R 1

kitefleaUrban and Civil

Nov 15, 2013 (3 years and 8 months ago)

128 views



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


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
-


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”