PHY 184
Spring 2007
Lecture 17
Title: Resistance and Circuits
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184 Lecture 17
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Announcements
 Homework Set 4 is done!
 Midterm 1 will take place in class on Thursday
• Chapters 16 - 19
• Homework Set 1 - 4
• You may bring one 8.5 x 11 inch sheet of equations, front and back, prepared
any way you prefer.
• Bring a calculator
• Bring a No. 2 pencil
• Bring your MSU student ID card
 We will post Midterm 1 as Corrections Set 1 after the exam
• You can re-do all the problems in the Exam
• You will receive 30% credit for the problems you missed
• To get credit, you must do all the problems in Corrections Set 1, not just the ones
you missed
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Seating Instructions Thursday




Please seat yourselves
alphabetically
Sit in the row (C, D,…)
corresponding to your last
name alphabetically
For example, Bauer would
sit in row C, Westfall in
Row O
We will pass out the exam
by rows
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Section 2
Name
Row Name
END
BEGIN
Breen
Abron
C
Burdick
D
Fall
Semester 2006 Clare
Filipiak
Coleman
E
Midterm 1
Green
Fink
F
Section 1
Huynh
Gruhl
G
Alphabetical Seating
Kantz
Ingham
H
Order
Lantzy
Kelly
I
Miller
Legg
J
Norris
Mislik
K
Pierce
Novak
L
Scafuri
Provines
M
Tran
Schleh
N
Work
Valentini
O
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Review
 The property of a particular device or object that
describes it ability to conduct electric currents is
called the resistance, R
V
 The definition of resistance R is
R
i
 The unit of resistance is the ohm, 
1V
1
1A
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Review (2)
 The resistance R of a device is given by
L
R
A
  is resistivity of the material from which the device is
constructed
 L is the length of the device
 A is the cross sectional area of the device
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Temperature Dependence of Resistivity
 The resistivity (and hence resistance) varies with temperature.
 For metals, this dependence on temperature is linear over a broad range
of temperatures.
 An empirical relationship for the temperature dependence of the
resistivity of metals is given by
  0  0 T  T0 
•
•
•
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Copper
 is the resistivity at temperature T
0 is the resistivity at some
standard temperature T0
 is the “temperature coefficient”
of electric resistivity for the
material under consideration
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Temperature Dependence of Resistance
 In everyday applications we are interested in the
temperature dependence of the resistance of various
devices.
 The resistance of a device depends on the length and the
cross sectional area.
 These quantities depend on temperature
 However, the temperature dependence of linear expansion
is much smaller than the temperature dependence of
resistivity of a particular conductor.
 So the temperature dependence of the resistance of a
conductor is, to a good approximation,
R  R0  R0 T  T0 
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Temperature Dependence
 Our equations for temperature dependence deal
with relative temperatures so that one can use °C
as well as K.
 Values of  for representative metals are shown
below
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Other Temperature Dependence
 At very low temperatures the resistivity of some materials
goes to exactly zero.
 These materials are called superconductors
• Many applications including MRI
 The resistance of some semiconducting materials actually
decreases with increasing temperature.
 These materials are often found in high-resolution
detection devices for optical measurements or particle
detectors.
 These devices must be kept cold to keep their resistance
high using refrigerators or liquid nitrogen.
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Ohm’s Law
 To make current flow through a resistor one must establish
a potential difference across the resistor.
 This potential difference is termed an electromotive force,
emf.
 A device that maintains a potential difference is called an
emf device and does work on the charge carriers
 The emf device not only produces a potential difference but
supplies current.
 The potential difference created by the emf device is
termed Vemf .
 We will assume that emf devices have terminals that we can
connect and the emf device is assumed to maintain Vemf
between these terminals.
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Ohm’s Law (2)
 Examples of emf devices are
• Batteries that produce emf through chemical reactions
• Electric generators that create emf from electromagnetic induction
• Solar cells that convert energy from the Sun to electric energy
 In this chapter we will assume that the source of emf is a
battery.
 A circuit is an arrangement of electrical components
connected together with ideal conducting wires (i.e., having
no resistance).
 Electrical components can be sources of emf, capacitors,
resistors, or other electrical devices.
 We will begin with simple circuits that consist of resistors
and sources of emf.
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Ohm’s Law (3)
 Consider a simple circuit of the form shown below
 Here a source of emf provides a voltage V across a resistor with
resistance R.
 The relationship between the voltage and the resistance in this circuit
is given by Ohm’s Law
Vemf  iR
 … where i is the current in the circuit
( agrees with def. of R = V / i )
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Ohm’s Clicker
 What is the resistance of the resistor in this Demo?
A) about 1 
B) about 100 
C) about 10 
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Ohm’s Clicker
 What is the resistance of the resistor in this Demo?
C) about 10 
V
R
i
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Ohm’s Law (4)
 Now let’s visualize the same circuit in a different way,
making it clearer where the potential drop happens and what
part of the circuit is at which potential.
 The top part of this drawing is just our
original circuit diagram.
 In the bottom part we show the same
circuit, but now the vertical dimension
represents the voltage drop around
the circuit.
 The voltage is supplied by the source
of emf and the entire voltage drop
occurs across the single resistor.
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Resistances in Series
 Resistors connected such that all the current in a circuit
must flow through each of the resistors are connected in
series.
 For example, two resistors R1 and R2 in series with one
source of emf with voltage Vemf implies the circuit shown
below
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Two Resistors in 3D
 To illustrate the voltage drops in this circuit we can
represent the same circuit in three dimensions.
 The voltage drop across resistor R1 is
V1 .
 The voltage drop across resistor R2 is
V2 .
 The sum of the two voltage drops
must equal the voltage supplied by
the battery
Vemf  V1  V2
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Resistors in Series
 The current must flow through all the
elements of the circuit so the current
flowing through each element of the
circuit is the same.
 For each resistor we can apply Ohm’s
Law
V  iR  iR  iR
emf
 … where
1
2
eq
Req  R1  R2
 We can generalize this result to a circuit
with n resistors in series
n
Req   Ri
i 1
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Example: Internal Resistance of a Battery
 When a battery is not connected in a circuit, the voltage
across its terminals is Vt
 When the battery is connected in series with a resistor
with resistance R, current i flows through the circuit.
 When current is flowing, the voltage, V, across the
terminals of the battery is lower than Vt .
 This drop occurs because the battery has an internal
resistance, Ri, that can be thought of as being in series with
the external resistor.
 We can express this relationship as
Vt  iReq  i R  Ri 
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Example: Internal Resistance of a Battery (2)
 We can represent the battery, its internal resistance and
the external resistance in this circuit diagram
terminals of the
battery
battery
 Consider a battery that has a voltage of 12.0 V when it is
not connected to a circuit.
 When we connect a 10.0  resistor across the terminals, the
voltage across the battery drops to 10.9 V.
 What is the internal resistance of the battery?
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Example: Internal Resistance of a Battery (3)
 The current flowing through the external resistor is
V 10.9 V
i 
 1.09 A
R 10.0 
 The current flowing in the complete circuit must be the
same so
Vt  iReq  i R  Ri 
V
R  Ri  
i
V
12.0 V
Ri   R 
 10.0   1.0 
i
1.09 A
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Resistances in Parallel
 Instead of connecting resistors in series so that all the current must
pass through both resistors, we can connect the resistors in parallel
such that the current is divided between the two resistors.
 This type of circuit is shown is below
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Resistance in Parallel (2)
 In this case the voltage drop across each resistor is equal
to the voltage provides by the source of emf.
 Using Ohm’s Law we can write the current in each resistor
i1 
Vemf
i2 
R1
Vemf
R2
 The total current in the circuit must equal the sum of these
currents
i  i1  i2
 Which we can rewrite as
 1
1
i  i1  i2 

