Lect06

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EEE 498/598

Overview of Electrical

Engineering

Engineering Lecture 6:

Lecture 6:

Electromotive Force;

Kirchoff’s Laws;

Redistribution of Charge;

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Lecture 6 Objectives Lecture 6 Objectives

 To study electromotive force; To study electromotive force; Kirchoff’s laws; charge

Kirchoff’s laws; charge

redistribution in a conductor; redistribution in a conductor;

and boundary conditions for and boundary conditions for

steady current flow. steady current flow.

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Electromotive Force Electromotive Force

 Steady current flow requires a Steady current flow requires a closed circuit.

closed circuit.

 Electrostatic fields produced by Electrostatic fields produced by stationary charges are

stationary charges are

conservative. Thus, they cannot conservative. Thus, they cannot

by themselves maintain a steady by themselves maintain a steady

current flow. current flow.

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(Cont’d) (Cont’d)

 The current The current II must must

be zero since the

electrons cannot

gain back the

energy they lose in

traveling through

the resistor.

  



I

increasing potential

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 To maintain a To maintain a

steady current, steady current,

there must be an there must be an

element in the element in the

circuit wherein circuit wherein

the potential

the potential risesrises along the

along the

direction of the direction of the

  



I

+

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-Electromotive Force Electromotive Force

 For the potential to rise along the For the potential to rise along the

direction of the current, there must be a direction of the current, there must be a

source

source which converts some form of energy _{which converts some form of energy} to electrical energy.

to electrical energy.

 Examples of such sources are:Examples of such sources are:

 batteriesbatteries  generatorsgenerators

 thermocouplesthermocouples

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Inside the Voltage Inside the Voltage

Source Source

• E_emf is the electric field established by the energy conversion.

• This field moves positive charge to the upper plate, and negative charge to the lower plate.

• These charges establish an electrostatic field E.

emf E

+ + +

-E

In equilibrium:

0



 E

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At all points in the circuit, we must have

E E

E J

emf

total  





exists only in battery

E

  



I

+

-E E_emf

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 Integrate around the circuit in the Integrate around the circuit in the

direction of current flow



          C emf C C C total l d J l d E l d E l d J l d E   1 1

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 Define the Define the electromotive forceelectromotive force ( (emfemf) or ) or “voltage” of the battery as

“voltage” of the battery as









 E dl

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_{We also note that}_{We also note that}

_{Thus, we have the circuit relation}_{Thus, we have the circuit relation}

RI I

A l l

d J

C

 





_1 _

RI

V

_emf



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Overview of Electromagnetics

Maxwell’s equations

Fundamental laws of classical electromagnetics

Special

cases Electro-_statics Magneto-_statics _magnetic

Electro-waves

Kirchoff’s Laws

Statics: 0

 

t





d

Geometric Optics

Transmission Line Theory Circuit

Theory

Input from other disciplines

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Kirchhoff’s Voltage Law Kirchhoff’s Voltage Law  For a closed circuit containing For a closed circuit containing

voltages sources and resistors, we

have



V_emf  I



R

• “the algebraic sum of the emfs around a closed circuit equals the algebraic sum of the voltage drops over the resistances around the circuit.”

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Kirchhoff’s Voltage Law Kirchhoff’s Voltage Law

 Strictly speaking KVL only Strictly speaking KVL only

applies to circuits with steady applies to circuits with steady

currents (DC). currents (DC).

 However, for AC circuits having However, for AC circuits having dimensions much smaller than a dimensions much smaller than a

wavelength, KVL is also wavelength, KVL is also

approximately applicable. approximately applicable.

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Conservation of Charge Conservation of Charge

 Electric charges can neither be Electric charges can neither be

created nor destroyed. created nor destroyed.

 Since current is the flow of charge Since current is the flow of charge

and charge is conserved, there and charge is conserved, there

must be a relationship between the must be a relationship between the

current flow out of a region and current flow out of a region and the rate of change of the charge the rate of change of the charge

contained within the region. contained within the region.

