Lect05

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EEE 498/598

Overview of Electrical

Engineering

Engineering Lecture 5:

Lecture 5:

Electrostatics: Dielectric

Breakdown, Electrostatic

Boundary Conditions,

Electrostatic Potential

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Lecture 5 Objectives Lecture 5 Objectives

 To continue our study of To continue our study of

electrostatics with dielectric electrostatics with dielectric

breakdown, electrostatic breakdown, electrostatic

boundary conditions and boundary conditions and

electrostatic potential energy. electrostatic potential energy.

 To study steady conduction To study steady conduction

current and Ohm’s law. current and Ohm’s law.

(3)

Dielectric Breakdown Dielectric Breakdown

 If a dielectric material is placed If a dielectric material is placed

in a very strong electric field, in a very strong electric field,

electrons can be torn from their electrons can be torn from their

corresponding nuclei causing corresponding nuclei causing

large currents to flow and large currents to flow and

damaging the material. This damaging the material. This

phenomenon is called

phenomenon is called dielectric dielectric breakdown

(4)

(Cont’d) (Cont’d)

 The value of the electric field at The value of the electric field at

which

which dielectric breakdowndielectric breakdown occurs is occurs is called the

called the dielectric strengthdielectric strength of the _{of the} material.

material.

 The The dielectric strengthdielectric strength of a material of a material

is denoted by the symbol

(5)

 The dielectric strength of a material The dielectric strength of a material may vary by several orders of

may vary by several orders of

magnitude depending on various

factors including the exact

composition of the material.

 Usually dielectric breakdown does not Usually dielectric breakdown does not permanently damage gaseous or

permanently damage gaseous or

liquid dielectrics, but does ruin solid

dielectrics.

(6)

 Capacitors typically carry a Capacitors typically carry a

maximum voltage rating. Keeping maximum voltage rating. Keeping

the terminal voltage below this the terminal voltage below this

value insures that the field within value insures that the field within

the capacitor never exceeds

the capacitor never exceeds EE_BR_BR

for the dielectric. for the dielectric.

 Usually a safety factor of 10 or so Usually a safety factor of 10 or so

is used in calculating the rating. is used in calculating the rating.

(7)

Fundamental Laws of Fundamental Laws of

Electrostatics in Integral Electrostatics in Integral

Form Form



 

V

ev S

C

dv q

s d D

l d

E 0

E D 



Conservative field Gauss’s law

(8)

Fundamental Laws of

Electrostatics in

Differential Form

ev

q D

E  



 

 0

E D 



Conservative field Gauss’s law

(9)

Fundamental Laws of Fundamental Laws of

Electrostatics Electrostatics

 The integral forms of the fundamental laws The integral forms of the fundamental laws

are more general because they apply over

regions of space. The differential forms are

only valid at a point.

 From the integral forms of the fundamental From the integral forms of the fundamental

laws both the differential equations

governing the field within a medium and

the boundary conditions at the interface

between two media can be derived.

(10)

Boundary Conditions Boundary Conditions

 Within a homogeneous medium, Within a homogeneous medium,

there are no abrupt changes in there are no abrupt changes in EE

or

or DD. However, at the interface . However, at the interface between two different media

between two different media

(having two different values of (having two different values of



, it is obvious that one or both _{, it is obvious that one or both} of these must change abruptly. of these must change abruptly.

(11)

 To derive the boundary To derive the boundary

conditions on the normal and conditions on the normal and

tangential field conditions, we tangential field conditions, we

shall apply the integral form of shall apply the integral form of

the two fundamental laws to an the two fundamental laws to an

infinitesimally small region that infinitesimally small region that

lies partially in one medium and lies partially in one medium and

partially in the other. partially in the other.

(12)

 Consider two semi-infinite media separated by Consider two semi-infinite media separated by a boundary. A surface charge may exist at the

a boundary. A surface charge may exist at the

interface.

