# Boundary Value Conditions in Electrostatics

In electrostatics, which deals with stationary electric charges and their interactions, electric boundary conditions describe the behavior of the electric field at the interfaces between different media or regions. The two main types of boundary conditions are the tangential electric field continuity and the normal electric flux density continuity.

## Tangential Electric Field Continuity:

At the interface between two different materials or regions, the tangential component of the electric field must be continuous. Mathematically, this is expressed as:

Et1= Et2

Here, Et1 and ​ Et2 are the tangential components of the electric field in regions 1 and 2, respectively. This condition ensures that there is no abrupt change in the tangential electric field across the interface.

## Normal Electric Flux Density Continuity:

The normal component of the electric flux density (D) must also be continuous across the boundary. Mathematically, this is expressed as:

Dn1​=Dn2

Here, Dn1 ​and Dn2​ are the normal components of the electric flux density in regions 1 and 2, respectively. The electric flux density is related to the electric field by the equation D=εE, where ε is the permittivity of the material. This condition ensures that the normal displacement of electric flux is continuous across the boundary.

These boundary conditions are essential for solving electrostatic problems involving different materials or regions. They help in establishing a smooth transition of the electric field across interfaces and are derived from the fundamental laws of electrostatics, such as Gauss’s law and the conservation of charge.

## Numerical Example

The xy-plane is a charge free boundary separating two dielectric media with the permittivities ε1 and  ε2. If the electric field in medium 1 is E1xax+ E1yay+ E1zaz. Find the Electric Field E2 in medium 2.

With reference to the figure 3.19, find E1 if E2=4ax-3ay+3az (V/m), where ε1=2 ε0 and ε2=8 ε0. Assume the boundary to be charge free.