Electric Currents and Resistance
(Chapter 31)
What is Electric Current?
• Electric current is the motion of electric charge
• The amount of charge that passes a point in a some
time is the current.
• For electric current, charge carriers must be mobile
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–
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Metals
Semiconductors (Si, GaAs)
Carbon Materials (Carbon nanotubes and graphene)
Electrolytic solutions (ions dissolved in water)
Plasmas (ionized gases)
etc...
• We will discuss METALS only here. In metals, the charge
carriers are ELECTRONS.
Electron Current in Metals
•
•
•
How do we know that moving electrons are the source of electric currents in
metals?
In 1879 Edwin Hall (a graduate student at Johns Hopkins U.) devised an
experiment that distinguishes between positive and negative charge carriers
in a conductor.
A magnetic field perpendicular to an electric current produces a electric
potential difference that depends on the sign of the charge carriers.
Setup
!VH across the conductor for
+ charge carriers is opposite
to - charge carrier
Discovery of Electron Came Later...
• J. J. Thompson discovered the electron in 1897 using
a Crookes tube by measuring the charge and mass of
the particles emitted as “cathode rays”
Electrons from ionized gas
molecules accelerated by applied
voltage. Electric and Magnetic
fields bend their trajectory
- Primitive TV set
Tolman-Stewart Experiment
• Tolman-Stewart experiment of 1916 showed that
when a metal bar is accelerated the rear surface
becomes negatively charged and the front surface
positively.
• Proves that the electrons are mobile in a metal.
The electron current
• The electrons in conductor behave similarly to molecules in a
gas/liquid
– Electrons move randomly because of the heat they have but
there is no net motion (thermal motion)
– A force on the electrons can cause the random motion to have a
net flow in the direction of the force
A force causes the electrons to have a net velocity on top of the
random thermal motion, this is the drift velocity vd
The ‘Geometry’ of the Electron Current
ie = N e / !t
ie = (ne A!x) / !t = Ane vd
The Electron Current
• Pushing on the sea of electrons with an electric field
causes the entire sea of electrons to move in one direction
like a gas or liquid flowing through a pipe.
• This net motion, which takes place at the drift speed vd,
is superimposed on top of the random thermal motions of
the individual electrons.
• The electron current is
How to Make an Electron Current
•
•
•
A current is a nonequilibrium
motion of charges created by an
electric field in the conductor.
Electrons collide with each other
and with ions. This is like friction
and slows electrons down.
Because of ‘friction’, an electron
current can only be sustained by a
constantly applied E field: battery
or generator.
The Electric Field in a Wire
• A nonuniform distribution of
surface charges along a wire
creates a net electric field inside
the wire.
• Internal E field points from +
terminal of battery to - terminal.
• Potential difference across wire =
emf of battery
Drude Model of Conduction: Origin of
Resistance
•
•
•
Drude model is an early model of conductance from before quantum mechanics.
It gives a good intuitive picture of how electrons behave in a metal.
However, quantum mechanics has shown that many of the details of the model are incorrect.
Let’s consider the classical motion of an electron in
a conductor:
Analogy to Gravity: Resistance is like Air Drag
Resistance = Electron Friction
• Where does the friction come from ? Collisions !
• Two types of collisions in metals:
– Electrons collide with the positive metal ions that
form the rigid lattice of the metal
– Electrons collide with each other.
• Collisions scatter electrons into new random
direction. Average time between collisions (at room
temperature):
! ~ 10
"14
s
After each collision, the velocity
is completely random
In between collision, the electron
is accelerated by the E field
Averaging Over Collisions: Drift Velocity
!
dv e !
= E
dt m
e !
e !
!
!
v ! (velocity after collision) + E"t = vC + E"t
m
m
Now average the velocity over the velocity after collisions and time between collisions:
e !
!
!
v = vC + E "t
m
e !
!
!
vd = v = E#
# = "t = average time between collisions
m
Drift velocity is how much the electric field accelerates
an electron in between collisions.
vC =
!
vc2 = 3kBT / m ! 10 5 m / s (room temperature thermal velocity of electrons)
vd " 10 "4 " 10 "5 m / s
ie = ne Avd =
ne e#
AE
m
Current
If Q is the total amount of charge that has moved past a
point in a wire, we define the current I in the wire to be the
rate of charge flow:
The SI unit for current is the coulomb per second, which is
called the ampere. 1 ampere = 1 A = 1 C/s.
The conventional current I and the electron current ie are
related by
Electron Current (ie) vs. Electrical Current (I)
•Electron current measures the number of electrons per unit time.
•Electrical current measures the charge per unit time that is flowing.
