President UniversityErwin SitompulEEM 9/1 Dr.-Ing. Erwin Sitompul President University Lecture 9...

President University Erwin Sitompul EEM 9/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 9

Engineering Electromagnetics

http://zitompul.wordpress.com


Boundary Conditions for Perfect Dielectric MaterialsConsider the interface between two dielectrics having

permittivities ε1 and ε2, as shown below.

Chapter 6 Dielectrics and Capacitance

We first examine the tangential components around the small closed path on the left, with Δw<< :

0d E Ltan1 tan 2 0E w E w

tan1 tan 2E E


Boundary Conditions for Perfect Dielectric MaterialsThe tangential electric flux density is discontinuous,


tan1 tan 2tan1 tan 2

1 2

D DE E

tan1 1

tan 2 2

D

D

1 2N N SD S D S Q S

1 2N N SD D

The boundary conditions on the normal components are found by applying Gauss’s law to the small cylinder shown at the right of the previous figure (net tangential flux is zero).

• ρS cannot be a bound surface charge density because the polarization already counted in by using dielectric constant different from unity

• ρS cannot be a free surface charge density, for no free charge available in the perfect dielectrics we are considering

• ρS exists only in special cases where it is deliberately placed there


Boundary Conditions for Perfect Dielectric MaterialsExcept for this special case, we may assume ρS is zero on the

interface:


1 2N ND D

1 1 2 2N NE E

The normal component of electric flux density is continuous. It follows that:


Boundary Conditions for Perfect Dielectric MaterialsCombining the normal and the tangential

components of D,

1 1 1 2 2 2cos cosN ND D D D

tan1 1 1 1

tan 2 2 2 2

sin

sin

D D

D D


2 1 1 1 2 2sin sinD D

1 1

2 2

tan

tan

After one division,1 2 1 2


Boundary Conditions for Perfect Dielectric MaterialsThe direction of E on each side of

the boundary is identical with the direction of D, because D = εE.


tan1 tan 2E E1 1 2 2N NE E

1 2 1 2

2E

1E


Boundary Conditions for Perfect Dielectric MaterialsThe relationship between D1 and D2 may be found from:


2

2 222 1 1 1

1

cos sinD D

2

2 212 1 1 1

2

sin cosE E

The relationship between E1 and E2 may be found from:


Boundary Conditions for Perfect Dielectric MaterialsChapter 6 Dielectrics and Capacitance

out 0 xEE a

out 0 0 xED a

out 0P

ExampleComplete the previous example by finding the fields within the Teflon.

in out 0 0 xE D D a

in 0 in1.1P E

inin

r 0 D

E 0 0

r 0

xE

a

00.476 xE a

0 00

r 0

1.1 xE

a

0 00.524 xE a

• E only has normal component


Boundary Conditions Between a Conductor and a Dielectric

The boundary conditions existing at the interface between a conductor and a dielectric are much simpler than those previously discussed.

First, we know that D and E are both zero inside the conductor.Second, the tangential E and D components must both be zero

to satisfy:


0d E LD E

Finally, the application of Gauss’s law shows once more that both D and E are normal to the conductor surface and that DN = ρS and EN = ρS/ε.

The boundary conditions for conductor–free space are valid also for conductor–dielectric boundary, with ε0 replaced by ε.

0t tD E

N N SD E


We will now spend a moment to examine one phenomena: “Any charge that is introduced internally within a conducting material will arrive at the surface as a surface charge.”

Given Ohm’s law and the continuity equation (free charges only):



J E

v

t

J

We have:v

t

E

v

t

D


If we assume that the medium is homogenous, so that σ and ε are not functions of position, we will have:



v

t

D

Using Maxwell’s first equation, we obtain;

vv t

Making the rough assumption that σ is not a function of ρv, it

leads to an easy solution that at least permits us to compare different conductors.

The solution of the above equation is:( )

0t

v e • ρ0 is the charge density at t = 0• Exponential decay with time constant of ε/σ


Good conductors have low time constant. This means that the charge density within a good conductors will decay rapidly.

We may then safely consider the charge density to be zero within a good conductor.

In reality, no dielectric material is without some few free electrons (the charge density is thus not completely zero), but the charge introduced internally in any of them will eventually reach the surface.



( )0

tv e

t

ρv

ρ0

ρ0/e

ε/σ


HomeworkNo homework this week.


President UniversityErwin SitompulEEM 9/1 Dr.-Ing. Erwin Sitompul President University Lecture 9...

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