norah ali al-moneef 1 conductors & semiconductors in conductors, the valence band is only...

Norah Ali Al-moneef1

Conductors & Semiconductors• In conductors, the valence band is only partially-full, so

electrons can easily move from being near one atom to being near another

• In semiconductors and insulators, the valence band is completely full, so electrons must gain extra energy to move

• In semiconductors, the band gap between the full valence band and the empty conduction band is small, so electrons move easily with only thermal energy

• In insulators, the band gap is larger, so electrons will not easily move into the conduction band


Conductors & Insulators• Electric current moves easily through some materials and less

easily through other materials• Materials that have very “tightly bound” electrons have few free

electrons when an electric force is applied. These materials are insulators (e.g. rubber, glass, dry wood)

• Materials that allow the movement of a large number of free electrons are called conductors (e.g., silver, copper, aluminum)

– Electrical energy is transferred through a conductor by means of the movement of free electrons that move from atom to atom

– Displaced electrons continue to “bump” each other

– The electrons move relatively slowly but this movement creates electrical energy throughout the conductor that is transferred almost instantaneously throughout the wire (e.g., billiard ball example, wind vs. sound example)


Electrons in an Electric Field


Conduction electrons move randomly in all directions in the absence of a field.

If a field is applied, the electric force results in acceleration in a particular direction:

F=ma= –eE a = –eE/mAs the charges accelerate, the potential

energy stored in the electric field is converted to kinetic energy which can be converted into heat and light as the electrons collide with atoms in the wire

This acceleration produces a velocityv = at = –eEt/m

ELECTRON MOTION IN A CONDUCTOR WITH AND WITHOUT AN ELECTRIC FIELD


27.1 Electric Current


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

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

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

Charge in motion through an area A. The time rate of the charge flow through A defines the current (=charges per time):

Units:1 C/s= 1 A SI unit of the current: Ampere

Definition of the current:

t

QI av

Electrical current


If an electric field points from left to right, positive charge carriers will move toward therightwhile negative charges will move toward theleft

The result of both is a net flow of positive charge to the right.

Current is the net change in positive charge per time

t

QI av

Instantaneous current i = d q / d t

• Coulomb (C) – represents the total charge of approximately 6.25 x 1018 electrons


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

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

the current is due to the motion of the negatively charged electrons

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

Charge Carrier Motion in a Conductor

The electric field force F imposes a drift on an electron’s random motion (106 m/s) in a conducting material. Without field the electron moves from P1 to P2. With an applied field the electron ends up at P2’; i.e., a distance vdt from P2, where vd is the drift velocity (typically 10-4 m/s).

Norah Ali Al-moneef 9

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

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


E

E

vd

vd

qvd

(-q)(-vd) = qvd

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

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

nAx=nV is the total number of charge carriers in V


The total charge is the number of carriers times the charge per carrier, q (elementary charge)

ΔQ = (nAΔx)q [unit: (1/m3)(m2 m)As=C]

Microscopic model of current


The drift speed, vd, is the speed at which the carriers movevd = Δx/Δt

Rewritten: ΔQ = (nAvdΔt)q

current, I = ΔQ/Δt = nqvdA

Δx

If the conductor is isolated, the electrons undergo (thermal) random motionWhen an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current


coulombs of charge pass a point in a wire every two seconds. Calculate current.

Coulomb (C) – represents the total charge of approximately 6.25 x 1018 electrons

Unit of Current – Ampere (A) = 1coulomb/second

A 1.5C/s 1.5s 2

C 3

t

QI

Example:

Example:


An 18-gauge copper wire (diameter 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. The density of free electrons is 8.51028 per cubic meter. Find the magnitudes of (a) the current density

(b) the drift velocity. (a) A=d 2/4=(0.00102 m)2/4=8.210-7 m2

J=I /A=1.67 A/(8.210-7 m2)=2.0106 A/m2

(b) From J=I /A=nqvd

)C1060.1)(m105.8(

m/A100.219328

26

d

nq

Jv

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

Example:

15 Norah Ali Al-moneef

• If 240 C of charge pass a point in a conductor in 5 min, what is the current through that point in the conductor?

