Electromagnetic Theory-ECE305_Chapter5 Current and Conductors

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5. Current and Conductors

Outline:

Introduction

Current and current density

Metallic Conductors

Conductor Properties and Boundary Conditions

The method of Images

Semiconductors

Summary

1 AURAK- ECEN 305

Electromagnetic Theory

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Introduction

Chapter aim: Define current and current density Apply the laws and methods that we have learnt to some materials Present Ohm’s law for a conducting material Calculate resistance values for few of geometric forms Study required conditions at conductor boundaries

2 AURAK- ECEN 305 Electromagnetic Theory

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5.1. Current and Current Density

Electric charges in motion constitute a current.

The unit of current is the ampere (A), defined as a rate of movement of charge passing a given reference point (or crossing a given reference plane).

Current is defined as the motion of positive charges, although conduction in metals takes place through the motion of electrons.

Current density (vector quantity), J is measured in amperes per square meter (A/m2).

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Electromagnetic Theory 3

dt

dQI

Current and ….

The increment of current ΔI crossing an incremental surface ΔS normal to the current density is:

If the current density is not perpendicular to the surface

Through integration, the total current is obtained

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SJI N

SJ I

S

dI SJ

3

Current and ….

Current density may be related to the velocity of volume charge density at a point.


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An element of charge ΔQ = ρvΔSΔL moves along the x axis.

In the time interval Δt, the element of charge has moved a distance Δx.

The charge moving through a reference plane perpendicular to the direction of motion is ΔQ = ρvΔSΔx.

QI

t

v

xS

t

Current and ….

The limit of the moving charge with respect to time is:

In terms of current density, we find:

This last result shows clearly that charge in motion constitutes a current. We name it here convection current.

J = ρvv is then called convection current density.


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v xI Sv

x v xJ v

vJ v

4

Current and ….

Exercise:

Ans: 180a -9a mA/m2, 3.26 A


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5.2. Continuity of Current

The principle of conservation of charge: “Charges can be neither created nor destroyed.”

But, equal amounts of positive and negative charge (pair of charges) may be simultaneously created, obtained by separation, destroyed, or lost by recombination.

Any outward flow of positive charge must be balanced by a decrease of positive charge (or perhaps an increase of negative charge) within the closed surface.

If the charge inside the closed surface is denoted by Qi, then the rate of decrease is –dQi/dt and the principle of conservation of charge requires:


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S

dI SJ • The Continuity Equation in Closed Surface

dt

dQdI i

S

SJ • The Integral Form of the Continuity Equation

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Continuity of Current…

The differential form (or point form) of the continuity equation is obtained by using the divergence theorem:

We next represent Qi by the volume integral of ρv:

If we keep the surface constant, the derivative becomes a partial derivative. Writing it within the integral,


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volS

dvd )( JSJ

vol vol( ) v

ddv dv

dt J

v

t

Jvol vol

( ) vdv dvt

J

( ) vv vt

J • The Differential Form (Point Form)

of the Continuity Equation


Example: Current density is given by

Total outward current at time instant t = 1 s and r = 5 m.

Total outward current at time instant t = 1 s and r = 6 m.

Finding volume charge density:


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2/1

mAer

r

taJ

r rI J S1 1 2

5( )(4 5 )r re a a 23.11A

r rI J S1 1 2

6( )(4 6 )r re a a 27.74 A

v

t

J 2

2

1 1( )tr e

r r r

2

1 ter

2

1 t

v e dtr

2

1( )te K r

r

, 0vt ( ) 0K r 3

2

1C mt

v er

r

r

v

Jv

m sr

J

r

J

rr

Jr

r

r

)sin(

1))(sin(

)sin(

1)(12

2JJdiv

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Exercise:

Ans: 39.7 A; -15.8 mC/m3; 29 m/s

Homework: Drill Problem 5.2 to be submitted


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5.3 Metallic Conductors

The energy-band structure of three types of materials at zero Kelvin (-

273 0C) is shown as follows:

Energy in the form of heat, light, or an electric field may raise the energy

of the electrons of the valence band, and in sufficient amount they will

be excited and jump the energy gap into the conduction band.


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Metallic Conductors…

First let us consider the conductor.

Here, the valence electrons (or free conductive electrons) move under

the influence of an electric field E.

An electron having a charge Q = –e will experience a force:

In the crystalline material, the progress of the electron is impeded by

collisions with the lattice structure, and a constant average velocity is

soon attained.

This velocity vd is termed the drift velocity. It is linearly related to the

electric field intensity by the mobility of the electron μe:


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EF e

d e v E

e e J E

J E

e e

The unit mobility of the

electron, μe= m2/volt-s.

e:-free electron charge

density (negative value).


The application of Ohm’s law in point form to a macroscopic region

leads to a more familiar form.

