Uploaded by cfurse on 21.10.2009

Transcript:

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.

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.