Convert 5 min to seconds 5.0min X 60s/1 min = 300s

A 0.80 s 300

C 240

t

Q I

Example:


electrons 1061.2N 101.6

s 5835.0 t Q Q )

53.860 00.2

101.6 106.4

t

t

Q )

20.04

101.6 105.6

t

t

Q

19

19-

19-21

-1914

A

e

I

eNNeb

ANe

Ic

mANe

I

To the left

Example:


electrons 1061.2N 101.6

s 5835.0 t Q Q )

19

19-

A

e

I

eNNeb

As

CIa 835.0

00.2

67.1

t

Q )

ANe

Ic 53.860 00.2

101.6 106.4

t

t

Q )

19-21

Example:


II. Electric current1. Definition

t

QI

Units: [ I ] = 1A = 1 C/s

Conventional current

Electron flow

1020 electrons passed through the electric conductor during 4 seconds. Find the electric current through this conductor.

As

C

t

qI 4

4

)10)(106.1( 2019

CsAItq 5.0)1)(5.0( CsAItq 30)60)(5.0(

Example: The electric current of 0.5 A is flowing through the electric conductor. a) What electric charge is passing through the conductor during each second b) What electric charge will pass through the conductor during 1 minute?

a)

b)


Example:

27.2 resistance

Norah Ali Al-moneef

Norah Ali Almoneef 20

• I = n q vd A– n = number of free charge carriers/unit volume

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

current density J) :

• Ohm's Law: E = J J = σ E– = resistivity– = 1/ = conductivity– Good conductor: low and high

• Ohm's Law: – R = resistance Measured in Volt/Ampere = Ohm ()

dd nev

A

Avne

A

IJ

)(

20

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


a

b

V=Va-Vb=EL The ratio of the potential drop to

the current is called resistance of the segment:

Unit: 1V/A= 1ohm

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

Ohm’s Law


V I V=const .I V=RIOhm’s Law is an empirical relationship that is

valid only for certain materialsMaterials that obey Ohm’s Law are said to be

ohmicI=V/R R, I0, open circuit; R0, I, short circuit

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

Unit: V/A=ohm

I

VR

Resistivity and ResistanceJ = E/ρ where ρ is the resistivity

I = V/R


Consider a bar or wire of cross-section A and length L, carrying current I and with potential difference V = Vb - Va between the ends.

I = V/R

and R = ρL/A is the resistance of the bar.

also called Ohm’s Law.

We know E = V/L so I/A = J = V/Lρ. Thus:

∆V = IR

Ohm’s Law, final

Plots of V versus I for (a) ohmic and (b) nonohmic materials. The resistance R=V/I is independent of I for ohmic materials, as is indicated by the constant slope of the line in (a). Norah Ali Al-moneef 24

Ohmic

Nonohmic


Ohmic Resistors• Metals obey Ohm’s Law linearly so long as their

temperature is held constant• Their resistance values do not fluctuate with

temperature• i.e. the resistance for each resistor is a constant

• Most ohmic resistors will behave non-linearly outside of a given range of temperature, pressure, etc.

On What Does Resistance Depend?

• If I increase the length of a wire, the current flow decreases because of the longer path

• If I increase the area of a wire, the current flowincreases because of the wider path

R = L/A• If I change to a material with better

conductivity, the current flowincreases because charge carriers move better

• If I change the temperature, the current flowchangesNorah Ali Al-moneef26

More on Resistance

I = V/R

Units of Resistivity : ρ is in Ω-m (ohm-meters), so R is in (Ω-m)(m)/(m2)

= Ω (ohms) = V/A (volts/ampere)

Resistivity ρ depends only upon the material (copper, silver…).

Resistance R depends upon the material and also upon the dimensions of the sample (L, A).

- R = ρL/A Note: Some devices (e.g.semiconductor diode) do not obey Ohm’s law!


Resistors are designed to have a specific resistance to reduce the amount of current going to a specific part of a circuit

To obey Ohm’s law means a conductor has a constant resistance regardless of the voltage.

V(Volts)

A(Amps)

R(Ohms)


V = IR = 2A x 3 = 6v

What voltage is required to produce 2a though a circuit with a 3 resistor.

V

3

I = 2a

Example:


Resistance

I

V

I

V

Nonohmic device 2. Ohm’s Law

I

VR

IRV

Units: [ R ] = 1Ω = 1 V/A

constR Ohm’s Law:

R

VI


Resistivity

L

AI

A

LR

L

AR

Definition:

Temperature dependence of resistivity

)(1 00 TTT

Example: What is the resistance of 1 m of nichrome wire of 2 mm diameter ?