Medium 1

Medium 2 x x

x x _s

(13)

 Locally, the boundary will look planarLocally, the boundary will look planar

1



2



n

a

ˆ

2 2, D

E

1 1, D

E

(14)

Boundary Condition on Boundary Condition on

Normal Component of Normal Component of

D D

• Consider an infinitesimal cylinder (pillbox) with cross-sectional area s and height h lying half in medium 1 and half in medium 2:

1



2



2 2, D E

1 1, D

E

s h/2

h/2

x x x x x x s

n

a

ˆ

(15)

Boundary Condition on Boundary Condition on Normal Component of D Normal Component of D

 Applying Gauss’s law to the pillbox, we have Applying Gauss’s law to the pillbox, we have

s q RHS s D s D s d D s d D s d D LHS dv q s d D n n side bottom top V ev S              



2 1 0

(16)

Boundary Condition on Boundary Condition on Normal Component of D Normal Component of D

The boundary condition isThe boundary condition is

If there is no surface chargeIf there is no surface charge

s n

n

D

₁



₂





n

D

₁



₂ For materials, non-conducting_s = 0 unless

(17)

Boundary Condition on Boundary Condition on

Tangential Component Tangential Component

of E of E

• Consider an infinitesimal path abcd with width w and height h lying half in medium 1 and half in medium 2:

1



2



n

a

ˆ

2 2, D E

1 1, D

E

h/2

w

a b c

(18)

Boundary Condition on

Tangential Component of

E

E (Cont’d)_(Cont’d)

n aˆ a b d s aˆ t aˆ path along boundary the to al r tangenti unit vecto ˆ ˆ ˆ contour by the defined direction in the path lar to perpendicu r unit vecto ˆ     n s t s a a a abcd a

(19)

Tangential Component of

Tangential Component of EE (Cont’d)

(Cont’d)

 Applying conservative law to the path, we have Applying conservative law to the path, we have

E E  w

w E h E h E w E h E h E l d E l d E l d E l d E LHS l d E t t t n n t n n a d d c c b b a C                                2 1 1 2 1 2 2 1 2 2 2 2 0

(20)

 The boundary condition isThe boundary condition is

t

E

₁



₂

Tangential Component of E

(Cont’d)

(21)

Electrostatic Boundary Electrostatic Boundary

Conditions - Summary Conditions - Summary

 At any point on the boundary,At any point on the boundary,

 the components of the components of EE₁₁ and and EE₂₂

tangential to the boundary are equal

 the components of the components of DD₁₁ and and DD₂₂ normal normal

to the boundary are discontinuous by

an amount equal to any surface

charge existing at that point

(22)

Conditions - Special Conditions - Special

Cases Cases

 Special Case 1: the interface Special Case 1: the interface

between two perfect

(non-conducting) dielectrics:

conducting) dielectrics:

 Physical principle:Physical principle: “there can be no “there can be no

free surface charge associated with

the surface of a perfect dielectric.”

 In practice:In practice: unless an impressed unless an impressed

surface charge is explicitly stated,

assume it is zero.

(23)

Conditions - Special Conditions - Special

Cases Cases

 Special Case 2: the interface between Special Case 2: the interface between

a conductor and a perfect dielectric:

 Physical principle:Physical principle: “there can be no “there can be no

electrostatic field inside of a conductor.”

 In practice:In practice: a surface charge always a surface charge always

exists at the boundary.

0 1



 _s

n

E

(24)

Potential Energy Potential Energy

 When one lifts a bowling ball and places it When one lifts a bowling ball and places it

on a table, the work done is stored in the

form of potential energy. Allowing the ball to

drop back to the floor releases that energy.

 Bringing two charges together from infinite Bringing two charges together from infinite

separation against their electrostatic

repulsion also requires work. Electrostatic

energy is stored in a configuration of

charges, and it is released when the charges

are allowed to recede away from each other.

(25)

Electrostatic Energy in a Electrostatic Energy in a

Discrete Charge Discrete Charge

Distribution Distribution

Q₁

 Consider a point Consider a point

charge

charge QQ₁₁ in an _{in an}

otherwise empty

universe.

 The system stores The system stores

no potential

energy since no

work has been

done in creating it.

(26)

Electrostatic Energy in a

Discrete Charge

Distribution (Cont’d)

Distribution (Cont’d)_ Now bring in Now bring in

from infinity from infinity

another point another point

charge

charge QQ₂₂._.

 The energy The energy

required to bring required to bring

Q

Q₂₂ into the system _{into the system} is

is

Q₁ Q₂ R12

12 2

2 Q V

W 

(27)

Discrete Charge

 Now bring in Now bring in

from infinity

another point

charge

charge QQ3₃..

