Lect03

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EEE 498/598

Overview of Electrical

Engineering

Engineering Lecture 3:

Lecture 3:

Electrostatics: Electrostatic

Potential; Charge Dipole;

Visualization of Electric Fields;

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Lecture 3 Objectives Lecture 3 Objectives

 To continue our study of To continue our study of

electrostatics with electrostatic

potential; charge dipole;

visualization of electric fields

and potentials; Gauss’s law and

applications; conductors and

conduction current.

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Electrostatic Potential of Electrostatic Potential of

a Point Charge at the a Point Charge at the

Origin Origin P r    

  r

Q r r d Q r d a r Q a l d E r V r r r r r 0 2 0 2 0 4 4 ˆ 4 ˆ               



   spherically symmetric

(4)

Electrostatic Potential Electrostatic Potential Resulting from Multiple Resulting from Multiple

Point Charges Point Charges

Q₁

P(R,)

r R₁

1

r

O

Q₂

2

r

 

_



 n

k _k

k R Q r

V

1 4₀

2

(5)

Electrostatic Potential Electrostatic Potential

Resulting from Continuous Resulting from Continuous

Charge Distributions Charge Distributions

 

_

 



           ev S es L el R v d r q r V R s d r q r V R l d r q r V 0 0 4 1 4 1 4 1  

  line charge

 surface charge

(6)

Charge Dipole Charge Dipole

 An An electric charge dipole_{electric charge dipole} consists of a pair of equal consists of a pair of equal

and opposite point charges separated by a small and opposite point charges separated by a small distance (i.e., much smaller than the distance at distance (i.e., much smaller than the distance at

which we observe the resulting field). which we observe the resulting field).

d

(7)

Dipole Moment Dipole Moment

• Dipole moment p is a measure of the strength of the dipole and indicates its direction

d

Q

p



+Q

-Q

d

p is in the direction from the negative point charge to the positive point

(8)

Due to Charge Dipole Due to Charge Dipole

observation point

d/2

+Q z

d/2

P

Qd

a

p



ˆ

_z 

R



R r

(9)

(Cont’d) (Cont’d)

 

 

 



R Q R

Q r

V r

V

0

0 4

4 ,

 



(10)

(Cont’d) (Cont’d) d/2 d/2    cos ) 2 / ( cos ) 2 / ( 2 2 2 2 rd d r R rd d r R         R  R r P

(11)

Electrostatic Potential Electrostatic Potential Due to Charge Dipole in Due to Charge Dipole in

the Far-Field the Far-Field

• assume R>>d

• zeroth order approximation:

R R

 





V



0

not good enough!

(12)

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the to Charge Dipole in the

Far-Field (Cont’d) Field (Cont’d)

• first order approximation from geometry:

 

cos 2

d r

R

d r

R

 

 

 

d/2 d/2



lines approximately



R



R r

(13)

Electrostatic Potential Due Electrostatic Potential Due to Charge Dipole in the to Charge Dipole in the

Far-Field (Cont’d) Field (Cont’d)

• Taylor series approximation:

                                       cos 2 1 1 1 cos 2 1 1 cos 2 1 1 cos 2

1 1 1

r d r R r d r r d r d r R

1  1 , 1

: Recall

 



(14)

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field

(Cont’d) (Cont’d)   2 0 0 4 cos 2 cos 1 2 cos 1 4 , r Qd r d r d r Q r V                             

(15)

Electrostatic Potential Due to Electrostatic Potential Due to Charge Dipole in the Far-Field Charge Dipole in the Far-Field

• In terms of the dipole moment:

2 0

ˆ 4

1

r a p

V   r

(16)

Electric Field of Charge Electric Field of Charge

Dipole in the Far-Field Dipole in the Far-Field



 



    sin ˆ cos 2 ˆ 4 1 ˆ ˆ 3 0 a a r Qd V r a r V a V E r r               

(17)

Visualization of Electric Visualization of Electric

Fields Fields

 An electric field (like any vector field) can An electric field (like any vector field) can

be visualized using

be visualized using flux linesflux lines (also called (also called

streamlines

streamlines or _orlines of forcelines of force)._).

 A A flux lineflux line is drawn such that it is everywhere is drawn such that it is everywhere

tangent to the electric field.

 A A quiver plot_{quiver plot} is a plot of the field lines is a plot of the field lines

constructed by making a grid of points. An

arrow whose tail is connected to the point

indicates the direction and magnitude of the

field at that point.

