  Vortex penetration affects the stability of a superconducting system and
limits the possible application of the system. However, the mathematical
description to this phenomenon is currently unavailable. Here I present a
mathematical model in which I consider the effects of bulk pinning and internal
field repulsive force on vortex hopping. Thereafter, I proposed a series
expansion to the activation energy and derived a general formula for describing
the time dependence of the vortex penetration process. With these formulas, I
can analyze the experimental data and calculate the activation energy of the
vortex penetration phenomenon. The results are accurate for the time dependence
of the internal field measurements in a $Bi_2Sr_2CaCu_2O_{8+x}$ superconductor.
