Chapter 27.

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Current Density and Drift Velocity


Perfect conductors

carry charge instantaneously from here to there


Perfect insulators

carry no charge from here to there, ever


Real substances
always

have

some density
n

of charges
q

that can move, however slowly


Usually electrons


When you turn on an electric

field, the charges start to move with average velocity

v
d


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
v
d

isn’t

d
nq

J v
J

Why did I draw
J

to the right?

CH 27

Current Density

Assume in each of the figures below, the number of charges drawn
represents the actual density of charges moving, and the arrows
represent equal drift velocities for any moving charges. In which
case is there the greatest current density going to the left?

+

+

+

+

+

-

-

-

-

-

+

+

+

+

+

-

-

-

-

-

+

+

+

+

+

-

-

-

-

-

+

+

+

+

+

-

-

-

-

-

A

B

C

D

d
nq

J v
JIT,
(magnitude)

Ans

a > c=b>d

Ohm’s Law: Microscopic Version


In general, the stronger the electric field, the faster the charge carriers
drift


The relationship is often proportional



J E

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:


=
1.59

10
-
8


m, a nearly perfect conductor


Fused Quartz:


=
7.5

10
17


m, a nearly perfect insulator


Silicon:


=
640

m, a semi
-
conductor

Ignore units for now

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

ˆ
n
ˆ
I 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

d
qnv A



2
3
C m/s m
m


The ampere or amp (A) is 1 C/s

C
A
s
I

dQ
I
dt


Current represents a change in charge


Almost always, this charge is being replaced somehow,
so there is no accumulation of charge anywhere

Warmup

10

Ex
-

Serway 27
-
5
. Suppose that the current through a conductor decreases
exponentially with time according to I = I
0
e
-
t/

, where I
0

is the initial current
(at t = 0), and


is a constant having dimensions of time. Consider a fixed
obs
ervation point within the conductor. (a) How much charge passes this
point between t = 0 and t =

? (b) How much charge passes this point
between t = 0 and t = 10

?

? (c) How much charge passes this point
between t = 0 and t =

?

Solve on
Board

Ohm’s Law for Resistors


Suppose we have a cylinder of material with conducting
endcaps


Length
L
, cross
-
sectional area
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 JA

V EL
 
JL


L
I
A



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

L
R
A


V IR
 
V
A
R
 
Circuit diagram
for resistor

Warmup

10

JIT

Ans

b, b

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

T
0


Normally 20

C


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 0
T
 



0
T
 



0
1
T T

 
 
 

For devices, it follows there will also be temperature dependence


The constants



and
T
0

will be the same for the device

L
R
A




0
0
1
L
T T
A


  
 
 


0 0
1
R R T T

  
 
 

for tungsten, 0.0045 1/K



for
carbon,
-
0.005
1/K


Warmup10b

This is basically Quick Quiz
4.
Ans

immediately (R lower).

Sample Problem

Platinum has a temperature coefficient of


=
0.00392/

C. A wire at
T

=
T
0

= 20.0

C has a resistance of
R

= 100.0

. What is the
temperature if the resistance changes to 103.9

?

A) 0

C


B) 10

C

C) 20

C

D) 30

C

E) 40

C

F) None of the above



0 0
1
R R T T

  
 
 


0
0
1
R
T T
R

  
103.9
100.0

1.039



0
0.039
T T

 
0
0.039
T T

 
0.039
0.00392/C


9.95 C
 
0
9.95 C
T T
  
30.0 C
 
Warmup10b

Non
-
Ohmic Devices

Some 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 carry current forever with no electric field

0

 
E J
Power and Resistors


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

dQ
I
dt

U Q V
   
U Q
V
t t
 
 
 

Q


V

dU
dt

P
dU dQ
V
dt dt
 


I V
 
P
V IR
 


2
2
V
I R
R

 
P

Where is the energy going?


The charge carriers are
bumping against atoms


They heat the resistor up

Warmup10b

CT


1 Two light bulbs Bulb 1 and Bulb 2of resistance R
1

and R
2
, such that
R
1
>R
2
, are connected to a battery in parallel. The Power dissipated

A.

By bulb 1 > than that of bulb 2

B.

By bulb 1 < than that of bulb 2

C.

Is the same for both bulbs

D.

Depends on what the
EMF of the battery is


CT
-

2 Two light bulbs Bulb 1 and Bulb 2of resistance R
1

and R
2
, such that
R
1
>R
2
, are connected to a battery in series. The Power dissipated

A.


By bulb 1 > than that of bulb 2

B.

By bulb 1 < than that of bulb 2

C.

Is the same for both bulbs

D.

Depends on what the EMF of the battery is.


Ans

B

Ans

A

Sample Problem

Two “resistors” are connected to the same
120 V circuit, but consume different
amounts of power. Which one has the larger
resistance, and how much larger?

A) The 50 W has twice the resistance

B) The 50 W has four times the resistance

C) The 100 W has twice the resistance

D) The 100 W has four times the resistance


V =
120 V



2
2
V
I R
R

 
P
P

㴠㔰=

P

= 100 W


The potential difference is the same across them both


The lower resistance one has more power


The one with twice the power has half the resistance



2
V
R


P
1
R

1
R

P
1
100 50
2
R R

JIT

Ans
: a=b, c=d, e =f

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

C3
1mF
Q7
2N3904
Q6
2N3904
Q5
2N3904
C4
0.06uF
Q4
2N3904
Q3
2N3904
C2
30uF
+V
V2
12V
Q2
2N3904
Q1
2N3904
C1
0.06uF
1kHz
V1
-1m/1mV
RL
50k
R7
25
R11
2.3Meg
R10
300k
R9
25k
R8
1k
R6
80
R5
1k
R4
25k
R3
300k
R2
2.3Meg
R1
15k