Boundary Conditions for Current Density

Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.

However, the tangential components of the current density are discontinuous across the interface.

Consider an interface separated by two conducting media with conductivities σ₁ and σ₂. The steady current density at the interface is , in medium 1 at a point P₁. It makes an angle α₁ with the normal. The current density  at point P₂ in medium 2 makes an angle α₂ with the normal.

The normal and tangential components of the current density give the equations as follows:

Taking the ratio of the two equations, the expression for the electrical conductivities in both the media is obtained.

If the electrical conductivity of medium 1 is greater than that of medium 2, the angle α₂ approaches zero. This implies that the current density  is normal to the surface of conductor 1.