Show how to generalize Ampere's circuital law to include the term due to displacement current?

Explanation

According to Ampere's circuital law,
`underset(s)( oint )vec(B)cdot vec(dl) = mu_(0)`
As ther current flows across the area bounded by loop `S_(1)`,so
`underset(s_(2))(oint )vec(B)cdot vec(dl) = mu_(0)`.
But the area bounded by `S_(2)` lies in the region between the plates capacitor where no current flows across it.
`underset(s_(2))(oint) vec(B)cdot vec(dl) =0.`
Consider that loops enclosing S1 & S2 are infinitesimally close to each other. then
`underset(s_(1))(oint )vec(B)cdot vec(dl) = underset(s_(2))(oint )vec(B)cdot vec(dl)`
This equation is inconsistent with equation (2) & (3). to remove this maxwell said that a changing electric field (during charging ) between the capacitor plates must induce a magnetic field which in turn must be associated with current `I_(d)`.
`I_(d) = epsilon_(0)((d phi_(E))/(dt)) [ (d phi_(E))/(dt) " change in electric flux"]`
The total current must be
`I = I_("conduction ") + I_("displacement")`
`I_(c ) = epsilon_(0) (d phi_(E))/(dt)`
Hence the generalised from of Ampere's circuital law is
`oint vec(B)cdot vec(dl) = mu_(0) [ I_(C ) + epsilon_(0) (d phi_(E))/(dt) ]`