e
1
I
1
e
2
e
3
R
R
R
I
2
I
3
loop
n
V
0
out
in
I
I
Devices
•
Capacitors:
Purpose is to store charge (energy).
•
We have calculated the capacitance of a system
•
We had to modify Gauss' Law to account for bulk
matter effects (dielectrics)
…
C
=
k
C
0
•
We calculated effective capacitance of series or
parallel combinations of capacitors
•
Batteries
(Voltage sources, sources of
emf):
Purpose is to provide a constant
potential
difference
between two points.
•
Cannot calculate the potential difference
from first principles... chemical
敬散瑲楣慬e
敮敲杹捯湶敲獩潮e⁎潮

楤敡i扡瑴敲楥猠w楬
扥敡汴bw楴栠楮i瑥牭猠潦o慮•a湴敲湡氠
牥獩獴r湣攢n
+

V
+

OR
Current is charge in motion
•
Charge, e.g. free electrons, exists in conductors
with a density,
n
e
(
n
e
approx 10
29
m

3
)
•
“Somehow” put that charge in motion:
–
effective picture

all charge moves with a velocity,
v
e
–
real picture

a lot of “random motion” of charges with
a small average equal to
v
e
•
Current density,
J
, is given by
J
q
e
n
e
v
e
–
unit of
J
is
C/m
2
sec or A/m
2
(A
≡
Ampere)
and
1A
ㅃ⽳
–
current,
I
, is
J
times cross sectional area,
I
J
p
r
2
–
for 10 Amp in 1mm x 1mm area,
J
+7
A/m
2
, and
v
e
is
about
10

3
m/s
(Yes, the average velocity is only 1mm/s!)
Devices
•
Resistors:
Purpose is to
limit current drawn in a circuit
.
•
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”).
•
The effective resistance of series and parallel
combinations of resistors will be calculated using
Kirchhoff's Laws (Notion of potential difference,
current conservation).
dQ
I
dt
UNIT: Ampere =
A
= C/s
Note:
Ohm’s
Law
•
Demo:
•
Vary applied voltage
V
.
•
Measure current
I
•
Does ratio remain
constant?
V
I
I
R
V
I
slope =
R
I
V
R
How to calculate the resistance?
Include “resistivity” of material
Include geometry of resistor
V
I
Resistance
•
What about acceleration?
V
I
I
R
•
V
E

field
捯c獴慮琠景牣攠故渠
electrons
捯湳瑡湴t慣捥汥牡瑩潮l
敶敲

楮捲敡獩湧畲牥湴i
•
These very large currents and “funny”
I
(
V
)
do
not
occur.
•
Charges
are
ballistically accelerated, but scatter in a very
short time (
t
=
10

14
s) from things that get in the way
–
defects, lattice vibrations (phonons), etc.
•
Average velocity attained in this time is
v
=
eE
t
/
m
•
Current density is
J
=
env
so current is proportional to
E
which is proportional to Voltage
•
OHM’s LAW
J
= (
e
2
n
t
/
m
)
E
or
J
=
s
E
s
㴠
conductivity
Resistance
•
Resistance
Resistance is defined to be the
ratio of the applied voltage to
the current passing through.
•
How do we calculate it?
Recall the case of capacitance:
(
C
=
Q
/
V
)
depended on the geometry
(and dielectric constant), not on
Q
or
V
individually
Similarly, for resistance
–
part depends on the geometry (length
L
and cross

sectional area
A
)
–
part depends on the “resistivity”
ρ
of the material
V
I
I
R
I
V
R
UNIT: OHM =
W
L
R
A
•
Increase the
length
晬fw 潦o敬散瑲潮猠業灥摥p
•
Increase the cross sectional
area
晬fw 晡捩c楴慴敤
•
What about
?
Resistivity
where
E
= electric field and
J
= current density in conductor.
L
A
E
J
•
Property of bulk matter related to
resistance of a sample is the
resistivity
(
)
摥晩湥搠慳a
2
1
E m
J e n
s t
e.g.,
for copper
†
縠

