Prof. David R. JacksonECE Dept.

Fall 2014

Notes 20

ECE 2317 Applied Electricity and Magnetism

1

Dielectrics

Water

0 r

Single H2O molecule:

+

-

+

H+

O-

H+

2

Dielectrics (cont.)

Water

0 r

“Dipole” model: p p

p qd

“Vector dipole moment” of molecule p:

d

+q

-q

+

-

+

-

+

H+

O-

H+

3

Dielectrics (cont.)

4

The dipoles representing the water molecules are normally pointing in random directions.

+

+

+

-

+

-

+-

--

-+

Dielectrics (cont.)

5

The dipoles partially align under an applied electric field.

+ -V

x

+

+

+

+

-

-

-

-

Ex

+- +-

+-

+-

+- +-

Define: P = total dipole moment/volume

1i

V

P pV

0D E P

Define the electric flux density vector D:

V+

-

+-

Dielectrics (cont.)

We can also write this as

1 avei v

V

NP p N p

V N

pave = average vector dipole moment

Nv = number of molecules per unit volume

N molecules (dipoles) inside V

6

/vN N V

Linear material: 0 eP E

Define:

0 r

0 rD E Then we have

Note: e > 0 for most materials

Dielectrics (cont.)

The term e is called the

“electric susceptibility.”

0 0

0 1e

e

D E E

E

1r e

7

D Eor where

Typical Linear Materials

Teflon

Water

Styrofoam

Quartz

2.2

81

1.03

5

r

r

r

r

(a very polar molecule, fairly free to rotate)

8

Note: r > 1 for most materials.

1 , 0r e e

Note: A negative value of e would

mean that the dipoles align against

the field.

Exotic MaterialsArtificial “metamaterials” that have been designed that have

exotic permittivity and/or permeability performance.

9

Negative index metamaterial array configuration, which was constructed of copper split-ring resonators and wires

mounted on interlocking sheets of fiberglass circuit board. The total array consists of 3 by 20×20 unit cells with overall dimensions of

10×100×100 mm.

http://en.wikipedia.org/wiki/Mhttp://en.wikipedia.org/wiki/Metamaterialetamaterial

0

0r

r

(over a certain bandwidth of operation)

Exotic Materials (cont.)

10

Cloaking of objects is one area of research in metamaterials.

The Duke cloaking device masks an object from one wavelength at

microwaves.

Image from Dr. David R. Smith.

Dielectrics: Bound Charge

0vb

Assume (hypothetically) that Px increases as

x increases inside the material:

A net volume charge density is created inside the material, called the “bound charge density” or “polarization charge density.”

vb = “bound charge density”: it is bound to the molecules.

v = “free charge density”: this is charge you place inside the

material. You can freely place it wherever you want.

Note: For simplicity, the dipoles are shown perfectly aligned, with the number of aligned dipoles increasing with x.

11

+-

+-

+-

+-

+-

+-

+-

+-

+-

Dielectrics: Bound Charge (cont.)

totalv v vb

Total charge density inside the material:

0vb

This total charge density may be viewed as being in free space (since there is no material left after the molecules are removed).

0sb 0sb

12

Bound surface charge density

Bound charge densities

Note: The total charge is zero since the object is assumed to be neutral.

+-

+-

+-

+-

+-

+-

+-

+-

+-

Formulas for the two types of bound charge

13

Dielectrics: Bound Charge (cont.)

The derivation of these formulas is included in the Appendix.

vb P

ˆsb P n

, ,r x y z

- - - - - -

-- - -

-

++

+

-

-

-

+

+ n̂

The unit normal points outward from the dielectric.

0 0 1e rP E E

E = electric field inside material

Dielectrics: Gauss’s Law

0totalv v vbE

This equation is valid, but we prefer to have only the free-charge density in the equation, since this is what is known.

The goal is to calculate (and hopefully eliminate) the bound-charge density term on the right-hand side.

14

Inside a material:

The free-space form of Gauss’s law is

0totalv v vb vE P

Hence

vD

Dielectrics: Gauss’s Law (cont.)

0 vE P

or

This is why the definition of D is so convenient!15

Dielectrics: Gauss’s Law (Summary)

Important conclusion:

Gauss’ law works the same way inside a dielectric as it does in vacuum, with only the free charge density (i.e., the charge that is

actually placed inside the material) being used on the right-hand side.

vD

ˆ encl

S

D n dS Q

16

0 r

D E

ExamplePoint charge inside dielectric shell

Dielectric spherical shell with r

S

r

q

r

ab

Gauss’ law: ˆ encl

S

D n dS Q q

(The point charge q is the only free charge in the problem.)

17

Find D, E

Example (cont.)

