  A novel boundary element formulation for solving problems involving eddy
currents in the thin skin depth approximation is developed. It is assumed that
the time-harmonic magnetic field outside the scatterers can be described using
the quasistatic approximation. A two-term asymptotic expansion with respect to
a small parameter characterizing the skin depth is derived for the magnetic and
electric fields outside and inside the scatterer, which can be extended to
higher order terms if needed. The introduction of a special surface operator
(the inverse surface gradient) allows the reduction of the problem complexity.
A method to compute this operator is developed. The obtained formulation
operates only with scalar quantities and requires computation of surface
operators that are usual for boundary element (method of moments) solutions to
the Laplace equation. The formulation can be accelerated using the fast
multipole method. The method is much faster than solving the vector Maxwell
equations. The obtained solutions are compared with the Mie solution for
scattering from a sphere and the error of the solution is studied. Computations
for much more complex shapes of different topologies, including for magnetic
and electric field cages used in testing are also performed and discussed.