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 Consider a volume Consider a volume

V

V bounded by a _{bounded by a}

closed surface

closed surface SS in _in

a homogeneous

medium of

permittivity

permittivity  and and conductivity

conductivity 

containing charge

density

density qq ._.

V S

ds

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(Cont’d) (Cont’d)  The net current The net current

leaving

leaving VV

through

through SS must _must

be equal to the

time rate of

decrease of the

total charge

within

within VV, i.e.,, i.e.,

V S

ds

q_ev

dt dQ

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 The net current leaving the The net current leaving the region is given by

region is given by

 The total charge enclosed within The total charge enclosed within the region is given by

the region is given by



 

S

s d J

I





V

ev dv

q Q

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 Hence, we haveHence, we have



  

V

ev S

dv q

dt d s

d J

net outflow

of current net rate of

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Continuity Equation Continuity Equation

 Using the Using the divergence theoremdivergence theorem, we , we have

have

 We also haveWe also have



   

V S

dv J

s d J



 _

V

ev

V

ev dv

t q dv

q dt

d

Becomes a partial

derivative when moved inside of

the integral because q_ev is a

function of position as well

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 Thus,Thus,

 Since the above relation must be Since the above relation must be true for

true for any and all regionsany and all regions, we have _{, we have}

0

 

 







V V

dv t

dv

J 

0

 

  



t

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 For steady currents,For steady currents,

 Thus, Thus,

0

 



t



0

 

 J

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Continuity Equation in Continuity Equation in

Terms of Electric Field Terms of Electric Field

 Ohm’s law in a conducting medium statesOhm’s law in a conducting medium states

 For a homogeneous mediumFor a homogeneous medium

 But from Gauss’s law,But from Gauss’s law,

 Therefore, the volume charge density, Therefore, the volume charge density, , must be , must be

zero in a homogeneous conducting medium

E

J  

0

0    

 

 



 J  E E

ev

q

E 

 

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Kirchhoff’s Current Law Kirchhoff’s Current Law

 Since Since J_J is solenoidal, is solenoidal,

we must have

 In a circuit, steady In a circuit, steady

current flows in

wires.

 Consider a “node” in Consider a “node” in

a circuit.

0

 



S

s d

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Kirchoff’s Current Law Kirchoff’s Current Law

 We have for a node in a circuitWe have for a node in a circuit



I  0

• “the algebraic sum of all currents leaving a junction must be zero.”

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Kirchoff’s Current Law Kirchoff’s Current Law

 Strictly speaking KCL only Strictly speaking KCL only

applies to circuits with steady applies to circuits with steady

currents (DC). currents (DC).

 However, for AC circuits having However, for AC circuits having dimensions much smaller than a dimensions much smaller than a

wavelength, KCL is also wavelength, KCL is also

approximately applicable. approximately applicable.

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Redistribution of Free Redistribution of Free

Charge Charge

 Charges introduced into the interior of an Charges introduced into the interior of an

isolated conductor migrate to the conductor

surface and redistribute themselves in such

a way that the following conditions are met:

 E_E = 0_{= 0} within the conductor within the conductor

 EE_t_t = 0 = 0 just outside the conductor just outside the conductor  qq_ev_ev = 0 = 0 within the conductor within the conductor

 qq_es_es  0 0 on the surface of the conductor on the surface of the conductor 

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Charge (Cont’d) Charge (Cont’d)

 We can derive the differential equation We can derive the differential equation

governing the redistribution of charge

from Gauss’s law in differential form and

the continuity equation.

 From Gauss’s law for the electric field, we From Gauss’s law for the electric field, we

have

ev ev

ev J q

q E

q D

      

 

 

 

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_{From the continuity equation, we have}_{From the continuity equation, we have} _{Combining the two equations, we obtain}_{Combining the two equations, we obtain}

t J

    

 

 , _ _{ }_, _ ₀

 

t r t

t

r _

 

 Describes the time _{evolution of the}

charge density at a given location.

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 The solution to the DE is given The solution to the DE is given by

by

 

_r _t



 

_r _e t _r 



/

0

,  

Initial charge distribution at t = 0

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 The initial charge distribution at any point in The initial charge distribution at any point in

the bulk of the conductor decays

exponentially to zero with a time constant

exponentially to zero with a time constant r_r..