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Potentials Potentials

 The scalar electric potential can be The scalar electric potential can be

visualized using

visualized using equipotential surfacesequipotential surfaces..

 An An equipotential surface_{equipotential surface} is a surface over which is a surface over which

V

V is a constant. is a constant.

 Because the electric field is the negative of Because the electric field is the negative of

the gradient of the electric scalar potential,

the electric field lines are everywhere

normal to the equipotential surfaces and

point in the direction of decreasing

potential.

(19)

Fields Fields

 Flux linesFlux lines are suggestive of the flow of are suggestive of the flow of

some fluid emanating from positive charges

(

(sourcesource) and terminating at negative charges ) and terminating at negative charges (

(sinksink)._).

 Although electric field lines do NOT Although electric field lines do NOT

represent fluid flow, it is useful to think of

them as describing the

them as describing the fluxflux of something of something that, like fluid flow, is conserved.

(20)

Faraday’s Experiment Faraday’s Experiment

charged sphere (+Q)

+ +

insulator metal

(21)

 Two concentric conducting spheres are Two concentric conducting spheres are

separated by an insulating material.

 The inner sphere is charged to The inner sphere is charged to ++Q_Q. . The The

outer sphere is initially uncharged.

 The outer sphere is The outer sphere is grounded_grounded

momentarily.

 The charge on the outer sphere is The charge on the outer sphere is

found to be

(22)

 Faraday concluded there was a Faraday concluded there was a

“

“displacementdisplacement” from the charge on the inner ” from the charge on the inner sphere through the inner sphere through sphere through the inner sphere through

the insulator to the outer sphere. the insulator to the outer sphere.

 The The electric displacement_{electric displacement} (or (or electric flux_{electric flux}) is ) is

equal in magnitude to the charge that equal in magnitude to the charge that

produces it, independent of the insulating produces it, independent of the insulating

material and the size of the spheres. material and the size of the spheres.

(23)

Electric Displacement Electric Displacement

(Electric Flux) (Electric Flux)

+Q

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Electric (Displacement) Electric (Displacement)

Flux Density Flux Density

 The density of electric displacement is the The density of electric displacement is the electric _electric (displacement) flux density

(displacement) flux density, , DD..

 In free space the relationship between In free space the relationship between flux densityflux density and and

electric field is

E

D





₀

(25)

Electric (Displacement) Electric (Displacement)

Flux Density (Cont’d) Flux Density (Cont’d)  The electric (displacement) flux The electric (displacement) flux

density for a point charge centered at density for a point charge centered at

the origin is the origin is

2

4 ˆ

r Q a

D _r



(26)

Gauss’s Law Gauss’s Law

 Gauss’s law states that “the net electric Gauss’s law states that “the net electric

flux emanating from a close surface

flux emanating from a close surface SS is is equal to the total charge contained within

equal to the total charge contained within

the volume

the volume VV bounded by that surface.” bounded by that surface.”

encl S

Q

s

d

D





(27)

Gauss’s Law (Cont’d) Gauss’s Law (Cont’d)

V

S

ds By convention, ds

is taken to be outward from the volume V.



 _ev

encl q dv

Q

Since volume charge density is the most

general, we can always write

(28)

Applications of Gauss’s Applications of Gauss’s

Law Law

 Gauss’s law is an Gauss’s law is an integral equationintegral equation for the for the

unknown electric flux density resulting

from a given charge distribution.

encl S

Q

s

d

D







known

(29)

Law (Cont’d) Law (Cont’d)

 In general, solutions to In general, solutions to integral integral

equations

equations must be obtained using _{must be obtained using}

numerical techniques.

 However, for certain symmetric However, for certain symmetric charge distributions closed form

charge distributions closed form

solutions to Gauss’s law can be

obtained.

(30)

Law (Cont’d) Law (Cont’d)

 Closed form solution to Gauss’s Closed form solution to Gauss’s

law relies on our ability to

construct a suitable family of

Gaussian surfaces

Gaussian surfaces._.

