Lecture 08

Current & Resistance

February 23, 2005

Happenings Yesterday (Monday)

We completed the topic of capacitance Today

We begin to study electric current and the concept of resistance.

Tomorrow (Friday) Continue with electric current QUIZ on the material from this week. WebAssign on C is due

Monday WebAssign on Resistance and current is due

New Topic

Current and Resistance

Commercial Resistors

Color Coded

Conductors In the past we decided that in a

conductor The Electric Field is ZERO because we

postulated that no charges were to move. Static Situation Any electric field must be at the surface That field must be normal to the surface

Let’s look at another situation.

Consider a conductor

The mobile electrons can “move” under the influence of an electric field.

We then have a “current” (to be defined) flowing in the wire. But WAIT … how can we????? Only if …

V1 V 2

Electric Field

Walla ….. a CIRCUIT

From the Past

+

-

Vo

Q Flows and then stops.

A Different Situation

Charge will begin to flow through the short. Charge can flow back into the battery and discharge

it. Wire can get warm, emit light or even burn our (fuse). The FLOW of charge is defined as a current.

Vo

circuit

Current isa good thing

Franklin’s Impact on Physics!!!

Positive charge will leave the battery from the positive terminal and flow through an external circuit to the negative terminal.

Electrons will go the other way.

Current is defined as the flow of POSITIVE CHARGE.

+ charge does not normally flow in a wire.

ELECTRONS

CURRENT

Definition

Current is the total amount of charge that flows through a “wire” in one second.

Current is measured in Coulombs per second.

A current of one coulomb per second is defined as an AMPERE. (Amp.)

CONCISE DEFINITION: CURRENT

2

1

)(t

t

dttiq

idtdqdt

dqi

Current will flow throughout the cross-section of the wire (usually).

Current through aa’ is the same as the current through bb’ and cc’.

What is DIFFERENT between aa’ and bb’ with respect to current?

What’s Different??

CURRENT DENSITY

areaunit

currentj

i=5 amps

A=.1 m2 A=0.05 m2

J=5 amps/.05 m2 = 100 amps/m2J=5 amps / .1 m2 = 50 amps/m2

Current

Current can’t “pile up” at a point n a circuit.

Example

210

BOTHFor

iii

First introduction to Kirchoff’s Node Equation

Consider

5A 3A ^2A v

6A

8 A

Question:A 5 Amp current is set up in a circuit for 6 minutes by a 6 Volt Battery. How much chemical energy is provided by the battery?

+

-

Vo

CIRCUIT OF SOME SORTWORK (Energy) per unitCharge = qV

KJVoltscoulqV

coulombscoul

coulamps

1061800

18006030min

sec60min6

sec5min65

Definition

i V

wire

SYMBOL

OHMAmpere

VoltUNIT

i

VR

11Current increases

with PotentialDifference (V)

Ohm’s Law

High Low

Observations Wires and Resistors are made from conducting

materials. These materials have some fundamental

properties associated with them. Electrons are attached to atoms.

Outer electrons weakly bound Small Force (Applied Electric Field) can easily push

them. They bump into things which retard their motion. The more things that retard their motion, the more

difficult it is to push a current via an applied potential difference.

Thus, the resistance goes up.

What kinds of things cause resistance?

Sudden Constrictions in the conductor including bends!

Underlying structure- Amorphous Crystalline

Defects Impurities Thermal Collisions

Temperature

Fundamental Property RESISTANCE vs. RESISTIVITY

Consider a wire made of some material.

Resistance is a property of the rire itself … the material and the shape.

New Quantity: RESISTIVITY is a property of the

material itself regardless of its shape/

Consider our wire:

V

A

L

What would happen to the current if we Increased the voltage? Increased the Area? Increased the length?

