where E_t and E_n are the tangential and normal components of the electric field respectively.

Let us assume that the closed path is very small so that over the elemental path length the variation of E can be neglected. Moreover very near to the interface, . Therefore

.......................(2.80)

Thus, we have,

or i.e. the tangential component of an electric field is continuous across the interface.

For relating the flux density vectors on two sides of the interface we apply Gauss’s law to a small pillbox volume as shown in the figure. Once again as , we can write

..................(2.81a)

i.e., .................................................(2.81b)