer and the resistance of the primary circuit. A method of finding the magnitude and correlation function of the fluctuations from a record of the throw of the secondary galvanometer consequent on passing a known charge through the primary is suggested. The magnitude of the Brownian fluctuations in a galvanometer amplifier was determined experimentally. The effects of external disturbances were reduced till the changes in zero, including overall drift, caused by them in 30 min. (say 1000 times the response time) were much less than the r.m.s. Brownian deflexion. The methods used to determine the quantities in the theoretical expression for the r.m.s. Brownian deflexion are described. Various conditions of damping were used and the average ratio of experimental to theoretical r.m.s. deflexions was found to be 100-0% with ± 1.1 % standard error. , An appendix describes the verification for the case of the galvanometer amplifier of Rice's formulae for the number of zeros and number of points of zero slope on the record of a one-dimensional Gaussian random process.

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