Prof. David R. Jackson ECE Dept. - University of Houstoncourses.egr.uh.edu/ECE/ECE3318/Class...

36
Prof. David R. Jackson Dept. Of ECE Spring 2018 Notes 20 Dielectrics ECE 3318 Applied Electricity and Magnetism 1

Transcript of Prof. David R. Jackson ECE Dept. - University of Houstoncourses.egr.uh.edu/ECE/ECE3318/Class...

Prof. David R. Jackson Dept. Of ECE

Spring 2018

Notes 20 Dielectrics

ECE 3318 Applied Electricity and Magnetism

1

Dielectrics

Water

0 rε ε ε=

Single H2O molecule:

2

+

-

H+

O-

H+

+

Dielectrics (cont.)

Water

0 rε ε ε=

“Dipole” model: p p

p qd

=

=

“Vector dipole moment” of molecule p:

3

+

-

d

q+

q−

+

-

H+

O-

H+

+

Dielectrics (cont.)

4

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

+

+ +

-

+

-

+ -

- -

- +

Note: The molecules are not floating in water – they make up the water.

Dielectrics (cont.)

5

The dipoles partially align under an applied electric field.

+ -

+

+

+

+

-

-

-

-

+ -

+ -

+ -

+ -

x

0V

xE

Define: P = total dipole moment/volume

1i

VP p

V ∆

≡∆ ∑

0D E Pε≡ +

Define the electric flux density vector D:

V∆

Dielectrics (cont.)

We can also write this as

( )( )1 avei v

V

NP p N pV N ∆

= = ∆ ∑

pave = average vector dipole moment

N molecules (dipoles) inside ∆V

6

/vN N V= ∆

Nv = number of molecules per unit volume

Linear material: 0 eP Eε χ=

Define:

0 rε ε ε=

0 rD Eε ε=Then we have

Note: χe > 0 for most materials

Dielectrics (cont.)

The term χe is called the “electric susceptibility.”

( )0 0

0 1e

e

D E EE

ε ε χε χ

= +

= +

1r eε χ≡ +

7

D Eε=or where

Note: A negative value of χe

would mean that the dipoles align against the field.

Typical Linear Materials

Teflon Water Styrofoam Quartz

2.2811.035

r

r

r

r

εεεε

====

(a very polar molecule, fairly free to rotate)

8

Note: εr > 1 for most materials: 1 , 0r e eε χ χ≡ + >

Dielectrics: Bound Charge

0vbρ <

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.”

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

ρv = “free charge density”: this is charge that 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.

9

+ -

+ -

+ -

+ -

+ -

+ - + - + - + -

Dielectrics: Bound Charge (cont.)

totalv v vbρ ρ ρ= +

Total charge density inside the material:

0vbρ <

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

0sbρ <0sbρ >

10

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

11

Dielectrics: Bound Charge (cont.)

The derivation of these formulas is included in the Appendix.

vb Pρ = −∇ ⋅

ˆsb P nρ = ⋅

( ), ,r x y zε

- - - - - - - - - - -

+ +

+

-

- -

+

+ n̂

The unit normal points outward from the dielectric.

( )0 0 1e rP E Eε χ ε ε= = −

E = electric field inside material

Dielectrics: Gauss’s Law

( )0totalv v vbEε ρ ρ ρ∇⋅ = = +

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.

12

Inside a material:

The free-space form of Gauss’s law is

( )0totalv v vb vE Pε ρ ρ ρ ρ∇⋅ = = + = −∇⋅

Hence

vD ρ∇ ⋅ =

Dielectrics: Gauss’s Law (cont.)

( )0 vE Pε ρ∇ ⋅ + =

or

This is the usual Gauss’s law: This is why the definition of D is so convenient! 13

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 ρ∇ ⋅ =

ˆ enclS

D n dS Q⋅ =∫

14

0 rε ε ε=

D Eε=

Example Point charge inside dielectric shell

Dielectric spherical shell with εr

Gauss’ law: ˆ enclS

D n dS Q q⋅ = =∫

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

15

Find D, E

qr

ba

r

S

Example (cont.)

22ˆ C/m

4qD rrπ

=

[ ]

[ ]

20

20

ˆ V/m ,4

ˆ V/m4 r

qE r r a r br

qE r a r br

πε

πε ε

= < >

= < <

16

( )24rD r qπ =

Hence

We then have

qr

ba

r

S

Example (cont.)

17

qrε

Flux Plot

Note that there are less flux lines inside the dielectric region (assuming that the flux lines represent the electric field).

Homogeneous Dielectrics

( )0

0

00

00

1

1

vb

e

e

er

er

rv

r

PE

E

D

D

ρε χ

ε χ

ε χε ε

ε χε ε

ε ρε

= −∇ ⋅

= −∇ ⋅

= − ∇ ⋅

= − ∇ ⋅

= − ∇ ⋅

−= −

( ), ,r x y zε = constant

Homogeneous dielectric No free charge density inside.

Note: All of the interior bound charges cancel from

one row of molecules to the next. 0vbρ =

18

0vρ =

Practical case:

Hence

Homogeneous Dielectrics (cont.)

( ), ,r x y zε = constant

19

0vρ =

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ - + -

0vbρ =

Blown-up view of molecules

Practical case (homogeneous)

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 ε.