8
W

活
景爠杬慳猬g
†
縠
+12
W

活m景爠
獥浩捯湤畣c潲猠
†
縠ㄠ
W

活m
for superconductors,
= 0 [see Appendix]
The resistivity varies greatly with the sort of material:
,
For uniform case:
I
J
A
EL
V
J
I
ρL
V EL L L I
A A
A
L
R
where
IR
V
1
Two cylindrical resistors,
R
1
and
R
2
, are made of identical material.
R
2
has twice the length of
R
1
but half the radius of
R
1
.
–
These resistors are then connected to a battery
V
as shown:
V
I
1
I
2
–
What is the relation between
I
1
,
the current flowing in
R
1
,
and
I
2
,
the current flowing in
R
2
?
(a)
I
1
<
I
2
(b)
I
1
=
I
2
(c)
I
1
>
I
2
1A
1B
•
A very thin metal wire patterned as
shown is bonded to some structure.
•
As the structure is deformed slightly,
this stretches the wire (slightly).
–
When this happens, the resistance of
the wire:
(a)
decreases
(b)
increases
(c)
stays the same
•
The resistivity of both resistors is the same (
⤮
•
Therefore the resistances are related as:
1
1
1
1
1
2
2
2
8
8
)
4
/
(
2
R
A
L
A
L
A
L
R
•
The resistors have the same voltage across them; therefore
1
1
2
2
8
1
8
I
R
V
R
V
I
•
Two cylindrical resistors,
R
1
and
R
2
, are made of identical material.
R
2
has twice the length of
R
1
but half the radius of
R
1
.
–
These resistors are then connected to a battery
V
as shown:
V
I
1
I
2
–
What is the relation between
I
1
,
the current flowing in
R
1
,
and
I
2
,
the current flowing in
R
2
?
(a)
I
1
<
I
2
(b)
I
1
=
I
2
(c)
I
1
>
I
2
1B
•
A very thin metal wire patterned as
shown is bonded to some structure.
•
As the structure is deformed slightly,
this stretches the wire (slightly).
–
When this happens, the resistance of
the wire:
(a)
decreases
(b)
increases
(c)
stays the same
1B
•
A very thin metal wire patterned as
shown is bonded to some structure.
•
As the structure is deformed slightly,
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, is slightly increased.
Also, because the overall volume of the wire is ~constant, increasing
the length decreases the area
A
, which also increases the resistance.
By carefully measuring the change in resistance, the strain in the
structure may be determined (we’ll see later how to do this optically).
~
R L A
Two cylindrical resistors are made
from the same material, and they
are equal in length. The first
resistor has diameter
d
,
and the
second resistor has diameter
2
d
.
2) Compare the resistance of the two cylinders.
a)
R
1
>
R
2
b)
R
1
=
R
2
c)
R
1
<
R
2
3) If the same current flows through both resistors, compare the
average velocities of the electrons in the two resistors:
a)
v
1
>
v
2
b)
v
1
=
v
2
c)
v
1
<
v
2
Preflight 9:
Resistors
in
Series
a
c
R
effective
a
b
c
R
1
R
2
I
1
IR
V
V
b
a
2
IR
V
V
c
b
)
(
2
1
R
R
I
V
V
c
a
The Voltage “drops”:
)
(
2
1
R
R
R
effective
Hence
:
Whenever devices are in SERIES, the
current is the same through both !
This reduces the circuit to:
Another
(intuitive)
way…
Consider two cylindrical resistors with
lengths
L
1
and
L
2
V
R
1
R
2
L
2
L
1
A
L
R
1
1
A
L
R
2
2
2
1
R
R
R
2
1
2
1
R
R
A
L
L
R
effective
Put them together, end to end to make a longer one...
Two resistors are connected in series to a
battery with emf
E
.
The resistances are
such that
R
1
= 2
R
2
.
5) Compare the current through
R
1
with the
current through
R
2
:
a)
I
1
>
I
2
b)
I
1
=
I
2
c)
I
1
<
I
2
6) What is the potential difference across
R
2
?
a)
V
2
=
E
b)
V
2
= 1/2
E
c)
V
2
= 1/3
E
Preflight 9:
The World’s Simplest
(and most useful)
circuit:
Voltage Divider
