PES 1120 Spring 2014, Spendier
Lecture 20/Page 1
Today:
- Resistance and Resistivity
- Ohm's Law
Last Lecture:
Average Current:
I av 
q
t
 Coulomb 
Unit 
  A ( Ampere)
 sec 
q dq

t  0 t
dt
Think about current as the rate at which charge moves through a circuit:
I  vd nAe
Instantaneous Current: I  lim
Current density: J 
I
 A
Unit  2 
A
m 
Resisting the flow of current:
Resistance is a measure of an object's opposition to the flow of electrons. This may sound
like a bad thing, but it's actually very useful. Resistance is what makes it possible to
generate heat and light, restrict the flow of electric current when necessary, and ensure
that the correct voltage is supplied to a device. For instance, as electrons travel through
the filament of a light bulb, they meet so much resistance that they slow down a lot. As
they fight their way through the filament, the atoms of the filament bump into each other
furiously, generating heat - which produces glow that you see from your light bulb.
Everything, even the best conductors - exhibits a certain amount of resistance to the flow
of electrons. (Well, actually, there are certain materials, called superconductors, that can
conduct current with zero electrical resistance. We will talk about these next lecture)
So the higher resistance, the more restricted the flow of current. We will now talk about
what determines how much resistance an object has.
Factors that determine the amount of current that flows though a material:
Two factors determine the amount of current that flows though a material:
(1) The force on the charges - Given by E
(2) The resistivity of a material. The resistivity is actually due to collisions of conduction
electrons with other particles (nuclei) in a wire. The number of collisions is quantified by
ρ = Resistivity
E

E…. more force results in more current
ρ…..more collisions result in less current
J
PES 1120 Spring 2014, Spendier
Lecture 20/Page 2
Definition of resistivity:
V 
Units   m =  m
A 
Resistivity determines the current density in a material with a given electric field.

E
J
V 
 A 
    = Ohm
(Greek symbol for omega)
This equations hold only for isotropic materials—materials whose electrical properties
are the same in all directions
We can write expression in vector form as


E  J
We often speak of the conductivity σ of a material. This is simply the reciprocal of its
resistivity, so
s
1
(  m)1 
r
Resistors
A resistor is an object (not a material) that is used to restrict the current. We can use the
water analogy again.
First, we have a pipe with water flowing through it.
• Then, we add a restriction in the pipe.
PES 1120 Spring 2014, Spendier
Lecture 20/Page 3
Making a Resistor
Resistivity determines the current density in a material with a given electric field. We
will find it useful to use this property to control the flow of current in circuits. So we now
need to make a “resistor” out of some material with resistivity ρ. In this case, we want to
control the entire current through the resistor.
What is the potential difference between the ends of the resistor?
I  JA 
E
A
r
Calculate E:
f
 
V   E  dl  EL
(constant electric field)
i
Va Vb  Vab  EL
V
E  ab
L
V
Hence I  ab A
Lr
Or
Lr
Vab  I
 IR
A
Or
V  IR We call this law Ohm’s law – more below
Where resistance R is defined as
V 
Lr
R
    = Ohm (Greek symbol for omega)
 A 
A
Georg Ohm (1789-1854) Bavarian math teacher.
PES 1120 Spring 2014, Spendier
Lecture 20/Page 4
Looking at the above expression how do we expect resistance R to change – does it make
sense?
- R increase with increasing length L of the object/wire. The longer the wire, the more
resistance it has because there are more opportunities for electrons to bump into other
particles along the way.
- R increase with increasing resistivity ρ. It can change with different materials.
Resistance depends on how easily electrons can drift inside the material. Some materials
allow their electrons to roam freely whereas others hold on tight to their electrons.
Conductors have relatively low resistivity; whereas insulators have relatively high
resistivity.
- R decreases with increasing cross-sectional area A of the object/wire. The larger the
wire the easier it is for electrons to move - that is, the lower the resistance to their
environment. Think of water flowing through a pipe: The wider the pipe, the easier the
water flows.
PES 1120 Spring 2014, Spendier
Lecture 20/Page 5
DEMO: Guess what metal the wire is made off
22 Gauge wire ( diameter is 0.6440 mm)
The basic procedure of this experiment is to send a known current through a wire and measure the
electric potential difference (or voltage) across a certain length of the wire. We can use this to
calculate the resistivity of the wire (i.e. material the wire is made of) if we know the diameter of the
wire.
PES 1120 Spring 2014, Spendier
Lecture 20/Page 6
After resistivity is calculated also calculate:
a) the electric-field magnitude in the wire of 1 m length
b) the resistance of a 50.0 m length of this wire
Ohm’s Law
In 1826, the German physicist Georg Ohm discovered that for metals at constant
temperature the resistivity is constant.
V = IR
Any device obeying Ohm’s Law is called a resistor. An “ohmic” material follows
proportionality between the electric field and the current density:
Caveats
The resistivity of materials in general depends on temperature.
Not all materials are ohmic. A resistor—obeys Ohm’s law (left). A pn junction
diode—does not (right).
PES 1120 Spring 2014, Spendier
Lecture 20/Page 7
Variation of resistivity with temperature:
Resistivity tends to get bigger as temperature increases. Why?
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:
The rule for how resistivity changes is given as
 r  r0 
rise
slope 
 r0 a 
run
T  T0 
hence
r  r0   r0a T  T0 
r  r0  r0 a T  T0 
T0 = reference temperature (usually room temp 20C)
ρ0 = resistivity at the reference temperature T0
ρ = resistivity as a function of temperature T
α = constant for a given material (temperature coefficient of resistivity)
Example:
resistivity of copper ρ = 1.7 x 10-8  m (at room temp = 293 K)
temperature coefficient of resistivity α = 4.3 x 10-3 K-1
For a copper wire, what was the change in resistivity using the high temp from this
Monday (73F or 296 K) and high temp from today (36F or 278 K)
a) Without any calculations, which day was resistivity higher and why?
b) Now find the % difference
PES 1120 Spring 2014, Spendier
Lecture 20/Page 8