Chapter 21-part1Chapter 21-part1

Current and ResistanceCurrent and Resistance

1 Electric Current1 Electric Current

Whenever electric charges move, Whenever electric charges move, an an electric currentelectric current is said to exist is said to exist

The current is the The current is the rate at which the rate at which the charge flows through a certain charge flows through a certain cross-sectioncross-section

For the current definition, we look For the current definition, we look at the charges flowing at the charges flowing perpendicularly to a surface of perpendicularly to a surface of area area AA

Definition of the current:Definition of the current: Charge in motion Charge in motion

through an area through an area A. A. TheThe time time raterate of the of the charge flow through charge flow through AA defines the defines the current (=charges current (=charges per time):per time):

II==QQ//tt Units: C/s=As/s=AUnits: C/s=As/s=A SI unit of the SI unit of the

current: current: Ampere Ampere

+ -

Electric Current, contElectric Current, cont

The direction of current flow is the The direction of current flow is the direction positive charge would flowdirection positive charge would flow This is known as This is known as conventional (technical) conventional (technical)

current flow, i.e., from plus (+) to minus (-)current flow, i.e., from plus (+) to minus (-) However, in a common conductor, such as However, in a common conductor, such as

copper, the current is due to the motion of the copper, the current is due to the motion of the negatively charged electronsnegatively charged electrons

It is common to refer to a moving It is common to refer to a moving charge as a mobile charge as a mobile charge carriercharge carrier A charge carrier can be positive or negativeA charge carrier can be positive or negative

2 Current and Drift Speed2 Current and Drift Speed

Charged particles Charged particles move through a move through a conductor of cross-conductor of cross-sectional area sectional area AA

nn is the number of is the number of charge carriers per charge carriers per unit volume unit volume V V (=“concentration”)(=“concentration”)

nAnAxx==nV nV is the is the total number of total number of charge carriers in charge carriers in VV

Current and Drift Speed, Current and Drift Speed, contcont

The total charge is the number of carriers times The total charge is the number of carriers times the charge per carrier, the charge per carrier, q q (elementary charge) (elementary charge) ΔΔQQ = ( = (nAnAΔΔxx))qq [unit: (1/m[unit: (1/m33)(m)(m2 2 m)As=C]m)As=C]

The drift speed, The drift speed, vvdd, is the speed at which the , is the speed at which the carriers movecarriers move vvdd = Δ = Δxx/Δ/Δtt

Rewritten: Rewritten: ΔΔQQ = ( = (nAvnAvddΔΔtt))qq

Finally, current, Finally, current, II = Δ = ΔQQ/Δ/Δtt = = nqvnqvddAA

ΔΔxx

Current and Drift Speed, Current and Drift Speed, finalfinal

If the conductor is isolated, the If the conductor is isolated, the electrons undergo (thermal) electrons undergo (thermal) random motionrandom motion

When an electric field is set up in When an electric field is set up in the conductor, it creates an the conductor, it creates an electric force on the electrons and electric force on the electrons and hence a currenthence a current

Charge Carrier Motion in a Charge Carrier Motion in a ConductorConductor

The electric field force The electric field force FF imposes a drift on an imposes a drift on an electron’s random motion electron’s random motion (10(1066 m/s) in a conducting m/s) in a conducting material. Without field material. Without field the electron moves from the electron moves from PP11 to P to P22. With an applied . With an applied field the electron ends up field the electron ends up at Pat P22’; i.e., a distance ’; i.e., a distance vvddtt from Pfrom P22, where , where vvdd is the is the drift velocity (typically drift velocity (typically 1010-4-4 m/s). m/s).

Does the direction of Does the direction of the current depend the current depend on the sign of the on the sign of the charge? charge? No!No!

((a) Positive charges a) Positive charges moving in the same moving in the same direction of the direction of the field produce the field produce the same positive same positive current as (b) current as (b) negative charges negative charges moving in the moving in the direction opposite direction opposite to the field.to the field.