22

ˆ C/m4

qD r

r

20

20

ˆ V/m ,4

ˆ V/m4 r

qE r r a r b

r

qE r a r b

r

18

S

r

q

r

ab

24rD r q

Hence

We then have

Example (cont.)

19

qr

Flux Plot

Note that there are less flux lines inside the dielectric region.

Homogeneous Dielectrics

0

0

00

00

00

1

1

vb

e

e

er

er

e vr

P

E

E

D

D

, ,r x y z constant

Assume a homogeneous dielectric with no free charge density inside.

Note: All of the interior bound charges cancel from one row of molecules to the next.

0v

0vb 20

r

Practical case:

Hence

Summary of Dielectric Principles

A dielectric can always be modeled with a bound volume charge density and a bound surface charge density.

A homogeneous dielectric with no free charge inside can always be modeled with only a bound surface charge density.

All of the bound charge is automatically accounted for when using the D vector and the total permittivity .

When using the D vector, the problem is solved using only the (known) free-charge density on the right-hand side of Gauss’ law.

Important points:

21

ExampleA point charge surrounded by a homogeneous dielectric shell.

22

Dielectric shell

qr

20

ˆ4

qE r

r

20

ˆ4 r

qE r

r

a

b

Find the bound surface charges densities and verify that we get the correct answer for the electric field by using them.

From Gauss’s law:

Example (cont.)

The alignment of the dipoles causes two layers of bound surface charge density.

23

Dielectric shell

q

r+-

+-

+-

+-

+- +-+ -+ -

+-

+-

+-

+-

+-

+-

+-

+-

sb

0 0v vb

Free-space model

24

0

0

ˆ

ˆ

ˆ1

1ˆ

sb

e

r

r

r

P n

E n

E n

D n

2ˆ

4

qD r

r

2

1

4a rsb

r

q

a

q

+

++

+

++

++

--

-

-

-

-

--

sb

n̂

n̂

2

1

4b rsb

r

q

b

Hence

Example (cont.)

Note: We use the electric field inside the material, at

the boundaries.

20

ˆ4

enclQE r

r

(free-space Gauss’s law)

Free-space model

25

1a rb

r

q q

1b rb

r

q q

20

ˆ:4

qr a E r

r

20

ˆ:4

abq q

a r b E rr

2

0

ˆ:4 r

qa r b E r

r

From Gauss’s law:

Example (cont.)

1a rb

r r

qq q q q

b ab bq qq

+

++

+

++

++

--

-

-

-

-

--

abq

bbq

20

ˆ4

enclQE r

r

24a ab sbq a

24b bb sbq b

Free-space model

26

1a rb

r

q q

1b rb

r

q q

20

ˆ:4

a bb bq q q

r b E rr

2

0

ˆ:4

qr b E r

r

From Gauss’s law:

Example (cont.)

0a bb bq q

b ab bq q

q

+

++

+

++

++

--

-

-

-

-

--

abq

bbq

27

Observations

We get the correct answer by accounting for the bound charges, and putting them in free space.

It is much easier to not worry about bound charges, and simply use the concept of r.

ˆ encl

S

D n dS Q

0 rD E E

Appendix

In this appendix we derive the formulas for the bound change densities:

28

ˆsb P n

vb P

ˆ ˆbn n outward normal to the dielectric boundary

where

Model:

Dipoles are aligned in the x direction end-to-end. The number of dipoles that are aligned (per unit volume) changes with x.

Nv = # of aligned dipoles

per unit volume

29

1vb bQ

V

V

x+d/2x-d/2

x

+-+-+-

+-

+-

d

Qb

Observation point

Appendix (cont.)

2 2

1vb b

d db v vx x

QV

Q q N V q N V

Hence

Nv = # of aligned dipoles

per unit volume

2 2

d dvb v vx x

v

q N N

q N

V

x+d/2x-d/2 x

+-+-+-

+-

+-

VV

Qb

30

Appendix (cont.)

vb vq N

Hence

Nv = # of aligned dipoles

per unit volume

V

x+d/2x-d/2x

+-+-+-

+-

+-

x v

x v

P pN

P p N

1

1

1

vb x

x

x

x

q Pp

q Pqd

Pd

P

x

or

xvb

dP

dx

31

Appendix (cont.)

In general,

yx zvb

dPdP dP

dx dy dz

or

vb P 32

xvb

dP

dx

For dipole aligned in the x direction,

Appendix (cont.)

33

After applying the divergence theorem, we have the integral form

ˆ bencl

S

P n Q

Applying this to a surface at a dielectric boundary, we have

ˆdiel air bb encl sbP P n S Q S

ˆb sbP n

DenotingdielP P

we have

S

Sˆbn

r

Appendix (cont.)