Download Physics 2102 Spring 2002 Lecture 8

Resistance Is
Futile!
Physics 2113
Jonathan Dowling
Physics 2102
Lecture 20: WED 04 MAR
Current & Resistance II
Georg Simon Ohm
(1789-1854)
Resistance is NOT Futile!
Electrons are not “completely free to move” in a conductor. They move
erratically, colliding with the nuclei all the time: this is what we call
“resistance”. The mechanical analog is FRICTION.
The resistance is related to the potential we need to apply to a device to
drive a given current through it. The larger the resistance, the larger the
potential we need to drive the same current.
Ohm’s laws
V
Rº
i
Units : [R] =
V
and therefore : i =
and V = iR
R
Volt
º Ohm (abbr. W)
Ampere
Georg Simon Ohm
(1789-1854)
"a professor who preaches such heresies
is unworthy to teach science.” Prussian
minister of education 1830
Devices specifically designed to have a constant value of R are called
resistors, and symbolized by
dq éCù
iº
= ê ú º [ Ampere] = [ A]
ësû
dt
Resistivity ρ vs. Resistance R
Metal
“field lines”
These two devices could have the same resistance
R, when measured on the outgoing metal leads.
However, it is obvious that inside of them different
things go on.
resistivity:
Resistivity is associated
( resistance: R=V/I )
r º [Wm] = [Ohm× meter]
with a material, resistance
with respect to a device
1
Conductivity : s =
constructed with the material.
r
Example:
A
-
L
V
+
V
E= ,
L
i
J=
A
r=
Makes sense!
For a given material:
V
L=RA
i
L
A
L
R=r
A
Longer ® More resistance
Wider ® Less resistance
Fluid Flow: An Analogy for the PETEs!
• Amount of Water = Charge
• Pressure = Potential = Voltage
• Flow Rate = Current = Amps
The pressure at the end of the
hose represents voltage.
The amount of water in the tank
represents charge.
The rate of flow gallons/minute out
the hose represents current or
amperage.
https://learn.sparkfun.com/tutorials/voltage-current-resistance-and-ohms-law
Fluid Flow: An Analogy for the PETEs!
R=r
• Amount of Water = Charge
• Pressure = Voltage
• Flow Rate = Current = Amps
Decrease hose width, decrease
A, increase resistance R.
Increase hose length L,
increase resistance R.
Put pebbles in the hose,
increase resistivity ρ, increase
resistance R.
https://learn.sparkfun.com/tutorials/voltage-current-resistance-and-ohms-law
L
A
26.4: Resistance and Resistivity:
The resistivity ρ of a resistor is defined as:
The SI unit for  is .m.
The conductivity  of a material is the reciprocal of its
resistivity:
Put pebbles in the hose,
increase resistivity ρ, increase
resistance R.
26.4: Resistance and Resistivity, Calculating Resistance from Resistivity:
Think pumping water through a
long hose. It is easier if L is short
and A is big (small R). It is harder
if L is long or A is small (big R).
If the streamlines representing the current
density are uniform throughout the wire,
the electric field E and the current density
J will be constant for all points within the
wire.
ICPP
The copper wire has radius r. What
happens to the Resistance R if you:
(a) Double the Length? R ® 2R
(b) Double the Area? R ® R / 2
(c) Double the Radius? R ® R / 4
A = pr2
What happens to the Resistivity ρ if you:
(a)Double the Length?
r®r
(b)Double the Area?
(c)Double the Radius?r ® r
r®r
Step I: The resistivity ρ is the same (all three are copper).
Find the Resistance R=ρL/A for each case:
r ( 3L / 2 )
r ( L / 2) r L
rL
rL
Rb =
=3
Rc =
=
Ra =
A
( A / 2)
( A / 2) A
A
Rb > Ra = Rc
Step II: Rank the current using V=iR or i=V/R with V constant!
ia = ic > ib
Ranking is reversed since R is downstairs.
Example
Two conductors are made of the same material and have the
same length. Conductor A is a solid wire of diameter r=1.0mm.
Conductor B is a hollow tube of outside diameter 2r=2.0mm and
inside diameter r=1.0mm. What is the resistance ratio RA/RB,
measured between their ends?
A
R= L/A
B
AA= r2
AB= (2r)2 - r2 =3r2
RA/RB= AB/AA= 3
LA=LB=L Cancels
Example, A material has resistivity, a block of the material has a resistance.:
26.4: Resistance and Resistivity, Variation with Temperature:
The relation between temperature and resistivity for copper—and for metals in general—
is fairly linear over a rather broad temperature range. For such linear relations we can
write an empirical approximation that is good enough for most engineering purposes:
Resistivity and Temperature
Resistivity depends on
temperature:
 =  0(1+ (T–T0) )
• At what temperature would the resistance of a
copper conductor be double its resistance at
20.0°C?
• Does this same "doubling temperature" hold for
all copper conductors, regardless of shape or
size?
Resistance is NOT Futile!
Electrons are not “completely free to move” in a conductor. They move
erratically, colliding with the nuclei all the time: this is what we call
“resistance”. The mechanical analog is FRICTION.
The resistance is related to the potential we need to apply to a device to
drive a given current through it. The larger the resistance, the larger the
potential we need to drive the same current.
Ohm’s laws
V
Rº
i
Units : [R] =
V
and therefore : i =
and V = iR
R
Volt
º Ohm (abbr. W)
Ampere
Georg Simon Ohm
(1789-1854)
"a professor who preaches such heresies
is unworthy to teach science.” Prussian
minister of education 1830
Devices specifically designed to have a constant value of R are called
resistors, and symbolized by
dq éCù
iº
= ê ú º [ Ampere] = [ A]
ësû
dt
26.5: Ohm’s Law:
L
A=r2
A
Current Density: J=i/A
Units: [A/m2]
Resistance: R= L/A
Resitivity:  depends only on
Material and Temperature.
Units: [•m]
V = iR
R = V / i = constant
Example
An electrical cable consists of 105 strands of fine wire, each
having r=2.35  resistance. The same potential difference is
applied between the ends of all the strands and results in a
total current of 0.720 A.
(a) What is the current in each strand?
i=I/105=0.720A/105=[0.00686] A
(b) What is the applied potential difference?
V=ir=[0.016121] V
(c) What is the resistance of the cable?
R=V/I=[.0224 ] 
Rd = 1.0x105
im = 1x10–3A
Rw = 1.5x103
im =
V1 = imRd
i1 = V1/Rw
V2 = imRw
Example
A human being can be electrocuted if a
current as small as i=100 mA passes near
the heart. An electrician working with
sweaty hands makes good contact with
the two conductors he is holding. If his
resistance is R=1500, what might the
fatal voltage be?
(Ans: 150 V) Use: V=iR
i=100 mA?
i=100 μA?
i=100 MA?
i = 100 ´10 -3 A
i = 100 ´10 -6 A
i = 100 ´10 +6 A
http://www.phys.lsu.edu/~jdowling/PHYS21132-SP15/lectures/NRL13.pdf