Chapter 5 Current and Conductors

President University Erwin Sitompul EEM 7/1

Lecture 7

Engineering Electromagnetics

Dr.-Ing. Erwin SitompulPresident University

http://zitompul.wordpress.com

2 0 1 3


Chapter 5Current and Conductors

Engineering Electromagnetics


Current and Current DensityElectric 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).

dQIdt

Chapter 5 Current and Conductors

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

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


Current and Current DensityChapter 5 Current and Conductors

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

NI J S

If the current density is not perpendicular to the surface,I J S

Through integration, the total current is obtained:

SI d J S



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

• 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

QIt

vxSt



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

x v xJ v In terms of current density, we find:

vJ v

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.


Continuity of CurrentChapter 5 Current and Conductors

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 or destroyed (lost) by recombination.

SI d J S

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:

iS

dQI ddt

J S

• The Continuity Equation in Closed Surface

• The Integral Form of the Continuity Equation


v

t

J


Continuity of CurrentThe differential form (or point form) of the continuity equation is

obtained by using the divergence theorem:

vol( )

Sd dv J S J

vol vol( ) v

ddv dvdt

J

We next represent Qi by the volume integral of ρv:

If we keep the surface (and thus the enclosed volume) constant, the derivative becomes a partial derivative,

vol vol( ) vdv dv

t

J

( ) vv vt

J

• The Differential Form (Point Form) of the Continuity Equation



Continuity of CurrentExample

The current density is given by .21 A mtre

rJ a

• Total outward current at time instant t = 1 s and r = 5 m.r rI J S 1 1 2

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

• Total outward current at time instant t = 1 s and r = 6 m.r rI J S 1 1 2

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

• Finding volume charge density:

v

t

J 22

1 1( )tr er r r

2

1 ter

2

1 tv e dt

r 2

1 ( )te K rr

, 0vt ( ) 0K r 32

1 C mtv er

rr

v

Jv

m sr



Metallic ConductorsThe energy-band structure of three types of materials at 0 K 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 ConductorsFirst 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 experiences a force:

eF E

d ev E

e e J E

J E

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:

e e • The Point Form of Ohm’s Law

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Metallic ConductorsThe 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:

V IR

LRS

S

I d JS J Sa

ab bV d E L

a

bd E L

ba E L ab E L

V ELIJ ES

VL

LV IS

abVRI

a

b

S

d

d

E L

E S

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Conductor Properties and Boundary ConditionsChapter 5 Current and Conductors

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).

Now our next concern is the fields external to the conductor.The external electric field intensity and electric flux density are

decomposed into the tangential component and the normal component, with respect to the conductor surface.


Conductor Properties and Boundary ConditionsChapter 5 Current and Conductors

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 thus it is 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 (inside the conductor) is zero. It must leave the surface normally.



Conductor Properties and Boundary Conditions

0d E L0

b c d a

a b c d

1 1,at b ,at a2 2 0t N NE w E h E h

0, 0th w E w

0tE

Sd Q D S

top bottom sidesQ

N SD S Q S

N SD

0t tD E 0N N SD E



Conductor Properties and Boundary ConditionsExample

Given the potential V = 100(x2–y2) and a point P(2,–1,3) that is predefined to lie on a conductor-to-free-space boundary, find V, E, D, and ρS at P, and also the equation of the conductor surface.

2 2100((2) ( 1) )PV 300 V

V E 200 200x yx y a a400 200 V mP x y E a a

0P PD = E23.542 1.771 nC mx y a a

2 2Conductor surface is equipotential

The surface equation is 300 100( )x y 2 2 3x y

23.96 nC mN PD = D =2

, 3.96 nC mS P ND =• Carefully examine

the surface direction


The Method of ImagesChapter 5 Current and Conductors

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.



Thus, we can replace the dipole configuration (left) with the single charge and conducting plane (right), without affecting the fields in the upper half of the figure.

Now, we begin with a single charge above a conducting plane. ► The same fields above the plane can be maintained by

removing the plane and locating a negative charge at a symmetrical location below the plane.

This charge is called the image of the original charge, and it is the negative of that value.



The same procedure can be done again and again.Any charge configuration above an infinite ground plane may

be replaced by an arrangement composed of the given charge configuration, its image, and no conducting plane.



ExampleFind 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.

• We remove the plane and install an image line charge

• The field at P may now be obtained by superposition of the known fields of the line charges



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

x = 0, z = –3

P(2,5,0)

2 3x z R a a

02L

RR

E a9

0

2 330 102 13 13

x z

a a

02L

RR

E a9

0

2 330 102 13 13

x z

a a

E E E9

0

180 102 (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


SemiconductorsChapter 5 Current and Conductors

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.

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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The conductivity of a semiconductor is described as:e e h h

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.

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The conductivity of the intrinsic semiconductor increases with temperature, while that of a metallic conductor decreases with temperature.

The intrinsic semiconductors also satisfy the point form of Ohm's law: the conductivity is reasonably constant with current density and with the direction of the current density.

J E

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Seminar on Embedded PC TechnologyChapter 5 Current and Conductors

Thursday, 30 May 2013. 09:00-17:00, At PU Auditorium, 5th Floor. Collaboration of TDS Technology,

Indotama, and EE PU . 7 sessions, 4 speakers.

Your task: Visit 2 sessions and write individual short summary of each session (@ 1 A4 page).

This task will be graded as Homework of EEM.

Originality (writing with your own words) will be highly regarded. Plagiarism means zero grade for giver and taker.

Due: Tuesday, 04 June 2013.


Homework 6D5.1D5.2. D5.5. D5.6. (Extra Question, + 20 points if correctly made)


All homework problems from Hayt and Buck, 7th Edition.Due: Monday, 03 June 2013.

Chapter 5 Current and Conductors

Documents

current densitychapter

total current

unit of current

convection current density

current density j

terms of current density

total outward current

moving charge