Boundary Conditions for Electrostatic Fields
As you study electromagnetism, 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,
we will simply state the boundary
conditions and try to explain how they are useful.
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 displacement), if we are
given a field distribution in one medium, we can calculate 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 manageable 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 description 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 power plane acts more like a
point charge.