)(

)(

00

000

TT

TTT

T

0

323

6 1018310

110

m

mm

A

LR

T

)( 00 TT 32 Norah Ali Al-moneef


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

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

Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s

Drift Velocity• In a conductor the free electrons are moving very

fast in random directions (v ~ 106 m/sec)• They collide with the atoms of the lattice and are

scattered in random directions

• If an electric field is present, there is a slow net drift of electrons in the direction opposite the electric field

• vDRIFT ~ mm/secNorah Ali Al-moneef34

• Amperes: the measure of the rate of current flow.– 6.24 × 1018 electrons passing a point per

second is equal to one amp.

• A current occurs whenever there is a source of electricity, conductors and a complete circuit.


Example


What is the current flow in a circuit with a voltage of 120 volts and a resistance of 0.23 ?

A 521.7= 0.23

V 120=

R

V=I

IR=V

ExampleWith the increase in the length of the wire, the current increases.A.TrueB.False

V = IR

4v = I x 2

I = 2A

A 4v battery is placed in a series circuit with a 2 resistor.

What is the total current that will flow through the circuit?

4v

2

I = ?

Example


V= IR

V = 2A x 3

V = 6v

What voltage is required to produce 2A though a circuit with a 3 resistor.

V

3

I = 2A

Example


V = IR

12 = 4 x R

R = 3

What resistance is required to limit the current to 4 A if a 12 V battery is in the circuit?

12v

3

I = 4a

Example


Example


A cylindrical copper rod has resistance R. It is reformed to twice its original length with no change of volume. Its new resistance is:

1. R

2. 2R

3. 4R

4. 8R

5. R/2


Two conductors are made of the same material and have the same length. Conductor A is a solid wire of diameter 1 mm. Conductor B is a hollow tube of inside diameter 1 mm and outside diameter 2 mm. The ratio of their resistances RA/RB is

1. 1/2

2. 1

3. 2

4. 3

5. 4B

A


Two cylinders are made of the same material and have the same length but different diameters. They are joined end-to-end and a potential difference is maintained across the combination. Which of the following quantities is the same for the two cylinders?

1. the potential difference

2. the current

3. the current density

4. the electric field

5. none of the above

Two cylindrical resistors, R1 and R2, are made of identical material. R2 has twice the length of R1 but half the radius of R1. They are connected to a battery V as shown. Compare the currents flowing through R1 and through R2.

A. I1 < I2 B. I1 = I2 C. I1 > I2

?/

2/

2

12

2

12

12

12

II

rA

rr

LL

4/12 AAA

LR

11

1

2

22 8

4/

2R

A

L

A

LR

VI1 I2

112

2 8

1

8I

R

V

R

VI


Example

Voltage and Current Relationship for Linear Resistors


Voltage versus Current for a 10 ohm Resistor

00.10.20.30.40.50.6

0 1 2 3 4 5 6

Voltage (V)

Cu

rre

nt

(A)

Voltage and current are linear when resistance is held constant.

Which one of the following graphs correctly represents Ohm's law, where V is the voltage and I is the current?

(a)A(b)B(c)C(d)D(e)A and C


If a piece of wire has a certain resistance, which wire made of the same material will have a lower resistance?

A )a hotter wire B ) a thicker wire C ) a longer wire D) a thinner wire

ANS: B



27.4 Resistance and Temperature

The resistivity (and hence resistance) varies with temperature.

For metals, this dependence on temperature is linear over a broad range of temperatures.

An empirical relationship for the temperature dependence of the resistivity of metals is given by

Copper

)](1[ 00 TT

•Resistance (R) is proportional to resistivity (): R = L / A The resistivity () depends on temperature and the physical properties of the material, so it has a different value for each material


• Some materials, when very cold, have a resistivity which abruptly drops to zero. Such materials are called superconductors.

)(1 00 TTT )( 000 TTT

T

0

)( 00 TT

0

T

• is the resistivity at temperature T• 0 is the resistivity at some standard temperature T0

• is the “temperature coefficient” of electric resistivity for the material under consideration

• The temperature coefficient of resistivity can be expressed as.


• In everyday applications we are interested in the temperature dependence of the resistance of various devices.

• The resistance of a device depends on the length and the cross sectional area.

• These quantities depend on temperature• However, the temperature dependence of linear

expansion is much smaller than the temperature dependence of resistivity of a particular conductor.

