  Analytically the forces and torques on a coil in a field of magnetic flux
density can be calculated one of two ways. The line integral can be conducted
along the coil's wire, summing up the differential force contribution. For each
differential line segment, the force is obtained as a cross product with the
magnetic flux density. Alternatively, the coil's energy in the field is the
current times the flux threading the coil. Hence, the energy is obtained by
executing a surface integral over the coil's open surface. Here, a dot product
of the differential surface element with the magnetic flux density is executed
under the integral sign. The forces and torques can then be obtained from 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 simple method that allows the calculation
of the position and orientation difference between the coil and the magnet from
three measurements.
