#### Transcript Lect18

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Today…
• Current and Current Density
• Devices
– Batteries
– Resistors
• Remember: Quiz on Thursday and Friday.
– Covers chapters 24-25-12,
– Potential, Capacitors, Gravity (and
everything before)
Devices and Circuits
• We are now finished with electrostatics: the
study of fields and potentials produced by
static charge distributions.
• Next topic: Devices and Circuits
• We have studied one device so far: the
capacitor. For the next week we will investigate
circuits composed of the following devices:
• Capacitors
• Batteries
• Resistors
}
and combinations in “DC” circuits,
(Direct Current circuits)
Current is charge in motion
• Electrons exists in conductors with a density, ne (ne
approx 1029 m-3) and are constantly in random motion
– In general only electrons move, the heavy nucleii
remain fixed in the material lattice
• In the absence of electric fields there is no net motion of
the charge, electrons bounce around like atoms in a gas
• When an electric field is applied:
– a small average velocity, ve ,is added to the random
motion (an electric current)
+
E
• NOTE that the current direction is defined as the
direction of the field BUT the electrons move in the
opposite direction
Current is charge in motion
-
ne electrons/m3
+
Area S
E
Velocity ve
• Current density, J, is given by J = qeneve
• unit of J is C/m2sec or A/m2 (A ≡ Ampere) and 1A ≡ 1C/s
• Current, I, is J times cross sectional area, I = J S
– for 10 Amp in 1mm x 1mm area, J=10+7 A/m2,
– ve is about 10-3 m/s
– (Yes, the average velocity is only 1mm/s!)
Devices: Batteries
• Batteries (Voltage sources, sources of emf):
Purpose is to provide a constant potential
difference and source of current between two
points.
+ -
• Cannot calculate the potential difference from
first principles... chemical  electrical energy
conversion. Non-ideal batteries will be dealt with
in terms of an "internal resistance".
OR
+
• Positive terminal has the higher potential
• Current is defined as flowing from the positive
to the negative terminal
• Inside the battery chemical processes return
the charge from the negative to positive
terminals
• emf is the term for the electrical potential
provided by the battery
E
dW
dq
V
-
Devices: Resistors
• Resistors:
• Resistors limit the current drawn in a circuit.
Note:
I
dQ
dt
UNIT: Ampere = A = C/s
• Resistance is a natural property of almost all
materials which opposes the motion of charge
through the material
• Resistance can be calculated from knowledge of
the geometry of the resistor AND the “resistivity”
of the material out of which it is made (often
“conductors”).
Ohm’s Law
• Set up this circuit
• Vary applied voltage V.
I
R
I
• Measure current I
• Ratio
V
remains constant
I
V
• Resistance R
V
slope = R
I
V
R
I
Resistance
• What is happening in the resistance?
I
R
• Voltage means Potential Difference -> E-field
• E-field -> constant force on electrons
• Constant force on electrons -> constant acceleration
V
• Constant acceleration -> very large and increasing currents
• This does not happen large increasing currents are not observed
–what’s wrong with this picture???
• Simple constant acceleration isn’t happening….
• Electrons undergo a lot of rapid and random scattering
• No constant acceleration (acceleration proportional to Voltage)
• Instead velocity of electrons is proportional to Voltage
velocity proportional to current -> I=V/R
I
What gives rise to non-ballistic
behavior?
• E-field in conductor (resistor)
provided by a battery
• Charges are put in motion,
but scatter in a very short
time from things that get in
the way
– it’s crowded inside that metal
– defects, lattice vibrations
(phonons), etc
• Typical scattering time t = 10-14 sec
• Charges ballistically accelerated for
this time and then randomly scattered
What gives rise to non-ballistic behavior?
• Newton’s 2nd Law says F=ma
• So the acceleration of the electron is eE/m
• Average velocity attained between scatters is given by
v=at or v = eEt/m
• Current density is J = env so current is proportional to E
which is proportional to Voltage
• OHM’s LAW J = (e2nt/m)E or J = s E
• Or
V
I
R
s = conductivity
Resistance
R
• Resistance
Resistance is defined to be the
ratio of the applied voltage to
the current passing through.
R
V
I
UNIT: OHM = W
I
I
V
• Is this a good definition?
i.e., does the resistance belong only to the resistor?
Recall the case of capacitance: (C=Q/V) depended on the
geometry, not on Q or V individually
Does R depend on V or I ?
It seems as though it should, at first glance...
Calculating Resistance
• To calculate R, must calculate
current I which flows when voltage
V is applied.
• Applying voltage V sets up an
electric field in the resistor. What
determines the current?
I
R
I
V
R
V
I
• Current is charge flowing past a point per unit time,
which depends on the average velocity of the charges.
• Field gives rise to force on the charge carriers which
reach a terminal velocity.
L
• Resistance calculation splits into two parts R    
 A
– Part depends on the “resistivity” ρ, a property of the material
– Part depends on the geometry (length L and cross sectional area
A)
Resistivity
• Resistivity is a property of
bulk matter related to the
resistance of a sample.
•
E
The resistivity () is
defined as:
E

j
1
m
  2
s e nt
j
A
L
• Where E = electric field and j = current density in
conductor.
• For the case of a uniform material
I
j
A
V  EL
Resistivity

