ECE3300 Lecture 21-1 Boundary Conditions

Welcome to ECE 3300 at the University of Utah. In Lecture No. 21,
we're going to talk about the electric and magnetic field boundary
conditions. The boundary conditions are used when you know the
field in one location, in one region, and you want to find it in another region.
Let's first review the names of things we're going to be using in
today's lecture. The electric field is E. The magnetic field is
H. Electric flux density is D. Magnetic flux density is B.
Permittivity and permeability are shown as here. And we can see
these two constitutive relationships that tell us how E and D
relate and how B and H relate. Now as we said the boundary
conditions are used in order to find how the field in one region
changes to a field in another region. Let's say, for instance,
that we have region number one, which is air, and region number
two, which is water. There are two electric field boundary
conditions: One for the tangential electric field and one for the
normal electric field. The tangential electric field equation
boundary condition says that the two electric field -- the
tangential components of the electric field are equal in region
number one and region number two. Is my picture drawn to scale?
That would say that E 2 T is equal to E 1 T. The tangential
fields are anything that is parallel to the surface. Now, I can't
say that this figure is exactly to scale, because E 1 T should be
equal to E 2 T. Now let's look at the normal components. The
normal components are controlled by the electric, by the charged
density. The charged density is the coulombs per meter squared
along the surface between the air and the water. The only time
that we can have a charged density is when we have some conductive
material. Is water conductive? As long as it is not pure
distilled water, it is indeed. So remember that row S is equal to
0 if the conductivity is equal to 0. So that's one thing that
will help you in these problems. But assuming that we have a
slightly conductive material such as water, we could have a
surface charged density given in coulombs per meter. Then we
would find the normal component of the electric flux density using
this equation right here. Now, we have one equation for the
electric field and the other for the electric flux density. In
order to get their other equation, remember to just use the
constitutive equation which says that D is equal to epsilon E. So
our first equation becomes the first equation shown here and our
second equation becomes the second equation that is shown. Now,
those are all that we need in order to find the electric field
from boundary conditions. What you do is you take the incoming
electric field, that would be E 2 in this case, and you break it
into its tangential components shown in red and its normal
components shown in blue. And you apply either the first equation
in either of its form and the second equation in either of its
forms. Now let's look at the magnetic field. Similarly, the
magnetic field will have a boundary condition on its tangential
component and another boundary condition on its normal component.
The tangential, the tangential magnetic field is equal in both
cases as shown here. Again, is this one drawn to scale? No,
because H 2 T should be equal to H 1 T. The normal magnetic flux
density is what is equal across this boundary. So B 1 N is equal
to B 2 N. Then we use our equations and we can see that the first
equation becomes the first equation, and the second equation
becomes the second equation. This way if we know the magnetic
field or the magnetic flux density in one of the regions, we can
find the magnetic field or flux density in the other.