•The direction of I and ie are opposite !! (e<0 for electron)
•I is in the direction of flow of positive charges
•ie is in the direction of flow of negative charges
Both situations produce
the SAME electrical current.
Negative charges moving to
the left and positive charges
moving to the right have the
same dQ/dt !
Conclusion: Measuring the
electrical current can not
tell you what the charge
carriers are!
Example (No. 47 in Text)
The starter motor of a car engine draws a current of 150A from
the battery. The copper wire to the motor is 5.0mm in diameter
and 1.2m long. The starter motor runs for 0.80s until the car
engine starts.
a) How much charge passes through the starter motor ?
b) How far does an electron travel along the wire while the motor
is on ?
Example (Solution)
The amount of charge that passes through the motor:
The drift velocity is:
I = nevd A
The electrons move a distance d=vd!t
Conductivity and Resistivity
! !
!
J = I / A = ne evd
The conductivity of a material is
e! !
!
vd = E
m
Independent of
sign of charge carriers
Conductivity, like density, characterizes a material as a
whole. The current density J is related to the electric field E
by:
The resistivity tells us how reluctantly the electrons move
in response to an electric field:
What determines the resistivity/conductivity?
• Resistivity "=1/# is a material property like density or
hardness
• Three parameters can be variable:
• ne: electron density depends on type of metal. In
semiconductors, the electron density (and sign of
charge carriers) are controlled by doping (addition of
impurities that supply extra negative or positive
charge).
• Mass of charge carriers
• scattering time $:
• Temperature: $ increases as T decreases.
• Purity of sample: defects and disorder increase the
amount of scattering and reduce $
Example 2
A metal cube 1.0 cm on each side is sandwiched between two
electrodes. The electrodes create 0.0050V/m electric field in the
metal. A current of 9.0A passes through the cube from the
positive to the negative electrode.
Identify the metal.
Example 2 (cont.)
I = JA = ! EA
! = I / EA = 9.0A / (10 "4 m 2 )(0.005V / m) = 1.8 # 10 7 $"1m "1
Look this number up in a table!
It’s tungsten.
A=1 cm2, L=1cm
Temperature is the Enemy of Conductors
Gedanken (Thought) Experiment
Consider a light bulb connected
to wires and a battery. The light
bulb glows....
A Cold Bright Place...
Wires are cold--> Less thermal motion-->Less scattering-->
More current--> Bulb glows brighter !
Relating the Current to the Potential Difference
!V !V V+ " V"
E=
=
=
!s
L
L
I = AE / #
A
I=
"V = "V / R
!L
R = !L / A
Resistance and Ohm’s Law
The resistance of a long, thin conductor of length L and
cross-sectional area A is
The SI unit of resistance is the ohm. 1 ohm = 1 Ω = 1 V/A.
The current through a conductor is determined by the
potential difference ΔV along its length:
Ohm’s law is limited to those materials whose resistance
R remains constant—or very nearly so—during use.
Resistance vs. Resistivity
• Resistivity (") is a bulk material property. It depends only on
the type of material (copper, gold, silver, etc...) and
the temperature.
• Resistance (R) depends also on the size of the material
1. Larger cross-sectional area means smaller R
2. Longer wire means larger R
EXAMPLE: Two copper wires with different
lengths have the same resistivity but different resistances !
Know When to Use Ohm’s Law:
Diode is a semiconductor
device with many applications.
Capacitors are also nonohmic ! We can not use Ohm’s Law
for a capacitor.
Resistors
Wires are made of metal, which are very good conductors.
The resistances of typical wires are less than a few Ohms.
Currents would be so large as to be dangerous even
with a small battery.
A resistor is made from a very poor conductor
(often carbon) with the purpose of increasing circuit resistance.
Resistors are used in virtually every electric device!
Circuit Symbol:
Resistor in a Circuit
Kirchoff's Voltage Law:
!Vbattery + !Vwire1 + !Vresist + !Vwire2 = 0
For each segment, we can use Ohm's Law:
!Vwire1,2 = IRwire1,2
!Vresist = IRresist
I(Rwire1 + Rresist + Rwire2 ) = !Vbattery
Kirchhoff’s Junction Law
For a junction, the law of conservation of current requires
that
where the Σ symbol means summation.
This basic conservation statement – that the sum of the
currents into a junction equals the sum of the currents
leaving – is called Kirchhoff’s junction law.
Two Reasons:
1. The number of electrons is conserved
-->One cannot create or destroy electrons in the wire
2. The drift velocity is the same everywhere. If not, electrons
would accumulate in some regions but their mutual repulsion
prevents this.
Example
1A
2A
3A
5A
11A
I=?