R 1

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”