 At the same time, surface charge is building At the same time, surface charge is building

up on the surface of the conductor.

 The The relaxation time_{relaxation time} decreases with increasing decreases with increasing

conductivity.

 For a good conductor, the time required for For a good conductor, the time required for

the charge to decay to zero at any point in the

bulk of the conductor (and to build up on the

surface of the conductor) is very small.

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Relaxation Times for Relaxation Times for

Some Materials Some Materials days 50 hrs 20 to 10 s 10 4 s 10 s 10 5 . 1 3 5 19 2          quartz r mica r amber r O H r copper r     

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Electrical Nature of Electrical Nature of

Materials as a Function Materials as a Function

for Frequency for Frequency

 The concept of The concept of relaxation timerelaxation time is is

also used to determine the

electrical nature (conductor or

insulator) of materials at a given

frequency.

 A material is considered to be a A material is considered to be a

good conductor if

1

1 _ _



 T  f

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Electrical Nature of

Materials as a Function

for Frequency (Cont’d)

for Frequency (Cont’d)  A material is considered to be a good A material is considered to be a good

insulator if

 A good conductor is a material with a A good conductor is a material with a

relaxation time such that any free charges

deposited within its bulk migrate to its

surface long before a period of the wave has

1

1 _ _



 f

f

T _r

r 

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Boundary Conditions Boundary Conditions for Steady Current Flow for Steady Current Flow

 The behavior of current flow across The behavior of current flow across the interface between two different the interface between two different

materials is governed by boundary materials is governed by boundary

conditions. conditions.

 The boundary conditions for The boundary conditions for

current flow are obtained from the current flow are obtained from the

integral forms

integral forms of the basic equations _{of the basic equations} governing current flow.

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Boundary Conditions for Boundary Conditions for

Steady Current Flow Steady Current Flow

1 1

,





2 2

,





n

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 The governing equations for The governing equations for steady

steady electric current (in a _{electric current (in a} conductor) are:

conductor) are:

0 0

0

 

 



 



S

l d J

l d E

s d J

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 The normal component of a The normal component of a

solenoidal

solenoidal vector field is continuous _{vector field is continuous} across a material interface:

across a material interface:

 The tangential component of a The tangential component of a

conservative

conservative vector field is continuous _{vector field is continuous} across a material interface:

across a material interface:

n n J

J₁  ₂

2 1t J t

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Conductor-Dielectric Conductor-Dielectric

Interface Interface

0 



0 



J

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Interface (Cont’d) Interface (Cont’d)

 The current in the conductor must flow The current in the conductor must flow

tangential to the boundary surface.

 The tangential component of the electric The tangential component of the electric

field must be continuous across the

interface.

 The normal component of the electric field The normal component of the electric field

must be zero at the boundary inside the

conductor, but not in the dielectric. Thus,

there will be a buildup of surface charge at

there will be a buildup of surface charge at the interface.

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no current flow  E_t = 0

+ + + + +

E

-+ -+ -+ -+ -+

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Conductor-Conductor Conductor-Conductor

Interface Interface

 The current bends as it cross the interface The current bends as it cross the interface

between two conductors

1



1

2 

 

2



1

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 The angles are related byThe angles are related by

 Suppose medium 1 is a good conductor and Suppose medium 1 is a good conductor and

medium 2 is a good insulator (i.e.,

medium 2 is a good insulator (i.e., 1₁ >> >> 22). ).

Then . In other words, the current

enters medium 2 at nearly right angles to

the boundary. This result is consistent with

the fact that the electric field in medium 2

should have a vanishingly small tangential

component at the interface.

        1 1 2 1

2 tan _ tan

 

0

2 

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 In general, there is a buildup of surface charge at In general, there is a buildup of surface charge at

the interface between two conductors.

  2

2 1 1 1 2 1 2 1 2 2 1 1 2 1 2 2 1 1 2 1 n n n n n n s n n n n n J E E E D D E E J J J                                     

 Only when the relaxation times of the two conductors are equal is there no buildup of surface charge at the