 Vemf   
R1
R2
 R1 R2 
Vemf
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Vemf
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Resistance in Parallel (3)
 We can then rewrite Ohm’s Law for the complete circuit as
i  Vemf
 .. where
1
Req
1
1
1


Req R1 R2
 We can generalize this result for two parallel resistors to n
parallel resistors
n
1
1

Req i 1 Ri
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Clicker Question
 A battery, with potential V across it, is connected to a
combination of two identical resistors and then has a
current i through it. What are the potential differences V
across and the current through either resistor if the two
resistors are in series?
A) V, 2i
B) V, i/2
C) V/2, i
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Clicker Question
 A battery, with potential V across it, is connected to a
combination of two identical resistors and then has a
current i through it. What are the potential differences V
across and the current through either resistor if the two
resistors are in series?
C) V/2, i
In series: The resistors have identical currents i
The sum of potential differences across the resistors is
equal to the applied potential difference:
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Clicker Question
 A battery, with potential V across it, is connected to a
combination of two identical resistors and then has a
current i through it. What are the potential differences V
across and the current through either resistor if the two
resistors are in parallel?
A) V, 2i
B) V, i/2
C) 2V, i
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Clicker Question
 A battery, with potential V across it, is connected to a
combination of two identical resistors and then has a
current i through it. What are the potential differences V
across and the current through either resistor if the two
resistors are in parallel?
B) V, i/2
In parallel: The resistors all have the same V applied
The sum of the currents through the resistors is equal to
the total current:
i  i1  i2
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