Summary

iRiA

LV

orL

VAi

R is called the RESISTANCEand is measured in

OHMS ()

Define CONDUCTIVITY

11 m

Resistivity – Ohm-Meters

Silver 1.62 x 10-8

Copper 1.69

Aluminum 2.65

Tungsten 5.25

Platinum 10.6

Silicon-Pure 2.5 x 103

Glass 1010 to 1014

Fused Quartz ~1016

WIRESAmerican Wire Gauge Diameter - mm

30 gauge 0.255

24 0.511

18 (typical household)

1.024

14 1.628

12 2.05

BACK TO OHM

V=iR

i=V/R

Not EVERYTHING is a resistor

The Semiconductor Diode

Effect of Temperature

Temperature

)1()(

)()(

0000

000

TTT

TT

Linear over a limited temperature range.

Question-For an 18 gauge wire of length L to have a resistance of 1 ohm what must L be if the material is copper?

2622

8

2

101

1069.1

4/

0.1

mmmd

dAL

RA

LR

(d for 18 gauge is about 1 mm)

A current of 6.5 Amps exists in a 9 Ohm resistor for 5 minutes. How many coulombs and how many electrons pass through the resistor in this time?

Coulombs: 6.5 COULOMBS per Second for 5 minutes6.5C x 5 min X 60 sec/min = 1950 coulombs

Number of electrons = # coulombs / electron charge =1950 / 1.6 x 10-19 = 1.22 x 10 +22

Microscopic Theory

J and E

EJ

and 0A

1

let

EL

VJ

L

V

AL

A

V

RA

V

A

iJ

JAi

Consider a wire

d

d

d

neJ

Anet

Necurrent

tAnN

Micro-View “Resistivity”

m

eEa

m

eEa

d

a

depends on the material and is the mean time between collisions

ease of motion – mobilityresistance to motion - scattering

m

ne

EEEm

nenevJ d

2

2 1

Power

V i

+

-

ELECTRONs

Battery supplies energy to the resistor which, in turn, dissipates it in the form of heat.

Work done on charge Q = Q x V

RiiRiiVVt

Q

t

QVP

PPOWERtimeWork

2

/

REMEMBER: P=iV and P=i2R

The Spectrum of Conductors

ENGINEERED MATERIALS!!!Semiconductors

When 105 volts are applied across a wire that is 12 meters long and has a 0.30 mm radius, the current density us 1.7 x 10 4 A/m2. What is the resistivity of the wire??

L

A

LR

RA

thatso

From the current density we can find the CURRENT.

i=JA = 1.7 x 104 amp/m2 X ( X 0.32) mm2 X (1m/1000mm)2

I = 4.8 ma

R= V/i = 105 volts / (4.8 x 10-3) amps = 2.18 x 104 ohms

= 2.18 x 104 ohms x 3 x 10-7 m2 / 12m = 0.005 ohm-meters

The figure below gives the electrical potential V(x) along a copper wire carrying a uniform current, from a point at higher potential (x=0m) to a point at a lower potential (x=3m). The wire has a radius of 2.45 mm. What is the current in the wire?

copper

12 volts 0 volts

What does the graph tell us??

*The length of the wire is 3 meters.*The potential difference across the

wire is 12 volts.*The wire is uniform.

Let’s get rid of the mm radius and convert it to area in square meters:A=r2 = 3.14159 x 2.452 x 10-6 m2

orA=1.9 x 10-5 m 2

Material is Copper so resistivity is (from table) = 1.69 x 10-8 ohm meters

We have all we need….

ma 49.41067.2

1012

R

Vi

:Law sOhm' From

67.2 109.1

0.3m-ohm 1069.1

3

6

5

8

ohms

volts

mx

mx

A

LR

Series CombinationsR1 R2

i i

V1 V2V

iiRseriesR

general

RRR

iRiRiRVVV

and

iRV

iRV

)(

:21

2121

22

11

Parallel Combination??

R1, I1

R2, I2

V

i iRR

general

RRR

so

R

V

R

V

R

Viii

iRV

11

111

..

21

2121

Materials

VERY IMPORTANT MATERIALSILICON

Silicon Crystal Lattice

Electron Freed!

Add an impurity

Special Impurities

Effect of Impurities on

Importance

P N

P N

P N P

Diode

voltage

current

Who Cares??

Silicon Transistor CHIPS

History

Thinking Chips?

That’s it for Resistance