Gradient coils and reciprocity

It is observed that the problem of optimizing the design of gradient coils is identical to the problem of optimizing the design of vibrating sample magnetometer (VSM) pick-up coils. The problem of optimizing the design of the gradient coils is that of maximizing the appropriate field gradient at the position of the magnetic sample depending on the design of the alternating gradient magnetometer (AGM). The equivalence of the two problems may be proved directly by reciprocity. In the AGM case, electric current is supplied to the coil system, and it produces, with a certain coefficient of coupling, a mechanical force on the magnetic sample. In the VSM case, mechanical displacement is supplied to the magnetic sample, and it produces, with a certain coefficient of coupling, an electric voltage in the coil system. Since electrical voltage and current and mechanical force and displacement are the conjugate variables of electrical and mechanical systems, respectively, the theorem of reciprocity asserts that the two coupling coefficients must be identical.<
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