?
V
0
2 2
1 2
V
V IR R
R R
0
2 1
V
V=
2
R R
2 1 0
V=V
R R
2 1
V=0
R R
By varying R
2
we can
controllably adjust
the output voltage!
V
0
R
1
R
2
V
Kirchhoff’s First Rule
“Loop Rule” or “Kirchhoff’s Voltage Law (KVL)”
"When any closed circuit loop is traversed, the algebraic
sum of the changes in potential must equal zero."
loop
n
V
0
KVL:
•
This is just a restatement of what you already know: that the
potential difference is independent of path!
e
1
R
1
e
2
R
2
I
e
1
IR
1
IR
2
e
2
0
Rules of the Road
Note: In the ECE convention, voltage drops enter with a + sign
and voltage gains enter with a
獩杮s
e
1
R
1
e
2
R
2
I
e
1
IR
1
IR
2
e
2
0
Our convention:
•
Voltage gains enter with a + sign, and voltage drops enter with a
獩杮.
•
We choose a direction for the current and move around the circuit in that
direction.
•
When a battery is traversed from the negative terminal to the positive
terminal, the voltage increases, and hence the battery voltage enters KVL
with a + sign.
•
When moving across a resistor, the voltage drops, and hence enters KVL
with a
獩杮⸠
Loop Demo
a
d
b
e
c
f
e
1
R
1
I
R
2
R
3
R
4
I
e
2
4
3
2
1
2
1
R
R
R
R
I
e
e
1 2 2 3 4 1
0
IR IR IR IR
e e
loop
n
V
0
KVL:
2
•
Consider the circuit shown.
–
The switch is initially open and the current
flowing through the bottom resistor is
I
0
.
–
After the switch is closed, the current
flowing through the bottom resistor is
I
1
.
–
What is the relation between
I
0
and
I
1
?
(a)
I
1
<
I
0
(b)
I
1
=
I
0
(c)
I
1
>
I
0
R
12V
12V
R
12V
I
a
b
•
Consider the circuit shown.
–
The switch is initially open and the current
flowing through the bottom resistor is
I
0
.
–
After the switch is closed, the current
flowing through the bottom resistor is
I
1
.
–
What is the relation between
I
0
and
I
1
?
(a)
I
1
<
I
0
(b)
I
1
=
I
0
(c)
I
1
>
I
0
•
Write a loop law for original loop:
12V
I
1
R
= 0
I
1
= 12V/
R
•
Write a loop law for the new loop:
12V +12V
I
0
R
I
0
R
= 0
I
0
= 12V/
R
R
12V
12V
R
12V
I
a
b
Summary
•
When you are given a circuit, you must first
carefully
analyze circuit topology
–
find the nodes and distinct branches
–
assign branch currents
•
Use KVL for all independent loops in circuit
–
sum of the voltages around these loops is zero!
Appendix: Superconductivity
•
1911
: H. K. Onnes, who had figured
out how to make liquid helium, used it
to cool mercury to 4.2 K and looked at
its resistance:
•
1957
: Bardeen (
UIUC!
), 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)
–
Current can flow, even if
E
=0
.
–
Current in superconducting rings can flow for years with no
decrease!
•
At low temperatures the resistance of
some metals
〬0浥慳畲敤 瑯t扥敳猠
than
10

16
•
ρ
conductor
(i.e.,
ρ
<
10

24
Ω
m
)!
Appendix: Superconductivity
•
1986
: “High” temp. superconductors (77K) discovered
–
Important because liquid N (77 K) is
much
cheaper than liquid He
–
Highest critical temperature to date
138 K
(

135
˚ C =

211˚ F)
•
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 lines,
magnetic levitation trains, quantum computers?? ...
•
2003
: UIUC Prof. Tony Leggett shares Nobel prize for
helping to explain the related phenomenon of
superfluids
.
–
Superconductivity arises from quantum correlations between pairs
of electrons in the metal, resulting a total loss of “friction”
–
In some materials at low temperature, a similar effect allows them
to flow with no viscosity
“superfluidity”
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