 The energy The energy

required to

bring

bring QQ3₃ into into

the system is

the system isW3  Q3



V13 V23



Q₁ Q₂ R12

Q₃ _R

13

(28)

Distribution (Cont’d) Distribution (Cont’d)

 The total energy required to assemble The total energy required to assemble the system of three charges is

the system of three charges is



₁₃ ₂₃



3 12

2

3 2

V V

Q V

Q

W W

W_e

 



 

(29)

Discrete Charge

 Now bring in from infinity a fourth Now bring in from infinity a fourth

point charge

point charge QQ4₄..

 The energy required to bring The energy required to bring QQ₄₄

into the system is

 The total energy required to The total energy required to

assemble the system of four

charges is



₁₄ ₂₄ ₃₄



4

4 q V V V

W   

 ₁₃ ₂₃ ₄ ₁₄ ₂₄ ₃₄ 

3 12 2 4 3 2 V V V Q V V Q V Q W W W W_e         

(30)

Discrete Charge

 Bring in from infinity an Bring in from infinity an iith point th point

charge

charge QQ_i_i into a system of _{into a system of}i-1i-1 point _point

charges.

 The energy required to bring The energy required to bring QQ_i_i into into

the system is

 The total energy required to The total energy required to

assemble the system of

assemble the system of NN charges is_{charges is}







 1

1 i

j

ji i

i Q V

W



     

 N _i N _i i _ji N i _i _ji

e W Q V QV

(31)

Distribution (Cont’d) Distribution (Cont’d)  Note thatNote that

ij j

ij i

j ji

j i

ji i

V Q

R Q Q

V Q



 

0

0 4

4 

 Physically, the above means that the partial

energy associated with two point charges is equal no matter in what order the charges are assembled.

(32)

Discrete Charge Discrete Charge Distribution (Cont’d) Distribution (Cont’d)





  ... 2 1 2 1 34 4 24 4 14 4 43 3 42 2 41 1 23 3 13 3 32 2 31 1 12 2 21 1 2 1 1 2 1 1                         V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q V Q W N i i j ji i ij j N i i j ji i e

(33)

Discrete Charge Discrete Charge Distribution (Cont’d) Distribution (Cont’d)             _                        N i i i e V Q V Q V Q V Q V V V Q V V V Q V V V Q V V V Q W 1 3 3 2 2 1 1 34 24 14 4 43 23 13 3 42 32 12 2 41 31 21 1 2 1 ... 2 1 ... ... ... ... ... 2 1

(34)

Distribution (Cont’d) Distribution (Cont’d)



 



N

i j j

ji

i

V

1

where

 Physically, V_i is the potential at the location of the ith point charge due to the other (N-1) charges.

(35)

Continuous Charge Continuous Charge

Distribution Distribution

   r V r dv

q V

Q

dv q

Q

V

ev n

i

i i

ev







1

   

r V r dv q

W

V

ev

e 



2 1

(36)

Continuous Charge

D q_ev   



D



dv V

W

V

e 



 

2 1

 D V D D V V      

: identity vector



V D



dv D V dv

W_e 



  





2 1 2

(37)

Divergence theorem and

dv E

D s

d D

V W

V S

e 



 





2 1 2

1

V

(38)

 Let the volume Let the volume V_V be all of space. Then the closed surface be all of space. Then the closed surface S_S

is sphere of radius infinity. All sources of finite extent look

like point charges. Hence,

0 lim

1

1 ₂

2

 

 





V D d s

R ds

R D

R V

(39)

dv E

D W

V

e 





2 1

2

2 1 2

1

E E

D w

dv w

W _e

V

e

e 



    

Electrostatic energy density in J/m3_.

(40)

dv E

P dv

E dv

E D

W

V V

V

e 



 









2 1 2

1 2

1 ₂

0

 energy required to

set the field up in free space

energy required to polarize the dielectric

P E

(41)

Capacitor

   

 ₂ ₁ ₁₂

2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 QV Q V V ds r q V ds r q V dv r V r q W c c S es S es V ev e          V₂ V_ + -+Q -Q V₁₂

(42)

Capacitor

2

2 1 2

1

CV QV

W_e  

(43)

Electrostatic Forces: The Electrostatic Forces: The

Principle of Virtual Work Principle of Virtual Work

 Electrostatic forces acting on bodies Electrostatic forces acting on bodies

can be computed using the

can be computed using the principle of principle of virtual work

virtual work._.

 The force on any conductor or The force on any conductor or

dielectric body within a system can be dielectric body within a system can be

obtained by assuming a differential obtained by assuming a differential

displacement of the body and displacement of the body and

computing the resulting change in the computing the resulting change in the

electrostatic energy of the system. electrostatic energy of the system.