 A A Gaussian surfaceGaussian surface is a surface to is a surface to

which the electric flux density is

normal and over which equal to a

(31)

Electric Flux Density of a Electric Flux Density of a

Point Charge Using Point Charge Using

Gauss’s Law Gauss’s Law

Consider a point charge at the origin:

(32)

Electric Flux Density of a

Point Charge Using Gauss’s

Law (Cont’d)

(1) Assume from symmetry the form of

the field

(2) Construct a family of Gaussian

surfaces

 

r D

a

D  ˆ_r _r

spheres of radius r where





r

0

spherical symmetry

(33)

Law (Cont’d)

(3) Evaluate the total charge within the (3) Evaluate the total charge within the

volume enclosed by each Gaussian surface volume enclosed by each Gaussian surface





V

ev

encl q dv

(34)

Law (Cont’d)

Q R

Gaussian surface

Q Q_encl 

(35)

Law (Cont’d)

(4) For each Gaussian surface,

evaluate the integral

DS s

d D

S

 



 _r ₄ _r2

D s

d

D   



magnitude of D

on Gaussian surface.

surface area of Gaussian

(36)

Law (Cont’d)

(5) Solve for

(5) Solve for DD on each Gaussian _{on each Gaussian}

surface

S Q

D  encl

2

4 ˆ

r Q a

D _r



 ₂

0

0 4

ˆ

r Q a

D

E _r

  

 

(37)

Spherical Shell of Charge

Using Gauss’s Law

Consider a spherical shell of uniform charge density:

 

  



otherwise ,

0 ,

0 a r b

q q_ev

(38)

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Spherical Shell of Charge Using Gauss’s Law (Cont’d) Using Gauss’s Law (Cont’d)

the field

surfaces

 

R D

a

D  ˆ_r _r

spheres of radius r where





r

(39)

 Here, we shall need to treat Here, we shall need to treat

separately 3 sub-families of Gaussian

surfaces:

a

r 



0

1)

b r

a  

2)

b

r 

3)

(40)

Spherical Shell of Charge Using

Gauss’s Law (Cont’d)

Gaussian surfaces for which

a r 



0

b r

a  

(41)





V

ev

encl q dv

(42)

0



encl

Q

 For For

a r   0 b r

a  



3 3



3 0 3 0 0 4 3 4 3 4 a r q a q r q dv q Q r a encl     



  

(43)

 For For



3 3



0 3 0 3 0 3 4 3 4 3 4 a b q a q b q dv q Q b a ev encl     



   b r 

(44)

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Spherical Shell of Charge Using Gauss’s Law (Cont’d) Using Gauss’s Law (Cont’d) (4) For each Gaussian surface,

DS s

d D

S

 



 _r ₄ _r2

D s

d

D   



magnitude of D on Gaussian

surface.

surface area of Gaussian

(45)

Electric Flux Density of a Electric Flux Density of a Spherical Shell of Charge Spherical Shell of Charge Using Gauss’s Law (Cont’d) Using Gauss’s Law (Cont’d) (5) Solve for

surface

S Q

(46)









_

_

                              b r r a b q a r a b q a b r a r a r q a r a r q a a r D r r r r , 3 ˆ 4 3 4 ˆ , 3 ˆ 4 3 4 ˆ 0 , 0 2 3 3 0 2 3 3 0 2 3 0 2 3 3 0    

(47)

 Notice that for Notice that for r > br > b

2

4 ˆ

r Q a

D tot

r _



Total charge contained in spherical shell

 3 3 

0 3 4

a b

q

(48)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

D r

(

C

/m

)

m 2

m 1

C/m

1 3

0

 



b a q

(49)

Electric Flux Density of an

Infinite Line Charge Using

Gauss’s Law

Consider a infinite line charge carrying charge per

unit length of

unit length of qqelel::

z

el

(50)

Electric Flux Density of an Electric Flux Density of an Infinite Line Charge Using Infinite Line Charge Using

the field

surfaces

 



 D

a

D  ˆ

cylinders of radius  where







(51)





L

el

encl q dl

Q

l q

(52)

Gauss’s Law (Cont’d) Gauss’s Law (Cont’d) (4) For each Gaussian surface,

DS s

d D

S

 



  l

D s

d

D    2 



magnitude of D on Gaussian

surface.

surface area of Gaussian

(53)

Gauss’s Law (Cont’d) Gauss’s Law (Cont’d) (5) Solve for

surface

S Q

D  encl

 



2 ˆ qel

a

(54)

Gauss’s Law in Integral Gauss’s Law in Integral

Form Form



  

V

ev encl

S

dv q

Q s

d D

V

S

s d

(55)

Recall the Divergence Recall the Divergence

Theorem Theorem

 Also called Also called

Gauss’s theorem

Gauss’s theorem or _or

Green’s theorem

Green’s theorem._.

