Electric Current

1
Chapter 24

• Electric Current

2
Electric Current

• The electric current I is the rate of flow of
  charge through some region of space
• The SI unit of current is Ampere (A) 1 A 1 C/s
• Let us look at the charges flowing
  perpendicularly to a surface of area A
• The average current
• The instanteneous current

3
Electric Current

• The conventional direction of the current is the
  direction positive charge would flow
• In a common conductor (e.g., copper), the current
  is due to the motion of the negatively charged
  electrons
• It is common to refer to a moving charge as a
  mobile charge carrier
• A charge carrier can be positive or negative

4
Current and Drift Speed

• Charged particles move through a conductor of
  cross-sectional area A and a charge carrier
  density n
• The total number of charge carriers n A ?x
• The total charge is the number of carriers times
  the charge per carrier, q ?Q (n A ?x) q
• The drift speed, vd, is the speed at which the
  carriers move vd ?x / ?t
• ?Q (n A vd ?t) q
• Iave ?Q / ?t n q vd A

5
Current and Drift Speed

• If the conductor is isolated, the electrons
  undergo random motion (due to collisions with the
  atoms)
• When an electric field is set up in the
  conductor, it creates an electric force on the
  electrons and hence a current
• The zigzag line represents the motion of charge
  carrier in a conductor

6
Current and Drift Speed

• The drift speed is much smaller than the average
  speed between collisions
• When a circuit is completed, the electric field
  travels with a speed close to the speed of light
• Therefore, although the drift speed is on the
  order of 10-4 m/s the effect of the electric
  field is felt on the order of 108 m/s

7
Current Density

• The current density J of a conductor is defined
  as the current per unit area
• If the current density is uniform and A is
  perpendicular to the direction of the current
  then this expression is valid J I / A nqvd
• J has SI units of A/m2
• The current density is in the direction of the
  positive charge carriers

8
Conductivity

• A current density and an electric field are
  established in a conductor whenever a potential
  difference is maintained across the conductor
• For some materials, the current density is
  directly proportional to the field
• J s E
• The coefficient of proportionality, s, is called
  the conductivity of the conductor

9
Ohms Law

• Ohms law states that for many materials, the
  conductivity s is a constant that is independent
  of the electric field producing the current
• Most metals obey Ohms law
• Ohms law is not a fundamental law of nature, but
  an empirical relationship valid only for certain
  materials

10
Resistance

• In a conductor, the voltage applied across the
  ends of the conductor is proportional to the
  current through the conductor
• The constant of proportionality is the resistance
  of the conductor it arises due to collisions
  between the electrons carrying the current with
  the fixed atoms inside the conductor
• SI unit of resistance is ohm (O) 1 O 1 V / A

11
Resistivity

• The inverse of the conductivity is the
  resistivity of the material (see table 24.1)
• ? 1 / s
• Resistivity has SI units of ohm-meters (O . m)
• The resistance of an ohmic conductor is
  proportional to its length, L, and inversely
  proportional to its cross-sectional area, A

12
Ohmic and Nonohmic Materials

• Materials that obey Ohms Law are said to be
  ohmic (the relationship between current and
  voltage is linear, and the resistance is constant
  over a wide range of voltages)
• Not all materials follow Ohms law
• Materials that do not obey Ohms law are said to
  be nonohmic

13
Resistance and Resistivity, Summary

• Every material has a characteristic resistivity
  that depends on the properties of the material
  and on temperature, i.e., resistivity is a
  property of substances
• The resistance of a material depends on its
  geometry and its resistivity, i.e., resistance is
  a property of an object
• An ideal conductor would have zero resistivity
• An ideal insulator would have infinite resistivity

14
Chapter 24Problem 27

• A uniform wire of resistance R is stretched until
  its length doubles. Assuming its density and
  resistivity remain constant, whats its new
  resistance?