• So the temperature dependence of the resistance of a conductor is, to a good approximation,

)(1 00 TTRR

where R0 and T0 are the resistance and temperature at a standard temperature, usually room temperature or 20o C.

Reminder: Battery as a “ski lift for charges”:

Ski lift raises objects to higher potential energy - flow may vary, but potential energy difference fixedBattery also fixed potential diff. , but current may vary


27.6 Electrical Power

• The chemical energy of the battery is converted to U, electrical potential energy: Echem U

• The resulting electric field causes the electrons to accelerate: UK

• Collisions in the lattice structure transfer the energy to the lattice as thermal energy: KEth

• Thermal energy is a dissipative energy (i.e. can’t be recovered like mechanical energy.


the rate at which the system loses electric potential energy as the charge Q passes through the resistor:

V I V Q

) V Q ( dt

d

dt

d

dt

du


Electrical Energy = Voltage x Electrical Current x Time Interval

energy = V x I (amps) x t (sec)E = V x I x t

•The system regains this potential energy when the charge passes through the battery,• Since a resistor obeys Ohm’s Law:

PR = I2R = (∆VR)2/R


How is Electrical Power calculated?

Electrical Power is the product of the current (I) and the voltage (v)

The unit for electrical power is watt (W)

How much power is used in a circuit which is 110 volts and has a current of 1.36 amps?

P = I V Power = (1.36 amps) (110 V) = 150 W

Example


electrical energy: Electrical energy is a measure of the amount of power used and the time of use.Electrical energy is the product of the power and the time.

Example

energy = Power X time

P = I V

P = (2A) (120 V) = 240 W

E = (240 W) (4 h) = 960Wh = 0.96 kWh

Electrical Energy = Voltage x Electrical Current x Time Interval

energy = V x I (amps) x t (sec)E = V x I x t


a. 0.44 Ab. 2.25 Ac. 5 Ad. 36 A

ANS:

B

example

A 9-volt battery drives an electric current through a circuit with 4-ohm resistance. What is the electric current running through the circuit?


• Joule’s Law– States that the rate at which heat

produced in a conductor is directly proportional to the square of the current provided its resistance is constant

– i.e. P = I 2R In order to prevent power lines from

overheating, electricity is transmitted at a very high voltage

From Joule’s law the larger the current the more heat produced hence a transformer is used to increase voltage and lower current

i.e. P = V I 58 Norah Ali Al-moneef

Power dissipated by a bulb relates to the brightness of the bulb.

The higher the power, the brighter the bulb. For example, think of the bulbs you use at home. The

100W bulbs are brighter than the 50W bulbs.


If an electric fire uses 1.8 MJ of energy in a time of 10 minutes, calculate the power output of the fire.

Energy = 1.8 MJ = 1.8x106 Jt=10 minutes = 600 sPower = Energy / time p = 1.8x106 J / 600 =3 10 3 watt

example


Calculate the power of a vacuum cleaner if the operating voltage is 120v, and the current flowing

through it when it is used is 7.90A.

P = V x IP = 120V x 7.9AP = 948 W

example


Calculate the voltage of a computer that has 600W of power and 1.9A flowing into the

monitor?

V = P I

V = 600W 1.9A

V = 316V

example


If a 500 watt speaker need 10 amps to operate, what is the voltage requirement?

example

VV

V

5010

500


Example

• How much would you be charged for using a 60 Watt light bulb for 10 hours if electricity costs 0.07 $per kWh?

• E = PT= 0.06kW x 10h = 0.6kWh

• Cost = 0.6kWh x 0.07 $/kWh= 0.04$


If an electric fire uses 1.8 MJ of energy in a time of 10 minutes, calculate the power output of the fire.

E = 1.8 MJ = 1.8x106 Jt=10 minutes = 600 s


Power• Power is the rate of doing work.• Electrical power is usually expressed in watts or

kilowatts• In DC and AC circuits, with resistance loads, power can be determined by:

• Examples of resistance loads are heaters and incandescent lamps.

Volts=V

Amps=I

Watts=P

IV=P

example

• Determine the power consumed by a resistor in a 12 volt system when the current is 2.1 amps.

