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Boundary Conditions for Electrostatic Fields

As you study electromagnetics, you will probably find a lot of focus is on the boundary conditions of problems. This is because often by knowing how the fields behave at the boundary of a particular problem, you can often either calculate the fields everywhere(for example, applying Laplace's equation or Poisson's equation) or you can have an intuitive understanding of the behavior of the system as a whole.

Rather than repeat the derivation of the boundary conditions, which involves using Gauss's law to find the normal component and a line integral to find the tangential component of the electric field at the boundary of two surfaces, I will simply state the boundary conditions and try to explain how they are useful. If you would like to see the derivation, look in any good physics book or electromagnetics text.

Tangential Components

The tangential component of E across an interface is constant That is:

E   =   E
  t1       t2

Normal Components

The normal component of D across an interface is given by:

a    *  (D  -  D ) = density of free charges at surface
  n2       1      2

or , if there are no free charges at the surface

D   =   D
  n1      n2

Since D =(epsilon)E=(epsilon0)E + P, where P is the polarization vector, E is the electric field intensity, and D is the electric flux density(or electric dispalcement), if we are given a field distribution in one medium, we can calcuate how the second medium reacts to the same field by applying these relations across the boundary.

What may seem like insurmountable problems involving the response of a complicated system containing many different types of material can then be broken down into managable chunks. If you know how the fields behave in one area, you can determine how other areas behave by applying boundary conditions.

In addition, knowing how materials behave at boundaries allows you to intuitively solve some problems. For example, suppose you have a flat conducting plate with some surface charge sealed inside a dielectric material, such as part of a power plane in a printed circuit board. If you want to know how the fields act very close to the board and far away from the board , you can apply boundary conditions.

Viewing the conducting plate as a perfect conductor, you know that there will be no tangential component to the electric field. You can then calculate how this electric field will change in the different dielectric materials by applying the boundary condition for the normal component of the electric field.

The electric fields very near the surface of the board will then be some scaled version of the original electric field directed perpendicular to the plane of the board. As we go larger distances from the board, the finite distribution of charge looks more and more like a point charge. This simple, intuitive discription of the situation from boundary conditions allows you to get a basic understanding of how this component would interact with a larger system. For elements very near the printed circuit board (for example, on the surface of the board) the power plane acts as a plane of charge. Larger distances away (for example, other circuit boards in a large cabinet)the powet plane acts more like a point charge.

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