Download Electric current

Chapter 21
Current
and
Direct Current Circuits
21.1 Electric Current
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Electric current is the rate of flow of
charge through a surface
The SI unit of current is the Ampere (A)
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1A=1C/s
The symbol for electric current is I
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Average Electric Current
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Assume charges are
moving perpendicular
to a surface of area A
If DQ is the amount of
charge that passes
through A in time Dt, the
average current is
Iavg
DQ

Dt
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Instantaneous Electric Current
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If the rate at which the charge flows
varies with time, the instantaneous
current, I, can be found
DQ dQ
I  lim

Dt 0 Dt
dt
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Direction of Current
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The charges passing through the area could
be positive or negative or both
It is conventional to assign to the current the
same direction as the flow of positive charges
The direction of current flow is opposite the
direction of the flow of electrons
It is common to refer to any moving charge as
a charge carrier
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Model of current in a
Conductor
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The zig-zag black line represents the
motion of charge carrier in a conductor
The sharp changes in direction are due
to collisions
When an external electric field is
applied on the conductor, the electric
field exerts a force on the electrons
The force accelerates the electrons and
produces a current
The net motion of electrons is opposite
the direction of the electric field
Vd is the drift velocity, which is small
If there is no electric field, Vd is zero
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Drift Velocity, Example
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Assume a copper wire, with one free
electron per atom contributed to the
current
The drift velocity for a 12 gauge copper
wire carrying a current of 10 A is 2.22 x
10-4 m/s
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This is a typical order of magnitude for drift
velocities
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Current and Drift Speed
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Charged particles move
through a conductor of
cross-sectional area A
n is the number of
charge carriers per unit
volume
n A Δx is the total
number of charge
carriers
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Current and Drift Speed
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The total charge is the number of
carriers times the charge per carrier, q
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The drift speed, vd, is the speed at
which the carriers move
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ΔQ = (n A Δ x) q
vd = Δ x/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA
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Current Density
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J is the current density of a conductor
It is defined as the current per unit area
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J = I / A = n q vd
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
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Conductivity
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A current density J and an electric field
E are established in a conductor
whenever a potential difference is
maintained across the conductor
J=sE
The constant of proportionality, s, is
called the conductivity of the
conductor
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21.2 Resistance
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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
R 
DV
I
SI units of resistance are ohms (Ω)
1Ω=1V/A
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Ohm’s Law
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Ohm’s Law states that for many materials, the
resistance is constant over a wide range of applied
voltages
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Not all materials follow Ohm’s Law
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Most metals obey Ohm’s Law
Materials that obey Ohm’s Law are said to be ohmic
Materials that do not obey Ohm’s Law are said to be
nonohmic
Ohm’s Law is not a fundamental law of nature
Ohm’s Law is an empirical relationship valid only for
certain materials
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Ohmic Material, Graph
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An ohmic device
The resistance is
constant over a wide
range of voltages
The relationship
between current and
voltage is linear
The slope is related
to the resistance
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Nonohmic Material, Graph
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Non-ohmic materials
are those whose
resistance changes
with voltage or
current
The current-voltage
relationship is
nonlinear
A diode is a
common example of
a non-ohmic device
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Resistivity
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Resistance is related to the geometry of the
device:
Rr
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A
r is called the resistivity of the material
The inverse of the resistivity is the
conductivity:
 s = 1 / r and R = l / sA
Resistivity has SI units of ohm-meters (W . m)
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Resistance and
Resistivity, Summary
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Resistivity is a property of a material
Resistance is a property of an object
The resistance of a objector depends on
its geometry and its resistivity
An ideal (perfect) conductor would have
zero resistivity
An ideal insulator would have infinite
resistivity
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Temperature variation of
Resistivity of a metal
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For metals, the resistivity is
nearly proportional to the
temperature
A nonlinear region always
exists at very low
temperatures
As the temperature
approaches absolute zero, the
resistivity usually reaches
some finite value r0, called the
residual resistivity
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Residual Resistivity
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The residual resistivity near absolute
zero is caused primarily by the collisions
of electrons with impurities and
imperfections in the metal
High temperature resistivity is
predominantly characterized by
collisions between electrons and the
vibrations of the metal atoms
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This is the linear range on the graph
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Over a limited temperature range, the
resistivity of a conductor varies
approximately linearly with the
temperature
r  ro [1  (T  To )]
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ρo is the resistivity at some reference
temperature To
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To is usually taken to be 20° C
 is the temperature coefficient of resistivity
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SI units of  are oC-1
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Resistors
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Most circuits use
elements called
resistors
Resistors are used to
control the current level
in parts of the circuit
Resistors can be
composite or wirewound
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Resistor Values
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Values of resistors
are commonly
marked by colored
bands
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21.3 Superconductors
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A class of metals and
compounds whose
resistances go to zero
below a certain
temperature, TC
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TC is called the critical
temperature
The graph is the same
as a normal metal above
TC, but suddenly drops
to zero at TC
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Superconductors, cont
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The value of TC is
sensitive to
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Chemical composition
Pressure
Crystalline structure
Once a current is set up in
a superconductor, it
persists without any
applied voltage
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Since R = 0
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21.4 A Model of Electrical
Conduction
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The free electrons in a conductor move with
average speeds on the order of 106 m/s
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Not totally free since they are confined to the
interior of the conductor
The motion of an electron is random
The electrons undergo many collisions
The average velocity of the electrons is zero
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There is zero current in the conductor
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Conduction Model, 2
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As an external electric field is applied,
the field modifies the motion of the
charge carriers
The electrons drift in the direction
opposite of the field
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The average drift speed is on the order of
10-4 m/s, much less than the average
speed between two successive collisions
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Conduction Model, 3
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Assumptions:
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The excess energy acquired by the
electrons from the external field loses to
the atoms of the conductor during the
collisions
The energy given up to the atoms
increases their vibration and therefore the
temperature of the conductor increases
The motion of an electron after a collision
is independent of its motion before the
collision
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Conduction Model, 4
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The force experienced by an electron is
F  eE
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From Newton’s Second Law, the acceleration
is
F Fe
eE
a
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me