E
j
j 
I
A
V  EL
I
 ρL 
V  EL  jL   L  I 

A
 A
V  IR
where R  
E
j
A
L
L
A
So YES, the property belongs to the material and we
can calculate the resistance if we know the
resistivity and the dimensions of the object
e.g., for a copper wire,  ~ 10-8 W-m, 1mm radius, 1 m
long, then R  .01W;
for glass,  ~ 10+12 W-m; for semiconductors  ~ 1 W-m
Makes sense?
E
L
R
A
j
A
L
• Increase the length, flow of electrons impeded
• Increase the cross sectional area, flow facilitated
• The structure of this relation is identical to heat flow
through materials … think of a window for an intuitive
example
How thick?
or
How big?
Question 1
I1
• Two cylindrical resistors, R1 and R2,
are made of identical material. R2 has V
twice the length of R1 but half the
– The resistors are then connected
to a battery V as shown
– What is the relation between the currents I1 and I2
(a) I1 < I2
(b) I1 = I2
(c) I1 > I2
I2
Question 1
1. a
2. b
3. c
Question 1
I1
• Two cylindrical resistors, R1 and R2,
are made of identical material. R2 has V
twice the length of R1 but half the
– The resistors are then connected
to a battery V as shown
– What is the relation between the currents I1 and I2
(b) I1 = I2
(a) I1 < I2
(c) I1 > I2
•The resistivity of both resistors is the same ().
•Therefore the resistances are related as:
R2  
L2
2 L1
L

 8  1  8R1
A2
( A1 / 4)
A1
•The resistors have the same voltage across them;
therefore
V
V
1
I2 
R2

 I1
8R1 8
I2
Question 2
• A very thin metal wire patterned as
shown is bonded to some structure.
• As the structure is deformed this
stretches the wire (slightly).
– When this happens, the resistance
of the wire:
(a) decreases
(b) increases
(c) stays the same
Question 2
1. a
2. b
3. c
Question 2
• A very thin metal wire patterned as
shown is bonded to some structure.
• As the structure is deformed this
stretches the wire (slightly).
– When this happens, the resistance
of the wire:
(a) decreases
(b) increases
(c) stays the same
•Because the wire is slightly longer, R ~ L A is increased.
•Because the volume of the wire is ~constant, increasing the
length, decreases the area, which increases the resistance.
•By carefully measuring the change in resistance, the strain
in the structure may be determined
Is Ohm’s Law a good law?
• Our derivation of Ohm’s law
ignored the effects of
temperature.
– At higher temperatures the
random motion of electrons
is faster,
– time between collisions
gets smaller
– Resistance gets bigger
– Temperature coefficient of
resistivity ()
– Typical values for metals
410-3
1
m

 2
s
e nt
   0 1   T  T0  
Is Ohm’s Law a good law?
• Our derivation of Ohm’s law
ignored quantum mechanical
effects
• Many materials, only
conduct when sufficient
voltage is applied to move
electrons into a “conduction
band” in the material
• Examples are semiconductor
diodes which have very far
from linear voltage versus
current plots
Is Ohm’s Law a good law?
Superconductivity
• At low temperatures (cooled to liquid
helium temperatures, 4.2K)the
resistance of some metals0,
measured to be less than 10-16•ρconductor
(i.e., ρ<10-24 Ωm)!
–Current can flow, even if E=0.
–Current in superconducting rings can flow for years
with no decrease!
• 1957: Bardeen, Cooper, and Schrieffer (“BCS”) publish
theoretical explanation, for which they get the Nobel prize
in 1972.
– It was Bardeen’s second Nobel prize (1956 – transistor)
Is Ohm’s Law a good law?
Superconductivity
• 1986: “High” temperature superconductors are
discovered (Tc=77K)
– Important because liquid nitrogen (77 K) is much cheaper than
liquid helium
– Highest critical temperature to date ~140K
• Today: Superconducting loops are used to produce
“lossless” electromagnets (only need to cool them, not
fight dissipation of current) for particle physics.
[Fermilab accelerator, IL]
• The Future: Smaller motors, “lossless” power
transmission lines, magnetic levitation trains,
quantum computers?? ...
Is Ohm’s Law a good law?
• Ohm’s Law is not a fundamental law of physics
• However it is a good approximation for metallic
conductors at room temperature as used in electrical
circuits
Summary
•Ohm’s Law states
V 
I
R
•Ohm’s Law is not a physical law but an approximation
which works well enough in normal conditions