Boundary Conditions for Current Density

The current density becomes discontinuous across an interface having different electrical conductivities.

For steady currents, the divergence of the current density is zero. So, the normal component of the current density is continuous across the boundary.

Recall that the tangential component of the electric field is continuous across an interface. Expressing the electric field in terms of the current density and the electrical conductivity gives the boundary condition for the tangential component of the current density.

Now,  the normal component of the electric displacement is discontinuous across the interface. So, the normal component of the electric field is also discontinuous across an interface.

Again, using the expression of the electric field and current density, the boundary condition for the normal component of the current density is obtained in terms of permittivity and conductivity.

It shows that a surface charge density is created across an interface having  different conductivities and/or permittivities.

The surface charge density is zero for an interface with the same conductivity and permittivity values or with an equal ratio of permittivity to conductivity.