Assuming J and E to be uniform, in a cylindrical region shown below,

we can write:


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V IR

LR

S

SI d JS J S

a

abb

V d E L

a

bd E L

ba E L ab E L

V EL

IJ E

S

V

L

LV I

S

abVR

I

a

b

S

d

d

E L

E S

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Common metallic material constants:

Note that the conductance, G is defined as

G= 1/Resistance= 1/R=current/voltage=I/V

The unit of conductance= -1 = Siemens (S)


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Material Conductivity, (S/m) Mobility, e (m2/V.s)

Aluminum 3.82 107 0.0012

Copper 5.8 107 0.0032

Silver 6.17 107 0.0056


Example:


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132.0

Mobility for copper, e

= 0.0032 m2/v.sec

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Exercise:

Homework to be submitted: 5.3c, 5.3d, 5.4c, 5.4d


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5.4 Conductor Properties and Boundary Conditions

Property 1:

The charge density within a conductor is zero (ρv = 0) and the surface charge density resides on the exterior surface.

Property 2:

In static conditions, no current may flow, thus the electric field intensity within the conductor is zero

(E = 0).

What about he fields external to the conductor?

The external electric field intensity and electric flux density are decomposed into the tangential components and the normal components.


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Conductor Properties…

The tangential component of the electric field intensity is seen to be zero Et = 0 Dt = 0.

If not, then a force will be applied to the surface charges, resulting in their motion and no static conditions.

The normal component of the electric flux density leaving the surface is equal to the surface charge density in coulombs per square meter (DN = ρS).

According to Gauss’s law, the electric flux leaving an incremental surface is equal to the charge residing on that incremental surface.

The flux cannot penetrate into the conductor since the total field there is zero.

It must leave the surface normally.


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Boundary conditions proof:


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0b c d a

a b c d

1 1

,at b ,at a2 20t N NE w E h E h

top bottom sidesQ

N SD S Q S

N SD

0t tD E 0N N SD E

0 LE d Qd SD

finitebutsmallverywhAs &0

00 tt EwE

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In summary, for conductors in electrostatic fields:

The static electric field intensity inside a conductor is zero.

The static electric field intensity at the surface of a conductor is everywhere directed normal to that surface.

The conductor surface is equi-potential surface as E dL=0 at the surface.


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Example:


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Example continued:


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Example continued:


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Exercise:

Homework to be submitted:

Drill Problem 5.5


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5.6. The Method of Images

One important characteristic of the dipole field developed in Chapter 4

is the infinite plane at zero potential that exists midway between the

two charges.

Such a plane may be represented by a thin infinite conducting plane.

The conductor is an equipotential surface at a potential V = 0. The

electric field intensity, as for a plane, is normal to the surface.


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2

0

)cos(

4

dipoletheFor

r

dQV

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The Method of Images…

Any charge configuration above an infinite conducting

ground plane may be replaced by an arrangement

composed of the given charge configuration, its

image, and no conducting plane.


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Example:-Find the surface charge density at P(2,5,0) on the

conducting plane z = 0 if there is a line charge of 30 nC/m located at

x = 0, z = 3, as shown below.

The field at P may now be obtained by superposition of the known

fields of the line charges: + l and - l.

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Electromagnetic Theory 28

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A line parallel to the y axis and located at the points (x0, z0) has an electric field at any point P(x,y,z):


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2

0

2

0

00

00 )()(

)()(

22),(

zzxx

zzxx

Rzx zxlRl aaa

E

2 3x z R a ax = 0, z = 3

x = 0, z = –3

P(2,5,0)

2 3x z R a a

02

LR

R

E a9

0

2 330 10

2 13 13

x z

a a

02

LR

R

E a9

0

2 330 10

2 13 13

x z

a a

E E E9

0

180 10

2 (13)z

a 249 V mz a 0D E 22.20 nC mz a

• Normal to the plane

S ND 22.20nC m at P


Exercise:

Ans: 317 V, -45.3ax-99.2ay V/m

Homework to be submitted: Drill Problem 5.6


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5.7. Semiconductors

In an intrinsic semiconductor material, such as pure germanium or

silicon, two types of current carriers are present: electrons and

holes.

The electrons are those from the top of the filled valence band

which have received sufficient energy to cross the small forbidden

band into conduction band.

The forbidden-band energy gap in typical semiconductors is of the

order of 1 eV (one electronvolt).

The vacancies left by the electrons represent unfilled energy

states in the valence band. They may also move from atom to atom

in the crystal.

The vacancy is called a hole, and the properties of semiconductor

are described by treating the hole as a positive charge of e, a mobility

μh, and an effective mass comparable to that of the electron.


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Semiconductors…

The conductivity of a semiconductor is described as:

As temperature increases, the mobilities decrease, but

the charge densities increase very rapidly.

As a result, the conductivity of silicon increases by a

factor of 100 as the temperature increases from about

275 K to 330 K.

Note that conductivity decreases with temperature for metallic

conductors.

Number of charge carriers and conductivity may both

be increased by adding small amount of impurities:

Getting electrons from donors: n-type semiconductors

Acceptors furnish extra holes: p-type semiconductors


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e e h h

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5.8. Summary

Current and current density in conductors.

Drift velocity and conductivity.

Determination of resistance of conductors

Relation ship between current density, electric field intensity, mobility and charge density

Properties of metallic conductors in electrostatic equilibrium:

No electric field within and tangent to metallic conductors.

The electric field is only normal at the surface of the conductors.

The conductivity of semiconductors increases as temperature increases in contrast to metallic conductors.


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Homework

Drill Problems: 5.2

5.3c, 5.3d, 5.4c, 5.4d

5.5

5.6


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Electromagnetic Theory-ECE305_Chapter5 Current and Conductors