Current And Resistance
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 vd
•Called the drift velocity
J
Why did I draw
J to the right?
•There is a current density J associated with this motion of charges
•Current density represents a flow of charge
J  nqv d
•Note: J tends to be in the direction of E, even when vd isn’t
Ohm’s Law: Microscopic Version
•In general, the stronger the electric field, the faster the charge carriers drift
•The relationship is often proportional
E
J
•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1017 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 relationship
between 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
v  vi
•Between collisions, the velocity is constant
•On average, the velocity at any given time is zero v d  v  vi  0
a  F m   eE m
•Now turn on an electric field
•The electron still scatters in a random direction at each collision
v  vi  at
•But between collisions it accelerates
•Let  be the average time since the last collision
2
J

nq
v


ne
a

J

ne
E m
v d  v  vi  a t  0  a
d
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
I   nˆ  JdA
n̂
J
I  JA
•The total amount of charge flowing out of an object is called the current
2
C

m/s
m


•What are the units of I?
I  JA  qnvd A
3
m
•The ampere or amp (A) is 1 C/s
C
•Current represents a change in charge I
A
s
•Almost always, this charge is being
replaced somehow, so there is no
dQ
I

accumulation of charge anywhere
dt
Ohm’s Law for Resistors
•Suppose we have a cylinder of material
with conducting end caps
•Length L, cross-sectional
L
area A
•The material will be assumed to
I  JA
follow Ohm’s Microscopic Law
L


JL

I
V  EL
•Apply a voltage V across it
A
L
J E 
E  V L
R
A
•Define the resistance as
V  IR
•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
Circuit diagram
•Resistance has units of Volts/Amps
for resistor
V
R

•Also called an Ohm ()
A
•An Ohm isn’t much resistance
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
0   T0 
E  J
•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
 T   0 1   T  T0  
•For devices, it follows there will also be temperature dependence
•The constants  and T0 will be the same for the device
R
L
A

0 L
1   T  T0  
A
R  R0 1   T  T0  
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
E  J  0
Power and Resistors
•The charges flowing through a resistor are having their potential energy changed
Q
•Where is the energy going?
•The charge carriers are
bumping against atoms
•They heat the resistor up
U  QV
U Q

V
t
t
dU dQ

V
dt
dt
dQ
I
dt
P 
dU
dt
V
P  I  V 
V  IR
V 

2
P I R
R
2
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
V2
12V
+V
V1
-1m/1mV
1kHz
R1
15k
C1
R2
0.06uF2.3Meg
R3
300k
Q1
2N3904
R6
80
C2
30uF
R4 C4
R11
25k0.06uF2.3Meg
Q2
2N3904
R5
1k
R10
300k
Q4
2N3904
R9
25k
Q3
2N3904
R8
1k
Q5
2N3904
Q6
2N3904
Q7 C3
2N39041mF
R7
25
RL
50k
Equations for Test 1
Electric Fields:
k qq
F2  e 12 2 rˆ
r
Gauss’s Law:
 E  AE  nˆ
E 
ke q
E  2 rˆ
r
F  Eq
qin
0
Potential:
Capacitance:
Q  C V
C  C1  C2
U  qV
V
V    E  d s
ke q
r
1 1
1
 
C C1 C2
Units:
N V

C m
C
A
s
F
C
V
V

A
End of material
for Test 1
V
Ex  
x
V
Ey  
y
V
Ez  
z