Transcript Chapter 27

Chapter 27
Current and Resistance
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:
I avg
Q

t
• The instanteneous current:
dQ
I
dt
André-Marie Ampère
1775 – 1836
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
Current and Drift Speed
• Charged particles move through a conductor of crosssectional area A and a charge carrier density n
• The total number of charge carriers: n A Δx
• The total charge is the number of carriers times the
charge per carrier, q: ΔQ = (n A Δx) q
• The drift speed, vd, is the speed at which the carriers
move: vd = Δx / Δt
ΔQ = (n A vd Δt) q
Iave = ΔQ / Δt = n q vd A
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
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
Current Density
• The current density J of a conductor is defined as the
current per unit area: J = I / A = nqvd
• This expression is valid only if the current density is
uniform and A is perpendicular to the direction of the
current
• J has SI units of A/m2
• The current density is in the direction of the positive
charge carriers
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=σE
• The coefficient of proportionality, σ, is called the
conductivity of the conductor
Ohm’s Law
• Ohm’s law states that for many materials, the
conductivity σ is a constant that is independent of the
electric field producing the current
• Most metals obey Ohm’s law
• Ohm’s law is not a fundamental law of nature, but an
empirical relationship valid only for certain materials
Georg Simon Ohm
1787 – 1854
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
V
R
I
• SI unit of resistance is ohm (Ω): 1 Ω = 1 V / A
Georg Simon Ohm
1787 – 1854
Resistivity
• The inverse of the conductivity is the resistivity of
the material (see table 27.2) :
ρ=1/σ
• Resistivity has SI units of ohm-meters (Ω . m)
• The resistance of an ohmic conductor is
proportional to its length, L, and inversely
proportional to its cross-sectional area, A:
L
Rr
A
• ρ is the constant of proportionality
Ohmic and Nonohmic Materials
• Materials that obey Ohm’s 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 Ohm’s law
• Materials that do not obey Ohm’s law are said to be
nonohmic
Non-ohmic
Resistance and Resistivity, Summary
• Every ohmic 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
Chapter 27
Problem 13
Suppose that you wish to fabricate a uniform wire out of 1.00 g of
copper. If the wire is to have a resistance R = 0.500 Ω, and if all of
the copper is to be used, what will be (a) the length and (b) the
diameter of the wire?
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
A Model for Electrical Conduction
• We assume:
• 1) The electron’s 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)
A Model for Electrical Conduction


• The force experienced by an electron is
F  qE
• From Newton’s Second
Law,

 the acceleration is
  F qE

a
me
m
• Applying a motion equation

    qE
v f  vi  at  vi 
t
me
• Since the initial velocities are random, their average
value is zero
• If t is the average time interval between
 successive
collisions, then 
 qE
v f ,avg  vd 
me
t
A Model for Electrical Conduction
• The current density:
nq t
E
J  I / A  nAqvd / A  nqvd 
me
• Using Ohm’s Law:
2
nq t

me
2
J  E
• The conductivity and the resistivity do not depend
on the strength of the field (characteristic of a
conductor obeying Ohm’s Law) 

 qE
v f ,avg  vd 
t
me
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
r  ro [1  (T  To )]
• ρ0 – resistivity at some reference temperature T0
(usually taken to be 20° C); α – is the temperature
coefficient of resistivity
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
R  Ro [1 (T  To )]
Chapter 27
Problem 22
An aluminum rod has a resistance of 1.234 Ω at 20.0°C. Calculate the
resistance of the rod at 120°C by accounting for the changes in both
the resistivity and the dimensions of the rod.
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
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
Semiconductors
• Semiconductors are materials that exhibit a decrease
in resistivity with an increase in temperature, i.e. α is
negative
• The reason – an increase in the density of charge
carriers at higher temperatures
Electrical Energy and Power
• In a circuit, as a charge moves through the battery, the
electrical potential energy of the system is increased
by ΔQ ΔV (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
Electrical Energy and Power
• The rate at which the energy is lost is the power
Q

V  I V
t
• From Ohm’s Law, alternate forms of power are
V
 I R 
R
2
2
• The SI unit of power is Watt (W) (I must be in Amperes,
R in ohms and ΔV 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
Chapter 27
Problem 29
A 500-W heating coil designed to operate from 110 V is made of Nichrome
wire 0.500 mm in diameter. (a) Assuming the resistivity of the Nichrome
remains constant at its 20.0°C value, find the length of wire used. (b) Now
consider the variation of resistivity with temperature. What power is delivered
to the coil of part (a) when it is warmed to 1 200°C?
Chapter 27
Problem 63
Gold is the most ductile of all metals. For example, one gram of gold
can be drawn into a wire 2.40 km long. What is the resistance of such
a wire at 20°C?
Answers to Even Numbered Problems
Chapter 27:
Problem 10
0.130 mm/s
Answers to Even Numbered Problems
Chapter 27:
Problem 12
500 mA
Answers to Even Numbered Problems
Chapter 27:
Problem 18
1.44 × 103 °C
Answers to Even Numbered Problems
Chapter 27:
Problem 44
27.0 yr