Calculation of magnetic forces and torques on the Kibble coil

Analytically the force acting on a current-carrying coil in a magnetic field can be calculated in two ways. First, a line integral can be conducted along the coil's wire, summing up the differential force contributions. Each contribution results from a cross-product of the corresponding differential line segment with the magnetic flux density at that location. Alternatively, the coil's energy in the field is given as a product of three factors, the number of turns, the current, and the flux threading the coil. The energy can then be obtained by executing a surface integral over the coil's open surface using the scalar product of the differential surface element with the magnetic flux density as its integrand. The force on the coil is the negative derivative of the energy with respect to the appropriate coordinate. For yoke-based Kibble balances, the latter method is much simpler since most of the flux is contained in the inner yoke of the magnet and can be written as a simple equation. Here, we use this method to provide simple equations and their results for finding the torques and forces that act on a coil in a yoke-based magnet system. We further introduce a straightforward method that allows the calculation of the position and orientation difference between the coil and the magnet from three measurements.