Download Chapter 27

Chapter 21
Current
and
Direct Current Circuits
Electric Current


Electric current is the rate of flow of
charge through a surface
The SI unit of current is the Ampere (A)


1A=1C/s
The symbol for electric current is I
Average Electric Current


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
Instantaneous Electric Current

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
Direction of Current




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
Current and Drift Speed



Charged particles
move through a
conductor of crosssectional area A
n is the number of
charge carriers per
unit volume
n A Δx is the total
number of charge
carriers
Current and Drift Speed, cont

The total charge is the number of
carriers times the charge per carrier, q


The drift speed, vd, is the speed at
which the carriers move



ΔQ = (n A Δ x) q
vd = Δ x/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA
Charge Carrier Motion
in a Conductor

The zig-zag black line
represents the motion
of charge carrier in a
conductor



The net drift speed is
small
The sharp changes in
direction are due to
collisions
The net motion of
electrons is opposite
the direction of the
electric field
Motion of Charge
Carriers , cont



When the potential difference is applied,
an electric field is established in the
conductor
The electric field exerts a force on the
electrons
The force accelerates the electrons and
produces a current
Motion of Charge
Carriers, final

The changes in the electric field that drives
the free electrons travel through the
conductor with a speed near that of light




This is why the effect of flipping a switch is
effectively instantaneous
Electrons do not have to travel from the light
switch to the light bulb in order for the light to
operate
The electrons are already in the light filament
They respond to the electric field set up by
the battery
Drift Velocity, Example


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

This is a typical order of magnitude for drift
velocities
Current Density


J is the current density of a conductor
It is defined as the current per unit area




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
Conductivity



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
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
DV
R
I
Resistance, cont

SI units of resistance are ohms (Ω)


1Ω=1V/A
Resistance in a circuit arises due to
collisions between the electrons
carrying the current with the fixed atoms
inside the conductor
Ohm’s Law

Ohm’s Law states that for many
materials, the resistance is constant
over a wide range of applied voltages


Most metals obey Ohm’s Law
Materials that obey Ohm’s Law are said to
be ohmic
Ohm’s Law, cont

Not all materials follow Ohm’s Law



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
Ohmic Material, Graph




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
Nonohmic Material, Graph



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
Resistivity

Resistance is related to the geometry of the
device:
Rr



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)
Some Resistivity Values
Resistance and
Resistivity, Summary





Resistivity is a property of a substance
Resistance is a property of an object
The resistance of a material depends on
its geometry and its resistivity
An ideal (perfect) conductor would have
zero resistivity
An ideal insulator would have infinite
resistivity
Resistors



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
Resistor Values

Values of resistors
are commonly
marked by colored
bands
Resistance and Temperature

Over a limited temperature range, the
resistivity of a conductor varies
approximately linearly with the
temperature
r  ro [1  (T  To )]

ρo is the resistivity at some reference
temperature To


To is usually taken to be 20° C
 is the temperature coefficient of resistivity

SI units of  are oC-1
Temperature Variation
of Resistance

Since the resistance of a conductor with
uniform cross sectional area is
proportional to the resistivity, you can
find the effect of temperature on
resistance
R  Ro [1   (T  To )]
Resistivity and
Temperature, Graphical View



For metals, the resistivity
is nearly proportional to
the temperature
A nonlinear region always
exists at very low
temperatures
The resistivity usually
reaches some finite value
as the temperature
approaches absolute zero
Residual Resistivity


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 the electrons and the
metal atoms

This is the linear range on the graph
Superconductors

A class of metals and
compounds whose
resistances go to zero
below a certain
temperature, TC


TC is called the critical
temperature
The graph is the same
as a normal metal
above TC, but suddenly
drops to zero at TC
Superconductors, cont

The value of TC is sensitive to




Chemical composition
Pressure
Crystalline structure
Once a current is set up in a
superconductor, it persists without any
applied voltage

Since R = 0
Superconductor Application


An important
application of
superconductors is
a superconducting
magnet
The magnitude of
the magnetic field
is about 10 times
greater than a
normal
electromagnet
Electrical Conduction –
A Model

