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Chapter 27

Current and Resistance

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Electric CurrentElectric current is the rate of flow of charge through some region of spaceThe SI unit of current is the ampere (A)

1 A = 1 C / s

The symbol for electric current is I

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Average Electric CurrentAssume charges aremoving perpendicular

to a surface of area AIf Q is the amount of charge that passesthrough A in time t ,then the averagecurrent is

av Q

t I

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Instantaneous Electric CurrentIf the rate at which the charge flowsvaries with time, the instantaneouscurrent, I , can be found

I dQdt

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Direction of CurrentThe charges passing through the area couldbe positive or negative or both

It is conventional to assign to the current thesame direction as the flow of positive chargesThe direction of current flow is opposite thedirection of the flow of electronsIt is common to refer to any moving charge asa charge carrier

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Current and Drift SpeedCharged particlesmove through a

conductor of cross-sectional area An is the number of charge carriers per

unit volumenA x is the totalnumber of chargecarriers

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Current and Drift Speed, contThe total charge is the number of carriers times the charge per carrier, q

Q = ( nA x )qThe drift speed, v d , is the speed atwhich the carriers move

v d = x / t Rewritten: Q = (nAv d t )qFinally, current, I av = Q /t = nqv d A

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Charge Carrier Motion in a

Conductor The zigzag black linerepresents the motionof a charge carrier in a

conductor The net drift speed issmall

The sharp changes indirection are due tocollisionsThe net motion of electrons is opposite thedirection of the electricfield

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Motion of Charge Carriers,

cont.In spite of all the collisions, the chargecarriers slowly move along theconductor with a drift velocity, v d Changes in the electric field that drivesthe free electrons travel through theconductor with a speed near that of light

This is why the effect of flipping a switch iseffectively instantaneous

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Motion of Charge Carriers,

finalElectrons do not have to travel from the lightswitch to the light bulb in order for the light to

operateThe electrons are already in the light filamentThey respond to the electric field set up bythe batteryThe battery does not supply the electrons, itonly establishes the electric field

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Drift Velocity, ExampleAssume a copper wire, with one freeelectron per atom contributed to the

currentThe drift velocity for a 12-gauge copper wire carrying a current of 10.0 A is

2.22 x 10-4

m/sThis is a typical order of magnitude for driftvelocities

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Current DensityJ is the current density of a conductor It is defined as the current per unit area

J = I / A = nq v d This expression is valid only if the current densityis uniform and A is perpendicular to the directionof the current

J has SI units of A/m 2

The current density is in the direction of thepositive charge carriers

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ConductivityA current density J and an electric fieldE are established in a conductor

whenever a potential difference ismaintained across the conductor J = E

The constant of proportionality, , iscalled the conductivity of theconductor

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Ohms LawOhms law states that for manymaterials, the ratio of the current

density to the electric field is a constant that is independent of the electric fieldproducing the current

Most metals obey Ohms lawMathematically, J = EMaterials that obey Ohms law are said tobe ohmic

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Ohms Law, cont.Not all materials follow Ohms law

Materials that do not obey Ohms law aresaid to be nonohmic

Ohms law is not a fundamental law of nature

Ohms law is an empirical relationshipvalid only for certain materials

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ResistanceIn a conductor, the voltage appliedacross the ends of the conductor isproportional to the current through theconductor The constant of proportionality is called

the resistance of the conductor

I

V R

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Resistance, cont.SI units of resistance are ohms ()

1 = 1 V / A

Resistance in a circuit arises due tocollisions between the electronscarrying the current with the fixed atomsinside the conductor

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ResistivityThe inverse of the conductivity is theresistivity :

= 1 /

Resistivity has SI units of ohm-meters( . m)Resistance is also related to resistivity:

R A

l

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Resistivity

Values

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Resistance and Resistivity,

SummaryEvery ohmic material has a characteristicresistivity that depends on the properties of

the material and on temperatureThe resistance of a material depends on itsgeometry and its resistivityAn ideal conductor would have zero resistivityAn ideal insulator would have infiniteresistivity

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ResistorsMost circuits useelements calledresistorsResistors are usedto control the currentlevel in parts of thecircuitResistors can becomposite or wire-wound

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Resistor Values

Values of resistors

are commonlymarked by coloredbands

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Ohmic Material, GraphAn ohmic deviceThe resistance is

constant over a widerange of voltagesThe relationshipbetween current and

voltage is linear The slope is relatedto the resistance

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Nonohmic Material, GraphNonohmic materialsare those whoseresistance changeswith voltage or currentThe current-voltagerelationship isnonlinear A diode is acommon example of a nonohmic device

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Resistance of a Cable,

ExampleAssume the siliconbetween the

conductors to beconcentric elementsof thickness dr The resistance of the hollow cylinder of silicon is

