Assume a voltage source connected to a capacitor which filled with a dielectric material having non zero conductivity, then, displacement current=conduction current of the wire=CdV/dt while conduction current density of the dielectric=σE=σV/d But considering dc source, displacement current=conduction current of the wire=0, while conduction current density of the dielectric=σV/d and not equals to zero Why there will be a contradiction?
Welcome to the PF. The DC leakage current is continuous through the wire and the dielectric. If there is leakage current in the capacitor, it has to be supplied by the wire.
So the equation I=CdV/dt has not considered the dc leakage? the actual conduction current in the wire is=∫σEdS? How about at other frequency? Is the actual conduction current inside the wire equals to CdV/dt+∫σEdS(σ is the conductivity of the dielectric) but not just CdV/dt?
A practical capacitor can be modelled as an ideal capacitor with two resistances, one in series and the other in parallel.