W 25.2=V 12A x 2.1=IV=P


IVP

ELECTRIC POWER

When there is current in a circuit as a result of a voltage, the electricpower delivered to the circuit is:

SI Unit of Power: watt (W)

Many electrical devices are essentially resistors:

RIIRIP 2

R

VV

R

VP

2


Rank in order, from largest to smallest, the powers Pa to Pd dissipated in resistors a to d.

1. Pb > Pa = Pc = Pd 2. Pb = Pc > Pa > Pc 3. Pb = Pd > Pa > Pc 4. Pb > Pc > Pa > Pd 5. Pb > Pd > Pa > Pc


Example• Determine the amount of energy a 100

Watt light bulb will use when operated for 8 hours.

Energy=Power x Time

=100 watts x 8 hour

=800 wh

• What will it cost to operate the light bulb if the electrical energy costs 0.12 $/kWh?

$ =0.12 $

kWh x 800 W x

1 kW

1,000 W x 8 h =0.77 $


Energy Use Calculations

How much electrical energy will an electric blanket use per month if it is used 8 hours a day? The blanket is on a 120 V circuit and draws 1.5 amp.

kWh 43.2= W1,000

kW 1 x

month

day 30 x

day

h 8 W xA) 1.5 x V (120=(kWh)Energy



Energy = Power x Time

E = (100 W) (300 s)

E = 30,000 J

E = 30 kJ


Rated for 4.2 kWUsed 20 h/month

Cost of 12 $ per kWh

Energy = Power x Time

= (4.2 kW) x (20 h)

= 84 kWh

Cost = Energy x rate per kWh

= (84 kWh) x ($0.12)

= $10.08

ExampleGiven copper wire 1mm diameter . 100m long has a

potential diffidence of 12 V Find

a) resistance, b) current in wire, c) current density,

d) electric field in wire, e) concentration of electrons (assuming 1electron / atm), f) drift velocity,

g) amount of electric charge flowing in 1 minute

Resistivity ρ = 1.72x10-8 Ohm-mDensity D = 8.9 E 3 kg/m3

molecular weight M = 63.546 g/moleAvogadro's # 6.022x10 23

electric charge e = 1.6x10-19 Cr = 5x10-4m radius, L=100m, t=60sec


Equations: Answers:a) R= ρL/A, A=πr2 A=7.85x10-7, R=2.19

Ωb) V=IR I=V/R I=5.48 Ac) J=I/A J=6.977x106 A/m2

d) E = ρJ 0.12 V/me) n =D Na /M 8.434E28 e/m3

f) I = n q vd A vd = I/nqA = 5.17x10-4 m/s

g) I= dQ/dt Q = It = 329 C


Example100 W light bulb connected to 110V what is a) current b) resistance c) at

10cents/kwhour how much to illuminate for a year, d) how many can be connected to a 15 ampere circuit breaker, e) how much electric power consumed by all these bulbs, f) if the temperature is 4500K and made from tungsten (α = 0.0038/K) what is the room temperature resistance at 300K

Given P=100 W, V= 100V Imax = 15A price = 0.1 $/kW h T=4500 K To = 300 Kα = 0.0038/K


Equations Answersa) P = IV I=P/V = 0.909 Ab) V = IR R=V/I = 121 Ωc) cost = ($0.1)(.1KW)(24 x 365) = $87.60d) Imax> Nmax I Nmax = 16e) Pmax = Nmax P Pmax = 1600Wf) R=Ro(1 + α (T-To)) = Ro =

7.13 Ω


Norah Ali Al moneef78

OHM’S LAW FORMULAS

Current equalsvoltage dividedby resistance

Voltage equalscurrent multiplied

by resistance

Resistance equalsvoltage divided

by current

Find Current Find Voltage Find Resistance

summaryFind current: I=ΔQ/Δt I=nqAvd


Quantity Unit of Measure

FunctionName NameSymbol Symbol

Voltage V, emfor E

Voltage VPressure whichmakes currentflow

Current I Ampere A Rate of flowof electrons

Resistance R Ohm Opposition tocurrent flow

Resistance related to physical parameters

The dimensions and geometry of the resistor as well as the particularmaterial used to construct a resistor influence its resistance. Theresistance is approximately given by

A

LR


)(1 00 TTRR

)(1 00 TTT

IVP

RIP 2

R

VP

2

norah ali al-moneef 1 conductors & semiconductors in conductors, the valence band is only...

Documents

conduction electrons

current definition

electric charges

bound electrons

direction of current

movement of free electrons

charge flow

atom displaced electrons