2

dR dr

rL

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Resistance of a Cable,

Example, cont.The total resistance across the entirethickness is

This is the radial resistance of the cable

This is fairly high, which is desirablesince you want the current to flow alongthe cable and not radially out of it

2lnb

a bR dR L a

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Electrical Conduction

A ModelThe diagram shows adescription of themotion of freeelectrons in aconductor The motion is random

There is no netdisplacement after many collisionsThe drift velocity is

zero

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Conduction Model, 2An electric field isapplied

The field modifiesthe motion of thecharge carriersThe electrons drift inthe directionopposite of E

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Active Figure 27.9

(SLIDESHOW MODE ONLY)

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Conduction Model, 3Assumptions:

The motion of an electron after a collision

is independent of its motion before thecollisionThe excess energy acquired by theelectrons in the field is lost to the atoms of

the conductor during the collisionThe energy given up to the atomsincreases their vibration and therefore thetemperature of the conductor increases

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Conduction Model, 4The force experienced by an electron isF = qE

From Newtons Second Law, theacceleration is a = F / m e = qE / m eApplying a motion equation v f = v i + a t

or v f = v i + (qE/m e)t Since the initial velocities are random,their average value is zero

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Conduction Model, 5Let be the average time intervalbetween successive collisionsThe average value of v f is the driftvelocityv f avg = v d = (qE/m e)

This is also related to the currentdensity: J = nqv d = (nq 2E / m e)

n is the number of charge carriers per unitvolume

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Conduction Model, finalUsing Ohms Law, expressions for theconductivity and resistivity of a conductor canbe found:

Note, the conductivity and the resistivity do

not depend on the strength of the fieldThe average time is also related to the freemean path: = /v av

2

2

1e

e

nq m

m nq

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Resistance and TemperatureOver a limited temperature range, theresistivity of a conductor varies

approximately linearly with thetemperature

o

is the resistivity at some referencetemperature T o

T o is usually taken to be 20 C is the temperature coefficient of resistivity

SI units of areo

C-1

[1 ( )]o o T T

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Temperature Variation of

ResistanceSince the resistance of a conductor withuniform cross sectional area is proportional tothe resistivity, you can find the effect of temperature on resistance

R = R o[1 + (T - T o)]

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Resistivity and Temperature,

Graphical ViewFor metals, the resistivityis nearly proportional tothe temperatureA nonlinear region alwaysexists at very lowtemperatures

The resistivity usuallyreaches some finite valueas the temperatureapproaches absolute zero

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Residual ResistivityThe residual resistivity near absolutezero is caused primarily by the

collisions of electrons with impuritiesand imperfections in the metalHigh temperature resistivity is

predominantly characterized bycollisions between the electrons and themetal atoms

This is the linear range on the graph

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SemiconductorsSemiconductors arematerials that exhibit a

decrease in resistivitywith an increase intemperature is negativeThere is an increase inthe density of chargecarriers at higher

temperatures

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SuperconductorsA class of materialsand compounds whoseresistances fall tovirtually zero below acertain temperature, T C

T C is called the criticaltemperature

The graph is the sameas a normal metalabove T C , but suddenlydrops to zero at T

C

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Superconductors, contThe value of T C is sensitive to:

chemical composition

pressuremolecular structure

Once a current is set up in a

superconductor, it persists without anyapplied voltageSince R = 0

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Superconductor ApplicationAn importantapplication of

superconductors is asuperconductingmagnetThe magnitude of the

magnetic field isabout 10 timesgreater than a normalelectromagnet

Used in MRI units

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Electrical Power Assume a circuit asshown

As a charge movesfrom a to b , theelectric potentialenergy of the system

increases by Q V The chemical energyin the battery mustdecrease by this

same amount

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Active Figure 27.13

(SLIDESHOW MODE ONLY)

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Electrical Power, 2As the charge moves through theresistor ( c to d ), the system loses this

electric potential energy duringcollisions of the electrons with theatoms of the resistor This energy is transformed into internalenergy in the resistor

Corresponds to increased vibrationalmotion of the atoms in the resistor

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Electric Power, 3The resistor is normally in contact with the air,so its increased temperature will result in a

transfer of energy by heat into the air The resistor also emits thermal radiationAfter some time interval, the resistor reachesa constant temperature

The input of energy from the battery is balancedby the output of energy by heat and radiation

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Electric Power, 4The rate at which the system losespotential energy as the charge passes

through the resistor is equal to the rateat which the system gains internalenergy in the resistor

The power is the rate at which theenergy is delivered to the resistor

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Electric Power, finalThe power is given by the equation:

Applying Ohms Law, alternative expressionscan be found:

Units: I is in A, R is in , V is in V, andis in W

I V

22 I I

V V R

R

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Electric Power TransmissionReal power lineshave resistance

Power companiestransmit electricity athigh voltages andlow currents tominimize power losses