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Prof. David R. JacksonDept. of ECE

Fall 2013

Notes 3

ECE 6340 Intermediate EM Waves

Types of Current

cvJ v

iJ

cJ

impressed current

conduction (ohmic) current

Linear medium:cJ E (Ohm’s law)

Note: The electric field is set up in response to the impressed sources.

Note: The free-charge density v refers to those charge carriers (either positive or negative) that are free to move (usually electrons or ions). It is zero for perfect insulators.

iJcJ

v

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Types of Current

Η J j E

iΗ J E j E

source conduction displacement

Ampere’s law:

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iJcJ

v

Effective Permittivity

i

i

i

i

Η J j E

J E j E

J j E

J j Ej

J j j E

4

icΗ J j E

c j

Define:

Effective Permittivity (cont.)

Ampere’s law becomes:

Ampere’s law thus becomes in the same form as for free space:

0iH J j E

This "effective" permittivity accounts for the conductivity.

Note: If there is polarization loss (molecular or atomic friction), than will be complex in addition to c.

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Note: c is often called for simplicity in most books.

However, be careful!

cD E

D E

Effective Permittivity (cont.)

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Effective Permittivity Principle

0 c

This allows us to solve a problem involving a homogeneous (lossy) material, as long as we know how to solve the corresponding free-space problem.

icΗ J j E

0iH J j E (Free-space problem)

(Material problem)

The formulas for the fields remains the same: we simply make this simple substitution.

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y

z

x

,

1I

Ocean

Example

First examine problem in free space:

A dipole is embedded in an infinite medium of ocean water. What is the far-field of the dipole?

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Example (cont.)

Dipole in free space:

y

z

x

0 0,

1I

r As

000 0 0sin ,

4j k rj

E e kr

r

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r As

101 0 1 1sin ,

4j k r

c

jE e k k jk

r

0c r j

Example (cont.)

y

z

x

,

1I

Ocean

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Loss Tangent

c c cj

tan c

c

c j

Write this as:

Re

Im

c

c

Note: The loss tangent combines losses from atomic and molecular friction together with

loss from conductivity.

Note: In most books, the symbol is used to denote c in the time-harmonic steady state.

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Polarization Current

0 0

ic

i

i

H J j E

J E j E

J E j E j E

Conduction Free-spacedisplacement

PolarizationcJ

SourcepJ

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Four types of current density (nonmagnetic medium)

pJcJiJ

vNote: The free-space displacement current is not an actual current flow.

Polarization Current (cont.)

Model of polarization current:

qx

Ex

x dqx NP

xd d

d dxN q N q v

dt dt

P p q

v dv qN v xJ

p d

dt x

x

PJHence

As the electric field changes, we imagine that the position x of the positive charge changes, with the negative charge being stationary.

vNd dipoles per unit volume

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Polarization Current (cont.)

p d

dt

PJ

0 0 01pe rJ j P j E j E j E

In general,

Time-harmonic steady state:

0pJ j E

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Polarization Current (cont.)

0

0 0

1 ic

i

i

B M J j E

J E j E

J E j E j E

If magnetic material is present:

0 00

1 iB J E j E j E M

Polarization current from magnetic properties

Polarization current from dielectric properties

0

1

H B M

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0

1H B M

LHS is that of Ampere's law in free space.

Polarization Current

ConductionFree-spacedisplacement

PolarizationcJ

Source

pJ

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Five types of current density

0 00

1 iB J E j E j E M

Magnetic polarization

mpJ

pJcJiJ

vi

impJ

Note: The free-space displacement current is not an actual current flow.

Equivalent Current

iJ,

Bodyc j

Inside body,

0 0

c

c

H j E

j E j E

0eqcJ j E Define

Nonmagnetic body

The equivalent current combines the conduction current and the

polarization current.

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Equivalent Current (cont.)

iJ

Interpretation:

0 0,

eqJ

0eqH j E J

The body is replaced by its

equivalent current in free space.

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Note: The equivalent current is unknown, since the electric field inside the body is unknown.