Theory of electrical noise induced in a wire loop by the thermal motions of ions in solution

Continuous Markov process theory is used to model the electrical noise induced in a passive wire loop by the thermal motions of ions in a nearby solution. The ions, being charged particles in Brownian motion, generate a fluctuating magnetic field, and that in turn induces a fluctuating electromotive force (emf) that augments the loop's Johnson emf. It is shown that the spectral density function of the equilibrium current in the wire loop is thereby increased, for moderate cycle frequencies ν, by approximately a factor (1+αν2), where α is determined by the geometry of the system, the resistance of the loop, and the charges, diffusion coefficients, and concentrations of the solution ions. It is also shown that the temporal trajectory of the loop current becomes "thickened," in a randomly fuzzy way, by an approximate factor of the form (1+β)1/2, where β depends not only on the aforementioned parameters that determine α, but also on the hydrated masses of the ions. These findings may be useful for estimating the intrinsic background noise in the detector coil of a medical magnetic resonance imaging machine, or any other sensitive electronic circuit that is required to operate in an immediate "salt water" environment.