The time dependent transient lateral photovoltaic effect (T-LPE) has been studied with microsecond time resolution and with chopping frequencies in the kHz range, in lithographically patterned 21 nm thick, 5, 10 and 20 micron wide and 1500 micron long Co lines grown over naturally passivated p-type Si (100). We have observed a nearly linear dependence of the LPE transient response with the laser spot position. An unusual T-LPE dynamic response with a sign change in the laser-off stage has also been corroborated by numerical simulations. A qualitative explanation suggests a modification of the drift-diffusion model by including the influence of a local inductance. In addition, influence of anisotropic magnetoresistance of the Co line structure on dynamic response on T-LPE has been investigated. Specifically, we have experimentally investigated influence of the direction of the external magnetic field respect to the drift velocity of the photogenerated carriers on the T-LPE. We have observed notable dependence of the T-LPE on the magnetic field in the small field range (below 100 Oe), compatible with anisotropic magnetoresistance values. The strong influence of the magnetization alignment on the dynamic response of photogenerated carriers has been also observed through a phase sensitive lock-in experiment. These findings indicate that the microstructuring of the ferromagnetic line based position sensitive detectors (PSD) could improve their space-time resolution and add capability of magnetic field tuning of the main PSD characteristics.