Finding correct RC value for transient equations in DC RC circuit
Assuming the switch closing in \$ t= 0 \, s\$ and \$v_c(0) = 0\, V\$, the Thevenin equivalent between upper capacitor lead and reference is:
$$ V_{th} = 3.2753\, V $$
$$ R_{th} = 2.6934\, k \Omega $$
So (and taking in account the comment from @G36):
$$ RC = 0.05386\, s $$
The expression of capacitor voltage for \$t\ge 0\$ is:
$$ v_c(t) = 3.2753(1-e^{-18.56t}) V $$
Knowing that
$$ i_c(t) = C\frac{dv_c(t)}{dt} $$
$$ i_c(t) = 1.22\times10^{-3}e^{-18.56t} A $$