E

E

vd

vd

qvd

(-q)(-vd) = qvd

Current density:Current density: The current per unit cross-section is called the The current per unit cross-section is called the current current

density density JJ::

JJ==II//AA= = nqvnqvddAA//AA==nqvnqvdd

In general, a conductor may contain several different kinds In general, a conductor may contain several different kinds

of charged particles, concentrations, and drift velocities. of charged particles, concentrations, and drift velocities. Therefore, we can define a Therefore, we can define a vector current densityvector current density: :

JJ==nn11qq11vvd1d1++nn22qq22vvd2d2+…+…

Since, the product Since, the product qqvvdd is for positive and negative charges is for positive and negative charges in the direction of in the direction of EE, , the vector current density the vector current density JJ always always points in the direction of the field points in the direction of the field EE. .

Example: Example:

An 18-gauge copper wire (diameter An 18-gauge copper wire (diameter 1.02 mm) carries a constant 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. current of 1.67 A to a 200 W lamp. The density of free electrons is The density of free electrons is 8.58.510102828 per cubic meter. Find the per cubic meter. Find the magnitudes of (a) the current magnitudes of (a) the current density and (b) the drift velocity.density and (b) the drift velocity.

  

Solution:Solution:

(a) (a) AA==dd22/4=(0.00102 m)/4=(0.00102 m)22/4=8.2/4=8.21010-7-7 m m22

JJ==II//AA=1.67 A/(8.2=1.67 A/(8.21010-7-7 m m22)=2.0)=2.0101066 A/m A/m22

(b)(b) From From JJ==II//AA==nqvnqvdd, it follows:, it follows:

)C1060.1)(m105.8(

m/A100.219328

26

d

nq

Jv

vd=1.510-4 m/s=0.15 mm/s

3 Electrons in a Circuit3 Electrons in a Circuit

The drift speed is much smaller than The drift speed is much smaller than the average speed between collisionsthe average speed between collisions

When a circuit is completed, the When a circuit is completed, the electric field travels with a speed electric field travels with a speed close to the speed of lightclose to the speed of light

Although the drift speed is on the Although the drift speed is on the order of 10order of 10-4-4 m/s the effect of the m/s the effect of the electric field is felt on the order of 10electric field is felt on the order of 1088 m/sm/s

Meters in a Circuit – Ammeter

An ammeter is used to measure currentAn ammeter is used to measure current In line with the bulb, all the charge passing In line with the bulb, all the charge passing

through the bulb also must pass through the through the bulb also must pass through the meter (in series!)meter (in series!)

Meters in a Circuit - Voltmeter

A voltmeter is used to measure voltage A voltmeter is used to measure voltage (potential difference)(potential difference) Connects to the two ends of the bulb Connects to the two ends of the bulb

(parallel)(parallel)

QUICK QUIZ

Look at the four “circuits” shown below and select those that will light the bulb.

4 Resistance and Ohm’s 4 Resistance and Ohm’s lawlaw

In a In a homogeneous homogeneous conductor, the conductor, the current density is current density is uniform over any uniform over any cross section, and cross section, and the electric field the electric field is constant along is constant along the length. the length.

a

b

V=Va-Vb=EL

ResistanceResistance

The ratio of the potential drop to the The ratio of the potential drop to the current is called current is called resistanceresistance of the of the segment: segment:

Unit: V/A=Unit: V/A=ohmohm

I

VR

Resistance, contResistance, cont

Units of resistance are Units of resistance are ohmsohms (Ω) (Ω) 1 Ω = 1 V / A1 Ω = 1 V / A

Resistance in a circuit arises due to Resistance in a circuit arises due to collisions between the electrons collisions between the electrons carrying the current with the fixed carrying the current with the fixed atoms inside the conductoratoms inside the conductor

Ohm’s LawOhm’s Law

VV II VV=const.=const.II VV==RIRI Ohm’s Law is an empirical relationship Ohm’s Law is an empirical relationship

that is valid only for certain materialsthat is valid only for certain materials Materials that obey Ohm’s Law are said to Materials that obey Ohm’s Law are said to

be be ohmicohmic II==V/R V/R RR, , II0, open circuit0, open circuit; ; RR0, 0, II, short , short

circuitcircuit

Ohm’s Law, finalOhm’s Law, final

Plots of Plots of VV versus versus II for (a) ohmic and for (a) ohmic and (b) nonohmic (b) nonohmic materials. The materials. The resistance resistance RR==VV//II is is independent of independent of II for ohmic for ohmic materials, as is materials, as is indicated by the indicated by the constant slope of constant slope of the line in (a).the line in (a).