(44)

Electrostatic Forces: The

Principle of Virtual Work

(Cont’d)

 The electrostatic force can be The electrostatic force can be

evaluated as the gradient of the evaluated as the gradient of the

electrostatic energy of the electrostatic energy of the

system, provided that the energy system, provided that the energy

is expressed in terms of the is expressed in terms of the

coordinate location of the body coordinate location of the body

being displaced. being displaced.

(45)

(Cont’d)

 When using the principle of When using the principle of

virtual work, we can assume virtual work, we can assume

either that the conductors either that the conductors

maintain a constant charge or maintain a constant charge or

that they maintain a constant that they maintain a constant

voltage (i.e, they are connected voltage (i.e, they are connected

to a battery). to a battery).

(46)

(Cont’d)

 For a system of bodies with fixed For a system of bodies with fixed

charges, the total electrostatic

force acting on the body is given by

e

Q

W

(47)

(Cont’d)

 For a system of bodies with fixed For a system of bodies with fixed

potentials, the total electrostatic

force acting on the body is given by

e

V

W

(48)

Force on a Capacitor Force on a Capacitor

Plate Plate

 Compute the force on one plate of a Compute the force on one plate of a

charged parallel plate capacitor.

Neglect fringing

Neglect fringing of the field._{of the field.}

y

+Q

• The force on the

upper plate can be found assuming a system of fixed charge.

(49)

Plate (Cont’d) Plate (Cont’d)

_{The capacitance can be written as a function of the location of the upper plate:}_{The capacitance can be written as a function of the location of the upper plate:}

_{The electrostatic energy stored in the capacitor may be evaluated as a function of the charge on the upper plate and its location:}_{The electrostatic energy stored in the capacitor may be evaluated as a function of the charge on the upper plate and its location:}

 

y A y

C d

A

C  0   0

 y Q_{ } Q y W_e

2 2



 

(50)

_{The force on the upper plate is given by}_{The force on the upper plate is given by}

_Using_Using_{Q = CV}_{Q = CV}_,_,

  A Q a y y W a W

F e _y

y e Q 0 2 2 ˆ ˆ          d CV a

F _Q _y

2 ˆ

2

 

(51)

 Compute the force on one plate of a Compute the force on one plate of a

charged parallel plate capacitor.

Neglect fringing

Neglect fringing of the field._{of the field.}

y

V = V₁₂

• The force on the

upper plate can be found assuming a system of fixed potential.

(52)

_{The capacitance can be written as a function of the location of the upper plate:}_{The capacitance can be written as a function of the location of the upper plate:}

_{The electrostatic energy stored in the capacitor may be written as a function of the voltage across the plates and the location of the upper plate:}_{The electrostatic energy stored in the capacitor may be written as a function of the voltage across the plates and the location of the upper plate:}

 

y A y

C d

A

C  0   0

 y CV AV

W 1

2 0

2 _ 

(53)

_{The force on the upper plate is given by}_{The force on the upper plate is given by}

_{Manipulating, we obtain}_{Manipulating, we obtain}

  2 2 0 2 ˆ ˆ y AV a y y W a W

F e _y

y e V         d CV a

F _Q _y

2 ˆ

2

 

(54)

Steady Electric Current Steady Electric Current

 Electrostatics is the study of Electrostatics is the study of

charges at rest. charges at rest.

 Now, we shall allow the charges Now, we shall allow the charges

to move, but with a constant to move, but with a constant

velocity (no time variation). velocity (no time variation).

 ““steady electric currentsteady electric current” = ” =

“

(55)

Conductors and Conductors and

Conductivity Conductivity

 A A conductor_conductor is a material in which electrons is a material in which electrons are free to migrate over macroscopic

are free to migrate over macroscopic

distances within the material.

 Metals are good conductors because they Metals are good conductors because they have many free electrons per unit volume.

have many free electrons per unit volume.

 Other materials with a smaller number of Other materials with a smaller number of free electrons per unit volume are also

free electrons per unit volume are also

conductors.

 ConductivityConductivity is a measure of the ability of is a measure of the ability of the material to conduct electricity.

(56)

Semiconductor Semiconductor

 A A semiconductorsemiconductor is a material in is a material in

which electrons in the outermost

shell are able to migrate over

macroscopic distances when a

modest energy barrier is overcome.

