We studied the temperature and magnetic field dependence of vortex dissipation and critical current in the mixed-state of unconventional superconducting alloys Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ ($0.044 leq x leq 0.100$) through current-voltage measurements. Our results reveal that all the electric field $E$ vs current density $j$ curves in the Ohmic regime merge to one point ($j_0,E_0$) and that there is a simple relationship between the critical current density $j_c$ and flux-flow resistivity $rho_{rm ff}$: $rho_{rm ff}/rho_{rm n} = (1- j_{c}/j_{0})^{-1}$, where $rho_{rm n}=E_0/j_0$ is the normal-state resistivity just above the superconducting transition. In addition, $E_0$ is positive for all five dopings, reflecting the abnormal behavior of the flux-flow resistivity $rho_{rm ff}$: it increases with decreasing magnetic field. In contrast, $E_0$ is negative for the conventional superconductor Nb since, as expected, $rho_{rm ff}$ decreases with decreasing magnetic field. Furthermore, in the under-doped and over-doped single crystals of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, the parameter $E_0$ remains temperature independent, while it decreases with increasing temperature for the single crystals around optimal doping ($ 0.060leq xleq 0.072 $). This result points to the co-existence of superconductivity with some other phase around optimal doping.
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