Dropping the displacement current in Maxwell’s equations with conductivity term yields the so-called eddy current problem. The resulting parabolic system has better regularity properties, which can for example be utilized for dealing with nonlinearities. The nonlinearities enter the discussion in this paper via the Nimitzky type magnetic material relation \(B=B(H)\). The transmission problem of a conducting region \(\Omega_{c}\) embedded in a non-conducting complement \(\Omega_{nc}\) with known current density \(J_{s}\) in \(\mathbb{R}^{3}\) is considered. The weak problem is solved by approximation using the horizontal line method (Rothe’s method) and showing convergence to the weak solution. The approach uses a Helmholtz type decomposition of the magnetic field \(H\) involving a potential called \(\psi\) (referred to as \(H\)-\(\psi\) formulation of the eddy current problem in the title). Particular attention is given to the case of layered materials. Specifically silicon steel laminates are considered (ignoring the silicon insulation layers).