Current Density and Drift VelocityCurrent And Resistance

•Perfect conductors carry charge instantaneously from here to there•Perfect insulators carry no charge from here to there, ever

•Real substances always havesome density n of charges qthat can move, however slowly

•Usually electrons•When you turn on an electricfield, the charges start to move with average velocity vd

•Called the drift velocity

•There is a current density J associated with this motion of charges•Current density represents a flow of charge

•Note: J tends to be in the direction of E, even when vd isn’t dnqJ v

J

Why did I draw J to the right?

Ohm’s Law: Microscopic Version•In general, the stronger the electric field, the faster the charge carriers drift•The relationship is often proportional

EJ•Ohm’s Law says that it is proportional

•Ohm’s Law doesn’t always apply•The proportionality constant, denoted , is called the resistivity

•It has nothing to do with charge density, even though it has the same symbol•It depends (strongly) on the substance used and (weakly) on the temperature

•Resistivities vary over many orders of magnitude•Silver: = 1.5910-8 m, a nearly perfect conductor•Fused Quartz: = 7.51017 m, a nearly perfect insulator•Silicon: = 640 m, a semi-conductor

Ignore units for now

The Drude Model•Why do we (often) have a simple relationshipbetween electric field and current density?

•In the absence of electric fields, electrons are moving randomly at high speeds

•Electrons collide with impurities/imperfections/vibrating atoms and change their direction randomly

•When they collide, their velocity changes to a random velocity vi•Between collisions, the velocity is constant

•On average, the velocity at any given time is zero•Now turn on an electric field•The electron still scatters in a random direction at each collision•But between collisions it accelerates•Let be the average time since the last collision

iv v

ma F e m E

i t v v a

2ne mJ E

d v v 0i v

d v v i t v a 0 a dnqJ v ne a

Current•It is rare we are interested in the microscopic current density•We want to know about the total flow of charge through some object

J

n̂ ˆI dA n J

I JA

•The total amount of charge flowing out of an object is called the current•What are the units of I? I JA dqnv A 2

3

C m/s m

m

•The ampere or amp (A) is 1 C/s CA

sI

dQI

dt

•Current represents a change in charge•Almost always, this charge is beingreplaced somehow, so there is noaccumulation of charge anywhere

Ohm’s Law for Resistors•Suppose we have a cylinder of material with conducting end caps

•Length L, cross-sectionalarea A•The material will be assumed to follow Ohm’s Microscopic Law

•Apply a voltage V across it

L

E V L J E

I JAV EL JL L

IA

•Define the resistance as•Then we have Ohm’s Law for devices

•Just like microscopic Ohm’s Law, doesn’t always work•Resistance depends on composition, temperature and geometry

•We can control it by manufacture•Resistance has units of Volts/Amps

•Also called an Ohm ()•An Ohm isn’t much resistance

LR

A

V IR

V

AR

Circuit diagram for resistor

Ohm’s Law and Temperature•Resistivity depends on composition and temperature•If you look up the resistivity for a substance, it would have to give it at some reference temperature T0

•Normally 20C•For temperatures not too far from 20 C, we can hope that resistivity will be approximately linear in temperature

•Look up 0 and in tables

E J 0 0T

0T 01 T T

•For devices, it follows there will also be temperature dependence•The constants and T0 will be the same for the device

LR

A

0

01L

T TA

0 01R R T T

Non-Ohmic DevicesSome of the most interesting devices do not follow Ohm’s Law•Diodes are devices that let current through one way much more easily than the other way•Superconductors are cold materials that have no resistance at all

•They can carrycurrent foreverwith no electricfield

0 E J

Power and Resistors

•The charges flowing through a resistor are having their potential energy changed

dQI

dt

U Q V

U QV

t t

Q

V

dU

dtP

dU dQV

dt dt I V P

V IR

2

2 VI R

R

P

•Where is the energy going?•The charge carriers are bumping against atoms•They heat the resistor up

Uses for Resistors•You can make heating devices using resistors

•Toasters, incandescent light bulbs, fuses•You can measure temperature by measuring changes in resistance

•Resistance-temperature devices•Resistors are used whenever you want a linear relationship between potential and current

•They are cheap•They are useful•They appear in virtually every electronic circuit

C31mF

Q72N3904

Q62N3904

Q52N3904

C40.06uF

Q42N3904

Q32N3904

C230uF

+V

V212V

Q22N3904

Q12N3904

C10.06uF

1kHz

V1-1m/1mV

RL50k

R725

R112.3Meg

R10300k

R925k

R81k

R680 R5

1k

R425k

R3300k

R22.3MegR1

15k

Equations for Test 1

End of material for Test 1

1 22 2

ˆek q q

rF r

qF E

2ˆek q

rE r

ˆE AE n

in

0E

q

V dE s U qV

ek qV

r x

y

z

VE

xV

Ey

VE

z

Q C V

1 2

1 1 1

C C C

1 2C C C

Electric Fields: Gauss’s Law: Potential:

Capacitance:

Units:

N V

C m

CF

V

CA

s

V

A