  We discuss the generation of the directed flow $v_1(p_T,y_z)$ induced by the
electromagnetic field as a function of $p_T$ and $y_z$. Despite the complex
dynamics of charged particles due to strong interactions generating several
anisotropies in the azimuthal angle, it is possible at $p_T > m$ to directly
correlate the splitting in $v_1$ of heavy quarks with different charges to some
main features of the magnetic field, and in particular its values at formation
and freeze-out time.
  We further found that the splitting $d\Delta v_1/dy_z|_{y_z=0}$ of positively
and negatively charged particles at high $p_T$ can be formulated as $d\Delta
v_1/dy_z|_{y_z=0}=-\alpha \frac{\partial \ln f}{\partial
p_T}+\frac{2\alpha-\beta}{p_T}$, where the constants $\alpha$ and $\beta$ are
constrained by the $y$ component of magnetic fields and the sign of $\alpha$ is
simply determined by the difference $\Delta[tB_y(t)]$ in the center of
colliding systems at the formation time of particles and at the time when
particles leave the effective range of electromagnetic fields or freeze out.
The formula is derived from general considerations and is confirmed by several
related numerical simulations; it supplies a useful guide to quantify the
effect of different magnetic field configurations and provides an evidence of
why the measurement of $\Delta v_1$ of charm, bottom and leptons from $Z^0$
decay and their correlations are a powerful probe of the initial e.m. fields in
ultra-relativistic collisions.
