PHY 184

1
PHY 184

• Spring 2007
• Lecture 17

Title Resistance and Circuits
2
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

3
Seating Instructions Thursday
Section 2

• 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

• Fall Semester 2006
• Midterm 1
• Section 1
• Alphabetical Seating Order

4
Review

• The property of a particular device or object
  that describes it ability to conduct electric
  currents is called the resistance, R
• The definition of resistance R is
• The unit of resistance is the ohm, ?

5
Review (2)

• The resistance R of a device is given by
• ? 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

6
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

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

7
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,

8
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

9
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.

10
Ohms 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.

11
Ohms 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.

12
Ohms 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 Ohms Law
• where i is the current in the circuit

( agrees with def. of R V / i )
13
Ohms Clicker

• What is the resistance of the resistor in this
  Demo?
• A) about 1 ?
• B) about 100 ?
• C) about 10 ?

14
Ohms Clicker

• What is the resistance of the resistor in this
  Demo?
• C) about 10 ?

15
Ohms Law (4)

• Now lets 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.

16
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

17
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

18
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 Ohms Law
• where
• We can generalize this result to a circuit with n
  resistors in series

19
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

20
Example Internal Resistance of a Battery (2)

• We can represent the battery, its internal
  resistance and the external resistance in this
  circuit diagram

• 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?

21
Example Internal Resistance of a Battery (3)

• The current flowing through the external resistor
  is
• The current flowing in the complete circuit must
  be the same so

22
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

23
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 Ohms Law we can write the current in each
  resistor
• The total current in the circuit must equal the
  sum of these currents
• Which we can rewrite as

24
Resistance in Parallel (3)

• We can then rewrite Ohms Law for the complete
  circuit as
• .. where
• We can generalize this result for two parallel
  resistors to n parallel resistors

25
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

26
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

27
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

28
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