 SemiconductorsSemiconductors support the flow of support the flow of

both negative charges (electrons)

and positive charges (holes).

(57)

Conduction Current Conduction Current

 When subjected to a field, an electron in When subjected to a field, an electron in

a conductor migrates through the

material constantly colliding with the

lattice and losing momentum.

 The net effect is that the electron moves The net effect is that the electron moves

(drifts) with an average drift velocity

that is proportional to the electric field.

E

(58)

 Consider a conducting wire in which Consider a conducting wire in which

charges subject to an electric field are

moving with drift velocity

moving with drift velocity vvd_d..

E _v

d

electron

s

n aˆ

(59)

 If there are If there are nn_c_c free electrons per free electrons per

cubic meter of material, then the

charge density within the wire is

 Consider an infinitesimal volume Consider an infinitesimal volume

associated with

associated with ss::

c ev en

q  

s

l s v   

(60)

 The total charge contained within The total charge contained within vv

is

 This charge packet moves through This charge packet moves through

the surface

the surface s_s with speed with speed

 The amount of time it takes for the The amount of time it takes for the

charge packet to move through

charge packet to move through s_s is is

l s en

v q

Q  _ev   _c 



n e

d

n v E a

aˆ     ˆ

l t  

(61)

 CurrentCurrent is the rate at which charges is the rate at which charges

passes through a specified surface

area (such as the cross-section of a

wire).

 The incremental current through The incremental current through ss

is given by





n e

c s E a

en t

Q

I    ˆ

  

(62)

Current Density Current Density

 The component of the current density in the direction normal to The component of the current density in the direction normal to ss is is

 In general, the current density is given by_{In general, the current density is given by}

 _n 

e c

n en E a

s I a

J ˆ   ˆ

  

 

E en

(63)

Current Density (Cont’d) Current Density (Cont’d)

 The constant of proportionality The constant of proportionality

between the electric field and the

conduction current density is

called the

called the conductivityconductivity of the _{of the}

material:

 Ohm’s law at a pointOhm’s law at a point::

e c

en 

 

E

J





(64)

Current Density (Cont’d) Current Density (Cont’d)

 The The conductivityconductivity of the medium is the of the medium is the

macroscopic quantity which allows

us to treat conduction current

without worrying about the

microscopic behavior of conductors.

 In In semiconductorssemiconductors, we have both holes , we have both holes

and electrons





p p

e

e N

N

e  

  

hole mobility hole density

(65)

Current Density Current Density

 The total current flowing through a The total current flowing through a

cross-sectional area

cross-sectional area SS may be found may be found as

as

 If the current density is uniform If the current density is uniform

throughout the cross-section, we have







S

s d J

I



J a



A I   ˆ

cross-sectional area

(66)

Current Flow Current Flow

 Consider a wire of non-uniform Consider a wire of non-uniform

cross-section:

E

(67)

Current Flow (Cont’d) Current Flow (Cont’d)

 To maintain a constant electric field To maintain a constant electric field

and a steady current flow, both

and a steady current flow, both E_E and and

J

J must be parallel to the conductor must be parallel to the conductor boundaries.

boundaries.

 The total current passing through the The total current passing through the

cross-section

cross-section AA1₁ must be the same as must be the same as

through the cross-section

through the cross-section AA2₂. So the . So the

current density must be greater in

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Ohm’s Law and Resistors Ohm’s Law and Resistors

 Consider a conductor of uniform Consider a conductor of uniform

cross-section:

• Let the wires and the two exposed faces of the

“resistor” be perfect conductor.

l

A E  _A

V₂ _F

1

I • In a perfect conductor:

J is finite  is infinite

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Ohm’s Law and Ohm’s Law and Resistors (Cont’d) Resistors (Cont’d)

 To derive Ohm’s law for resistors from To derive Ohm’s law for resistors from

Ohm’s law at a point, we need to relate

the circuit quantities (

the circuit quantities (VV and _andII) to the field _{) to the field}

quantities (

quantities (EE and and JJ))

 The electric field within the material is The electric field within the material is

given by

 The current density in the wire isThe current density in the wire is

l V l

V V

l V

E  12  2  1 

A I J 

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Ohm’s Law and Ohm’s Law and Resistors (Cont’d) Resistors (Cont’d)

Plugging into Plugging into J = J = EE, we have, we have

Define the resistance of the device asDefine the resistance of the device as

Thus,Thus,

I A l V





A l R





RI V 

Ohm’s law for resistors