Magnetometer
Magnetic fields are vector quantities characterized by both strength and direction. The strength of a magnetic field is measured in units of tesla in the SI units, and in gauss in the cgs system of units. 10,000 gauss are equal to one tesla. Measurements of the Earth's magnetic field are often quoted in units of nanotesla (nT), also called a gamma. The Earth's magnetic field can vary from 20,000 to 80,000 nT depending on location, fluctuations in the Earth's magnetic field are on the order of 100 nT, and magnetic field variations due to magnetic anomalies can be in the picotesla (pT) range. "Gaussmeters" and "teslameters" are magnetometers that measure in units of gauss or tesla, respectively. In some contexts, magnetometer is the term used for an instrument that measures fields of less than 1 millitesla (mT) and gaussmeter is used for those measuring greater than 1 mT.
Lorentz force velocimetry
The secondary currents create a magnetic field formula_10, the secondary magnetic field. The interaction of the primary electric current with the secondary magnetic field gives rise to the Lorentz force on the magnet system
formula_11
The reciprocity principle for the Lorentz force velocimetry states that the electromagnetic forces on the fluid and on the magnet system have the same magnitude and act in opposite direction, namely
The general scaling law that relates the measured force to the unknown velocity can be derived with reference to the simplified situation shown in Fig. 2. Here a small permanent magnet with dipole moment formula_13 is located at a distance formula_14 above a semi-infinite fluid moving with uniform velocity formula_4 parallel to its free surface.
The analysis that leads to the scaling relation can be made quantitative by assuming that the magnet is a point dipole with dipole moment formula_16 whose magnetic field is given by
where formula_18 and formula_19. Assuming a velocity field formula_20 for formula_21, the eddy currents can be computed from Ohm's law for a moving electrically conducting fluid
subject to the boundary conditions formula_23 at formula_24 and formula_25 as formula_26. First, the scalar electric potential is obtained as
from which the electric current density is readily calculated. They are indeed horizontal. Once they are known, the Biot–Savart law can be used to compute the secondary magnetic field formula_28. Finally, the force is given by
where the gradient of formula_30 has to be evaluated at the location of the dipole. For the problem at hand all these steps can be carried out analytically without any approximation leading to the result
This provides us with the estimate