  Relying on our theoretical approach for the superconducting critical state
problem in 3D magnetic field configurations, we present an exhaustive analysis
of the electrodynamic response for the so-called longitudinal transport problem
in the slab geometry. A wide set of experimental conditions have been
considered, including modulation of the applied magnetic field either
perpendicular or parallel (longitudinal) to the transport current density. The
main objective of our work was to characterize the role of the macroscopic
material law that should properly account for the underlying mechanisms of flux
cutting and depinning. The intriguing occurrence of negative current patterns
and the enhancement of the transport current flow along the center of the
superconducting sample are reproduced as a straightforward consequence of the
magnetically induced internal anisotropy. Moreover, we show that related to a
maximal projection of the current density vector onto the local magnetic field,
a maximal transport current density occurs somewhere within the sample. The
elusive measurement of the flux cutting threshold (critical value of such
parallel component $J_{c ||}$) is suggested on the basis of local measurements
of the transport current density. Finally, we show that a high correlation
exists between the evolution of the transport current density and the
appearance of paramagnetic peak structures in terms of the applied longitudinal
magnetic field.