Ohmic

Nonohmic

5 Resistivity5 Resistivity Expected:Expected: R RLL//A A  The resistance of an The resistance of an

ohmic conductor is ohmic conductor is proportional to its proportional to its length, length, LL, and inversely , and inversely proportional to its proportional to its cross-sectional area, cross-sectional area, AA

ρρ (“rho”) in (“rho”) in m is m is the constant of the constant of proportionality and is proportionality and is called the called the resistivityresistivity of the materialof the material

A

LρR

ExampleExample

Determine the required length of Determine the required length of nichrome (nichrome (=10=10-6-6 m) with a m) with a radius of 0.65 mm in order to radius of 0.65 mm in order to obtain obtain RR=2.0 =2.0 ..

RR==LL//AALL==RARA//

m65.2Ωm10

(0.00065m))(2.06

2

L

The resistivity The resistivity depends on the depends on the material and the material and the temperature temperature

6 Temperature Variation 6 Temperature Variation of Resistivityof Resistivity

For most metals, resistivity For most metals, resistivity increases with increasing increases with increasing temperaturetemperature With a higher temperature, the With a higher temperature, the

metal’s constituent atoms vibrate metal’s constituent atoms vibrate with increasing amplitudewith increasing amplitude

The electrons find it more difficult to The electrons find it more difficult to pass the atoms pass the atoms (more scattering!)(more scattering!)

Temperature Variation of Temperature Variation of Resistivity, contResistivity, cont

For most metals, resistivity increases For most metals, resistivity increases approximately linearly with temperature approximately linearly with temperature over a limited temperature rangeover a limited temperature range

ρρoo is the resistivity at some reference is the resistivity at some reference temperature temperature TToo

TToo is usually taken to be 20° C is usually taken to be 20° C is the is the temperature coefficient of temperature coefficient of

resistivity resistivity [unit: 1/([unit: 1/(C)]C)]

)]([1 oo TTαρρ

Temperature Variation of Temperature Variation of ResistanceResistance

Since the resistance of a conductor Since the resistance of a conductor with uniform cross sectional area is with uniform cross sectional area is proportional to the resistivity, the proportional to the resistivity, the temperature variation of resistance temperature variation of resistance can be writtencan be written

)]([1 oo TTαRR

ExampleExample

The material of the wire has a resistivity The material of the wire has a resistivity of of 00=6.8=6.81010-5-5 m at m at TT00=320=320C, a C, a

temperature coefficient of temperature coefficient of =2.0=2.01010-3-3 (1/(1/C) and C) and LL=1.1 m. =1.1 m. Determine the Determine the resistance of the heater wire at an resistance of the heater wire at an operating temperature of 420operating temperature of 420C.C.

SolutionSolution

==00[1+[1+00)])] =[6.8=[6.81010-5-5 m]m][[1+1+((2.02.01010-3-3 ( (C)C)-1-1) ) ((420420C-320C-320CC))]]=8.2=8.21010-5-5

mm RR==LL//AA R=R=(8.2(8.21010-5-5 m)(1.1 m)/(3.1m)(1.1 m)/(3.11010-6-6 m m22)) RR=29 =29

7 Superconductors7 Superconductors

A class of materials A class of materials and compounds whose and compounds whose resistances fall to resistances fall to virtually zero below a virtually zero below a certain temperature, certain temperature, TTCC

TTCC is called the is called the critical temperature critical temperature (in the graph 4.1 K)(in the graph 4.1 K)

“normal”

Superconductors, contSuperconductors, cont

The value of The value of TTCC is sensitive to is sensitive to Chemical compositionChemical composition PressurePressure Crystalline structureCrystalline structure

Once a current is set up in a Once a current is set up in a superconductor, it persists without superconductor, it persists without any applied voltageany applied voltage Since Since RR = 0 = 0

Superconductor TimelineSuperconductor Timeline 19111911

Superconductivity discovered by H. Kamerlingh Superconductivity discovered by H. Kamerlingh OnnesOnnes

19861986 High-temperature superconductivity discovered High-temperature superconductivity discovered

by Bednorz and Müllerby Bednorz and Müller Superconductivity near 30 KSuperconductivity near 30 K

19871987 Superconductivity at 92 K and 105 KSuperconductivity at 92 K and 105 K

CurrentCurrent More materials and more applicationsMore materials and more applications

TTcc values for values for different different materials; materials; note note the high the high TTcc values for the values for the oxides. oxides.