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

Important points:

20

Example A point charge surrounded by a homogeneous dielectric shell.

21

20

ˆ4

qE rrπε

=

20

ˆ4 r

qE rrπε ε

=

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

From Gauss’s law:

Dielectric shell

b

aq

Example (cont.)

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

22

sbρ

0 0v vbρ ρ= ⇒ =

Dielectric shell

+ -

+ -

+ -

+ -

+ - + - + - + -

q

Free-space model

23

( )( )( )

( )

0

0

00

ˆˆ

ˆ1

ˆ1

1 ˆ

sb

e

r

rr

r

r

P nE n

E n

D n

D n

ρε χ

ε ε

ε εε ε

εε

= ⋅

= ⋅

= − ⋅

= − ⋅

= ⋅

2ˆ4

qD rrπ

=

2

14

a rsb

r

qa

ερε π

− = −

2

14

b rsb

r

qb

ερε π

− = +

Hence

Example (cont.)

Note: We use the electric field

inside the material, at the boundaries.

+

+ +

+

+ +

+ +

- -

-

-

-

-

- -

n̂ q

sbρ

Free-space model

24

1a rb

r

q qεε

−= −

1b rb

r

q qεε

−= +

20

ˆ:4

qr a E rrπε

< =

: a

encl ba r b Q q q< < = +

20

ˆ:4 r

qa r b E rrπε ε

< < =

From Gauss’s law:

Example (cont.)

1a rb

r r

qq q q q εε ε

−+ = − =

b ab bq q= −Note :

20

ˆ4

enclQE rrπε

=

24a ab sbq aπ ρ=

24b bb sbq bπ ρ=

where

+

+ +

+

+ +

+ +

- -

-

-

-

-

- -

abq

bbq

qS

a r b< <

Free-space model

25

1a rb

r

q qεε

−= −

1b rb

r

q qεε

−= +

: a bencl b br b Q q q q> = + +

20

ˆ:4

qr b E rrπε

> =

From Gauss’s law:

Example (cont.)

0a bb bq q+ =

b ab bq q= −Note :

where

+

+ +

+

+ +

+ +

- -

-

-

-

-

- -

abq

bbq

q

S

r b>

26

Observations

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

However, it is much easier to not worry about bound charges,

and simply use the concept of εr and the free charge.

ˆ enclS

D n dS Q⋅ =∫

0 rD E Eε ε ε= =

From the previous example, we have the following observations:

Free charge enclosed

Exotic Materials

27

Plasmas have a relative permittivity that is less than one (and can even be negative)

Plasmas

2

0 1 pωε ε

ω

= −

( )

2

0 1 p

ε εω ω υ

= − −

Lossless plasma:

Lossy plasma:

ωp = plasma resonance frequency

υ = plasma collision frequency (loss term)

(derived in ECE 6340)

Exotic Materials (cont.)

28

Low frequencies (f < 30 MHz) will reflect off the ionosphere.

Plasmas

Shortwave radio signals propagate around the earth by

skipping off the ionosphere.

The ionosphere has a relative permittivity that is less than one, so the waves will bend (refract) “away from the normal” and

travel back down to the earth, bouncing off of the earth.

https://en.wikipedia.org/wiki/Skywave

Exotic Materials (cont.) Artificial “metamaterials” that have been designed that have

exotic permittivity and/or permeability performance.

29

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

00

r

r

εµ<<

(over a certain bandwidth of operation)

Appendix

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

31

ˆsb P nρ = ⋅

vb Pρ = −∇ ⋅

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

32

Appendix (cont.)

1vb bQ

Vρ =

+ - + - + -

+ - + -

Observation point

d

x/ 2x d+/ 2x d−

bQ

V∆

2 2

1vb b

d db v vx x

QV

Q q N V q N V

ρ

+ −

=∆

= − ∆ + ∆

Hence

Nv = # of aligned dipoles per unit volume

2 2d dvb v vx x

v

q N N

q N

ρ+ −

= − − = − ∆

33

Appendix (cont.)

+ - + - + -

+ - + -

d

x/ 2x d+/ 2x d−

bQV∆

vb vq Nρ = − ∆

Hence

Nv = # of aligned dipoles per unit volume

x v

x v

P pNP p N=

⇒ ∆ = ∆

1

1

1

vb x

x

x

x

q Pp

q Pqd

Pd

Px

ρ

= − ∆

= − ∆

= − ∆

∆= −

∆or

xvb

dPdx

ρ = −34

Appendix (cont.)

+ - + - + -

+ - + -

d

x/ 2x d+/ 2x d−

bQV∆

In general,

yx zvb

dPdP dPdx dy dz

ρ = − − −

or

vb Pρ = −∇⋅35

xvb

dPdx

ρ = −

For dipole aligned in the x direction,

Appendix (cont.)

36

After applying the divergence theorem, we have the integral form

ˆ bencl

S

P n Q− ⋅ =∫

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

( ) ˆdiel air bb encl sbP P n S Q Sρ− ⋅ ∆ = = ∆

ˆb sbP n ρ⋅ =

Denoting dielP P=

we have

S∆

Sˆbn

Appendix (cont.)