 Holds for Holds for anyany

volume and

corresponding

closed surface.

dv D

s d D

V

S



   

V

S

s d

(56)

Applying Divergence Applying Divergence Theorem to Gauss’s Law Theorem to Gauss’s Law



    

V

ev V

S

dv q

dv D

s d D

 Because the above must hold for any

volume V, we must have

ev

q D 



 Differential form

(57)

Fields in Materials Fields in Materials

 Materials contain charged Materials contain charged

particles that respond to applied

electric and magnetic fields.

 Materials are classified Materials are classified

according to the nature of their

response to the applied fields.

(58)

Classification of Classification of

Materials Materials

 ConductorsConductors

 SemiconductorsSemiconductors  DielectricsDielectrics

(59)

Conductors Conductors

 A A conductorconductor is a material in which is a material in which electrons in the outermost shell

electrons in the outermost shell

of the electron migrate easily

from atom to atom.

 Metallic materials are in general Metallic materials are in general good conductors.

(60)

Conduction Current Conduction Current

 In an otherwise empty universe, In an otherwise empty universe, a constant electric field would

a constant electric field would

cause an electron to move with

constant acceleration.

-e

m E e a  

(61)

Conduction Current Conduction Current

 In a conductor, electrons are constantly In a conductor, electrons are constantly

colliding with each other and with the

fixed nuclei, and losing momentum.

 The net macroscopic effect is that the The net macroscopic effect is that the

electrons move with a (constant) drift

velocity

velocity vvd_d which is proportional to the which is proportional to the

electric field.

E

v

_d







_e

(62)

Conductor in an Conductor in an Electrostatic Field Electrostatic Field

 To have an electrostatic field, all To have an electrostatic field, all

charges must have reached their

equilibrium positions (i.e., they

are stationary).

 Under such static conditions, Under such static conditions,

there must be

there must be zero electric fieldzero electric field

within the conductor. (Otherwise

(63)

 If the electric field in which the conductor If the electric field in which the conductor

is immersed suddenly changes, charge is immersed suddenly changes, charge

flows temporarily until equilibrium is once flows temporarily until equilibrium is once again reached with the electric field inside again reached with the electric field inside

the conductor becoming zero. the conductor becoming zero.

 In a metallic conductor, the establishment In a metallic conductor, the establishment

of equilibrium takes place in about 10

of equilibrium takes place in about 10-19-19 s - s -

an extraordinarily short amount of time an extraordinarily short amount of time

indeed. indeed.

(64)

 There are two important consequences to There are two important consequences to

the fact that the electrostatic field inside

a metallic conductor is zero:

 The conductor is an _{The conductor is an}equipotential_{equipotential} body._body.  The charge on a conductor must reside _{The charge on a conductor must reside}

entirely on its surface.

• A corollary of the above is that the A corollary of the above is that the

electric field just outside the conductor

must be normal to its surface.

(65)

+

+ ₊ +

+

(66)

-Macroscopic versus Macroscopic versus

Microscopic Fields Microscopic Fields

 In our study of electromagnetics, we In our study of electromagnetics, we

use Maxwell’s equations which are

written in terms of

written in terms of macroscopicmacroscopic

quantities.

 The lower limit of the classical The lower limit of the classical

domain is about 10

domain is about 10-8-8 m = 100 m = 100

angstroms. For smaller dimensions,

quantum mechanics is needed.

(67)

Boundary Conditions on the

Electric Field at the Surface of a

Metallic Conductor

es n

n t

q

D

a

D

E







ˆ

0

+

+ + + +

- - - -

-n

a

ˆ

(68)

Induced Charges on Induced Charges on

Conductors Conductors

 The BCs given above imply that if a The BCs given above imply that if a

conductor is placed in an externally conductor is placed in an externally

applied electric field, then applied electric field, then

 the field distribution is distorted so the field distribution is distorted so

that the electric field lines are normal

to the conductor surface

 a surface charge is a surface charge is inducedinduced on the on the

conductor to support the electric field

(69)

Applied and Induced Applied and Induced

Electric Fields Electric Fields

 The The applied electric fieldapplied electric field ( (EE_app_app) is the field that ) is the field that

exists in the absence of the metallic exists in the absence of the metallic

conductor (

conductor (obstacleobstacle).).

 The The induced electric fieldinduced electric field ( (EE_ind_ind) is the field that ) is the field that

arises from the induced surface charges. arises from the induced surface charges.

 The The total field_{total field} is the sum of the applied and is the sum of the applied and

induced electric fields. induced electric fields.

E E