It’s magic!It’s magic!

8 Electrical Energy and 8 Electrical Energy and PowerPower

In a circuit, as a charge moves through In a circuit, as a charge moves through the battery, the electrical potential the battery, the electrical potential energy of the system is increased by energy of the system is increased by ΔΔQQΔΔV V [AsV=Ws=J][AsV=Ws=J] The chemical potential energy of the battery The chemical potential energy of the battery

decreases by the same amountdecreases by the same amount As the charge moves through a resistor, As the charge moves through a resistor,

it loses this potential energy during it loses this potential energy during collisions with atoms in the resistorcollisions with atoms in the resistor The temperature of the resistor will increaseThe temperature of the resistor will increase

Electrical Energy and Electrical Energy and Power, contPower, cont

IVVt

Q

t

WP

Δ

Δ

The rate of the energy transfer is power (P):

V

Units: (C/s)(J/C) =J/s=W

1J=1Ws=1Nm

W=AV

Electrical Energy and Electrical Energy and Power, contPower, cont

From Ohm’s Law, alternate From Ohm’s Law, alternate forms of power are (use forms of power are (use VV==IRIR and and II==VV//RR))

R

VRIIVP

22

Joule heat (I2R losses)

Electrical Energy and Electrical Energy and Power, finalPower, final

The SI unit of power is Watt (W)The SI unit of power is Watt (W) II must be in must be in AmperesAmperes, , RR in in OhmsOhms and and

VV in in VoltsVolts The unit of energy used by electric The unit of energy used by electric

companies is the companies is the kilowatt-hourkilowatt-hour This is defined in terms of the unit of This is defined in terms of the unit of

power and the amount of time it is power and the amount of time it is suppliedsupplied

1 kWh =(101 kWh =(1033 W)(3600 s)= 3.60 x 10 W)(3600 s)= 3.60 x 1066 J J

9 Electrical Activity in 9 Electrical Activity in the Heartthe Heart

Every action involving Every action involving the body’s muscles is the body’s muscles is initiated by electrical initiated by electrical activityactivity

Voltage pulses cause Voltage pulses cause the heart to beatthe heart to beat

These voltage pulses These voltage pulses ((1 mV) are large 1 mV) are large enough to be enough to be detected by detected by equipment attached equipment attached to the skinto the skin

Heart beat Initiation

Electrocardiogram (EKG)Electrocardiogram (EKG)

A normal EKGA normal EKG P occurs just before P occurs just before

the atria begin to the atria begin to contractcontract

The QRS pulse occurs The QRS pulse occurs in the ventricles just in the ventricles just before they contractbefore they contract

The T pulse occurs The T pulse occurs when the cells in the when the cells in the ventricles begin to ventricles begin to recoverrecover

Abnormal EKG, 1Abnormal EKG, 1

The QRS portion The QRS portion is wider than is wider than normalnormal

This indicates the This indicates the possibility of an possibility of an enlarged heartenlarged heart

Abnormal EKG, 2Abnormal EKG, 2

There is no constant relationship between P and QRS There is no constant relationship between P and QRS pulsepulse

This suggests a blockage in the electrical conduction This suggests a blockage in the electrical conduction path between the SA and the AV nodespath between the SA and the AV nodes

This leads to inefficient heart pumpingThis leads to inefficient heart pumping

Abnormal EKG, 3Abnormal EKG, 3

No P pulse and an irregular spacing between No P pulse and an irregular spacing between the QRS pulsesthe QRS pulses

Symptomatic of irregular atrial contraction, Symptomatic of irregular atrial contraction, called called fibrillationfibrillation

The atrial and ventricular contraction are The atrial and ventricular contraction are irregularirregular

Implanted Cardioverter Implanted Cardioverter Defibrillator (ICD)Defibrillator (ICD)

Devices that can Devices that can monitor, record monitor, record and logically and logically process heart process heart signalssignals

Then supply Then supply different corrective different corrective signals to hearts signals to hearts that are not that are not beating correctlybeating correctly

Monitor lead

Dual chamber ICD