Magnetic field screening in strong crossed electromagnetic fields

We consider crossed electric and a magnetic fields $\left(\vec{B}=B\,\hat{z},~\vec{E}=E\,\hat{y}\right)$, with $E/B<1$, in presence of some initial number of $e^{\pm}$ pairs. We do not discuss here the mechanism of generation of these initial pairs. The electric field accelerates the pairs to high-energies thereby radiating high-energy synchrotron photons. These photons interact with the magnetic field via magnetic pair production process (MPP), i.e. $\gamma+B\rightarrow e^{+}+e^{-}$, producing additional pairs. We here show that the motion of all the pairs around the magnetic field lines generates a current that induces a magnetic field that shields the initial one. For instance, for an initial number of pairs $N_{\pm,0}=10^{10}$, an initial magnetic field of $10^{12}$ G can be reduced of a few percent. The screening occurs in the short timescales $10^{-21}\leq t \leq 10^{-15}$ s, i.e. before the particle acceleration timescale equals the synchrotron cooling timescale. The present simplified model indicates the physical conditions leading to the screening of strong magnetic fields. To assess the occurrence of this phenomenon in specific astrophysical sources, e.g. pulsars or gamma-ray bursts, the model can be extended to evaluate different geometries of the electric and magnetic fields, quantum effects in overcritical fields, and specific mechanisms for the production, distribution, and multiplicity of the $e^{-}e^{+}$ pairs.