15
A Model for Electrical Conduction

• Treat a conductor as a regular array of atoms
  plus a collection of free electrons conduction
  electrons
• In the absence of an electric field, the motion
  of the conduction electrons is random, and their
  speed is on the order of 106 m/s
• When an electric field is applied, the conduction
  electrons are given a drift velocity

16
A Model for Electrical Conduction

• We assume
• 1) The electrons motion after a collision is
  independent of its motion before the collision
• 2) The excess energy acquired by the electrons in
  the electric field is lost to the atoms of the
  conductor when the electrons and atoms collide
  (causing the temperature of the conductor to
  increase)

17
A Model for Electrical Conduction

• The force experienced by an electron is
• From Newtons Second Law, the acceleration is
• Applying a motion equation
• Since the initial velocities are random, their
  average value is zero
• If t is the average time interval between
  successive collisions, then

18
A Model for Electrical Conduction

• The current density
• Using Ohms Law
• The conductivity and the resistivity do not
  depend on the strength of the field
  (characteristic of a conductor obeying Ohms Law)

19
Temperature Variation of Resistivity

• For most metals, resistivity increases with
  increasing temperature the atoms vibrate with
  increasing amplitude so the electrons find it
  more difficult to pass through the atoms
• For most metals, resistivity increases
  approximately linearly with temperature over a
  limited temperature range
• ?0 resistivity at some reference temperature T0
  (usually taken to be 20 C) a is the
  temperature coefficient of resistivity

20
Temperature Variation of Resistance

• Since the resistance of a conductor with uniform
  cross sectional area is proportional to the
  resistivity, the effect of temperature on
  resistance is similar

21
Chapter 24Problem 59

• The resistivity of copper as a function of
  temperature is given approximately by ? ?01
  a (T - T0), where ?0 is Table 24.1s entry for
  20C, T0 20C, and a 4.3 10-3 C-1. Find
  the temperature at which coppers resistivity is
  twice its room temperature value.

22
Residual Resistivity

• For some metals, the resistivity is nearly
  proportional to the temperature
• A nonlinear region always exists at very low
  temperatures, and the resistivity usually reaches
  some finite value as the temperature approaches
  absolute zero
• The residual resistivity near 0 K is caused
  primarily by the collisions of electrons with
  impurities and imperfections in the metal

23
Superconductors

• Superconductors a class of materials whose
  resistances fall to virtually zero below a
  certain temperature, TC (critical temperature)
• The value of TC is sensitive to chemical
  composition, pressure, and crystalline structure
• Once a current is set up in a superconductor, it
  persists without any applied voltage (since R
  0)
• One application is superconducting magnets

24
Semiconductors

• Semiconductors are materials that exhibit a
  decrease in resistivity with an increase in
  temperature, i.e. a is negative
• The reason an increase in the density of charge
  carriers at higher temperatures

25
Electrical Energy and Power

• In a circuit, as a charge moves through the
  battery, the electrical potential energy of the
  system is increased by ?Q ?V (the chemical
  potential energy of the battery decreases by the
  same amount)
• The charge moving through a resistor loses this
  potential energy during collisions with atoms in
  the resistor (the temperature of the resistor
  increases)

• When the charge returns to a, the net result is
  that some chemical energy of the battery has been
  delivered to the resistor and caused its
  temperature to rise

26
Electrical Energy and Power

• The rate at which the energy is lost is the power
• From Ohms Law, alternate forms of power are
• The SI unit of power is Watt (W) (I must be in
  Amperes, R in ohms and ?V in Volts)
• The unit of energy used by electric companies is
  the kilowatt-hour (defined in terms of the unit
  of power and the amount of time it is supplied)
  1 kWh 3.60 x 106 J

27
Chapter 24Problem 29

• A 4.5-W flashlight bulb draws 750 mA. (a) At what
  voltage does it operate? (b) Whats its
  resistance?

28
Answers to Even Numbered Problems Chapter 24
Problem 14 2.9 105 C
29
Answers to Even Numbered Problems Chapter 24
Problem 28 1.4 kW
30
Answers to Even Numbered Problems Chapter 24
Problem 52 840 km
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• Answers to Even Numbered Problems
• Chapter 24
• Problem 54
• 8.70 kA
• 15.1