me

me
The equation of motion between two collisions
eE
v  v o  at  v o 
t
me
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Since the initial velocities are random, their
average value is zero
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Conduction Model, 5
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Let t be the average time interval
between two successive collisions
The average value of the final velocity
is the drift velocity
e E
vd 
t
me
This is also related to the current:
I = n e vd A = (n e2 E / me) t A
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Conduction Model, final
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Using Ohm’s Law, an expression for the
resistivity of a conductor can be found:
me
r
ne2t
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Note, the resistivity does not depend on the
strength of the field
The collision time t is also related to the free
mean path lavg and average speed vavg
t = lavg / vavg
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21.5 Electrical Power
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As a charge moves
from a to b, the
electric potential
energy of the system
increases by QDV
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The chemical energy in
the battery must
decrease by this same
amount
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Electrical Power, 2
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As the charge moves through the
resistor (c to d), the system loses this
electric potential energy during
collisions of the electrons with the
atoms of the resistor
This energy is transformed into internal
energy in the resistor
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Corresponding to increased vibrational
motion of the atoms in the resistor
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Electric Power, 3
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The resistor is normally in contact with the air,
so its increased temperature will result in a
transfer of energy by heat into the air
The resistor also emits thermal radiation
After some time interval, the resistor reaches
a constant temperature
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The input of energy from the battery is balanced
by the output of energy by heat and radiation
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Electric Power, 4
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The rate at which the system loses
potential energy as the charge passes
through the resistor is equal to the rate
at which the system gains internal
energy in the resistor
The power is the rate at which the
energy is delivered to the resistor
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Electric Power, final
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The power is given by the equation:
  I DV
Applying Ohm’s Law, alternative
expressions can be found:
2
V
  I DV  I 2R 
R
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Units: I is in A, R is in W, V is in V, and P
is in W
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Electric Power Transmission
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Real power lines have
resistance
Power companies
transmit electricity at
high voltages and low
currents to minimize
power losses
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