PHY 184

1
PHY 184

• Spring 2007
• Lecture 16

Title Electric Current and Resistance
2
Announcements

• Homework Set 4 is due tomorrow at 800 am.
• Midterm 1 will take place in class Thursday,
  February 8
• Will cover 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
Review

• Electric current i is the net charge passing a
  given point in a given time
• The ampere is abbreviated as A and is given by
• The current per unit area flowing through a
  conductor is the current density J
• If the current is constant and perpendicular to a
  surface, then and we can write an expression for
  the magnitude of the current density

4
Electron Drift Velocity

• In a conductor that is not carrying current, the
  conduction electrons move randomly. (thermal
  motion)
• When current flows through the conductor, the
  electrons have an additional coherent motion.
  (drift velocity, vd )
• The magnitude of the velocity of random thermal
  motion is on the order of 106 m/s while the
  magnitude of the drift velocity is on the order
  of 10-4 m/s
• We can relate the current density J to the drift
  velocity vd of the moving electrons.

5
Electron Drift Velocity (2)

• Consider a conductor with cross sectional area A
  and electric field E.
• Suppose that there are n electrons per unit
  volume.
• The negatively charged electrons will drift in a
  direction opposite to the electric field.
• We assume that all the electrons have the same
  drift velocity vd and that the current density J
  is uniform.
• In a time interval dt, each electron moves a
  distance vddt .
• The volume that will pass through area A is then
  Avd dt the number of electrons is dn nAvd dt .

6
Electron Drift Velocity (3)

• Each electron has charge e so that the charge dq
  that flows through the area A in time dt is
• So the current is
• and the current density is
• The current density and the drift velocity are
  parallel vectors, pointing in opposite
  directions. As vectors,

7
Electron Drift Velocity (4)

• Consider a wire carrying a current
• The physical current carriers are negatively
  charged electrons.
• These electrons are moving to the left in this
  drawing.
• However, the electric field, current density and
  current are directed to the right.

Comments Electrons are negative charges! On top
of the coherent motion the electrons have random
(thermal) motion.
8
Clicker Question

• The figure shows positive charge carriers that
• drift at a speed vd to the left. In what
• directions are J and E?
• A) J and E point to the right
• B) J points to the left, E to the right
• C) J points to the right, E to the left
• D) J and E point to the left

9
Example - current through a wire (1)

• The current density in a cylindrical wire of
  radius R2.0 mm is uniform across a cross section
  of the wire and has the value 2.0 105 A/m2. What
  is the current i through the outer portion of the
  wire between radial distances R/2 and R?
• J current per unit area di / dA

R
10
Example - current through a wire (1)

• The current density in a cylindrical wire of
  radius R2.0 mm is uniform across a cross section
  of the wire and has the value 2.0 105 A/m2. What
  is the current i through the outer portion of the
  wire between radial distances R/2 and R?
• J current per unit area di / dA

R
Area A (outer portion)
Current through A
11
Resistance and Resistivity

• Some materials conduct electricity better than
  others.
• If we apply a given voltage across a conductor,
  we get a large current.
• If we apply the same voltage across an insulator,
  we get very little current (ideal none).
• The property of a material that describes its
  ability to conduct electric currents is called
  the resistivity, ?
• The property of a particular device or object
  that describes it ability to conduct electric
  currents is called the resistance, R
• Resistivity is a property of the material
  resistance is a property of a particular object
  made from that material.

12
Resistance and Resistivity (2)

• If we apply an electric potential difference V
  across a conductor and measure the resulting
  current i in the conductor, we define the
  resistance R of that conductor as
• The unit of resistance is volt per ampere.
• In honor of George Simon Ohm (1789-1854)
  resistance has been given the unit ohm, ?

13
Resistance and Resistivity (3)

• We will assume that the resistance of the device
  is uniform for all directions of the current
  e.g., uniform metals.
• The resistance R of a conductor depends on the
  material from which the conductor is constructed
  as well as the geometry of the conductor
• First we discuss the effects of the material and
  then we will discuss the effects of geometry on
  resistance.

14
Resistivity

• The conducting properties of a material are
  characterized in terms of its resistivity.
• We define the resistivity, ?, of a material by
  the ratio
• The units of resistivity are

E magnitude of the applied field J magnitude of
the current density
15
Typical Resistivities

• The resistivities of some representative
  conductors at 20 C are listed in the table below
• As you can see, typical values for the
  resistivity of metals used in wires are on the
  order of 10-8??m.

(mW-cm)
16
Resistance

• Knowing the resistivity of the material, we can
  then calculate the resistance of a conductor
  given its geometry. Derivation
• Consider a homogeneous wire of length L and
  constant cross sectional area A.
• the resistance is

17
Resistance and resistivity

• For a wire,

18
Clicker Question

• You have three cylindrical copper conductors.
  Rank them according to the current through them,
  the greatest first, when the same potential
  difference V is placed across their lengths.

A a, b, c
B a and c tie, then b
C b, a, c
D a and b tie, then c
19
Clicker Question

• You have three cylindrical copper conductors.
  Rank them according to the current through them,
  the greatest first, when the same potential
  difference V is placed across their lengths.

B a and c tie, then b
D a and b tie, then c
20
Example Resistance of a Copper Wire

• Standard wires that electricians put into
  residential housing have fairly low resistance.
• Question
• What is the resistance of a length of 100 m of
  standard 12-gauge copper wire, typically used in
  household wiring for electrical outlets?
• Answer
• The American Wire Gauge (AWG) size convention
  specifies wire cross sectional area on a
  logarithmic scale.
• A lower gauge number corresponds to a thicker
  wire.
• Every reduction by 3 gauges doubles the
  cross-sectional area.

21
Example Resistance of a Copper Wire (2)

• The formula to convert from the AWG size to the
  wire diameter is
• So a 12-gauge copper wire has a diameter of 2.05
  mm
• Its cross sectional area is then
• Look up the resistivity of copper in the table

22
Clicker Question

• A rectangular block of iron has dimensions 2.0cm
  x 2.0 cm x 10cm. A potential difference is to be
  applied to the block between parallel sides. What
  is the ratio of the resistances R(1)/R(2) of the
  block for the two arrangements (1) and (2).
• A)
• B)
• C)

-
(1)
(2)
23
Clicker Question

• A rectangular block of iron has dimensions 2.0cm
  x 2.0 cm x 10cm. A potential difference is to be
  applied to the block between parallel sides. What
  is the ratio of the resistances R(1)/R(2) of the
  block for the two arrangements (1) and (2).
• A)

-
(1)
(2)
24
Resistors

• In many electronics applications one needs a
  range of resistances in various parts of the
  circuits.
• For this purpose one can use commerciallyavailabl
  e resistors.
• Resistors are commonly made from carbon,inside a
  plastic cover with two wires sticking out at the
  two ends for electrical connection.
• The value of the resistance is indicated by four
  color-bands on the plastic capsule.
• The first two bands are numbers for the mantissa,
  the third is a power of ten, and the fourth is a
  tolerance for the range of values.

25
Resistors (2)

• The number associated with the colors are
• black 0
• brown 1
• red 2
• orange 3
• yellow 4
• green 5
• blue 6
• purple 7
• gray 8
• white 9
• In the tolerance band
• gold means 5
• silver means 10
• no tolerance band means 20

For example, the single resistor shown herehas
colors (top to bottom)brown, green, brown and
goldUsing our table, we can see that the
resistance is 15101 ? 150 ?with a tolerance
of 5
26
Summary
.. speed of an electron
.. resistance to current