The free electrons in a conductor move with
average speeds on the order of 106 m/s




Not totally free since they are confined to the
interior of the conductor
The motion is random
The electrons undergo many collisions
The average velocity of the electrons is zero

There is zero current in the conductor
Conduction Model, 2



An electric field is applied
The field modifies the motion of the
charge carriers
The electrons drift in the direction
opposite of the field

The average drift speed is on the order of
10-4 m/s, much less than the average
speed between collisions
Conduction Model, 3

Assumptions:



The excess energy acquired by the
electrons in the field is lost to the atoms of
the conductor during the collision
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
Conduction Model, 4

The force experienced by an electron is
F  eE

From Newton’s Second Law, the acceleration
is
F Fe
eE
a

me

me

me
Applying a motion equation
eE
v  v o  at  v o 
t
me

Since the initial velocities are random, their
average value is zero
Conduction Model, 5



Let t be the average time interval
between 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
Conduction Model, final

Using Ohm’s Law, an expression for the
resistivity of a conductor can be found:
me
r 2
ne t


Note, the resistivity does not depend on
the strength of the field
The average time is also related to the
free mean path: t = l avg/vavg
Conduction Model,
Modifications


A quantum mechanical model is needed
to explain the incorrect predictions of
the classical model developed so far
The wave-like character of the electrons
must be included

The predictions of resistivity values then
are in agreement with measured values
Electrical Power


Assume a circuit as
shown
As a charge moves
from a to b, the
electric potential
energy of the system
increases by QDV

The chemical energy
in the battery must
decrease by this same
amount
Electrical Power, 2


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

Corresponding to increased vibrational
motion of the atoms in the resistor
Electric Power, 3



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

The input of energy from the battery is balanced
by the output of energy by heat and radiation
Electric Power, 4


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
Electric Power, final


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

Units: I is in A, R is in W, V is in V, and P
is in W
Electric Power Transmission


Real power lines
have resistance
Power companies
transmit electricity at
high voltages and
low currents to
minimize power
losses
emf

A source of emf (electromotive force) is
an entity that maintains the constant
voltage of a circuit


The emf source supplies energy, it does
not apply a force, to the circuit
The battery will normally be the source
of energy in the circuit

Could also use generators
Sample Circuit



We consider the
wires to have no
resistance
The positive
terminal of the
battery is at a higher
potential than the
negative terminal
There is also an
internal resistance in
the battery
Internal Battery Resistance



If the internal
resistance is zero,
the terminal voltage
equals the emf
In a real battery,
there is internal
resistance, r
The terminal
voltage, DV = e - Ir
emf, cont

The emf is equivalent to the open-circuit
voltage



This is the terminal voltage when no
current is in the circuit
This is the voltage labeled on the battery
The actual potential difference between
the terminals of the battery depends on
the current in the circuit
Load Resistance

The terminal voltage also equals the
voltage across the external resistance



This external resistor is called the load
resistance
In the previous circuit, the load resistance
is the external resistor
In general, the load resistance could be
any electrical device
Power



The total power output of the battery is
P = IDV =Ie
This power is delivered to the external
resistor (I2 R) and to the internal resistor
(I2r)
P = Ie  I2 R + I2 r

The current depends on the internal and
external resistances
I
e
Rr
Resistors in Series



When two or more resistors are connected endto-end, they are said to be in series
For a series combination of resistors, the
currents are the same in all the resistors
because the amount of charge that passes
through one resistor must also pass through the
other resistors in the same time interval
The potential difference will divide among the
resistors such that the sum of the potential
differences across the resistors is equal to the
total potential difference across the combination
Resistors in Series, cont

Potentials add



ΔV = IR1 + IR2
= I (R1+R2)
Consequence of
Conservation of Energy
The equivalent
resistance has the
same effect on the
circuit as the original
combination of resistors
Equivalent Resistance –
Series



Req = R1 + R2 + R3 + …
The equivalent resistance of a series
combination of resistors is the algebraic
sum of the individual resistances and is
always greater than any of the individual
resistances
If one device in the series circuit creates
an open circuit, all devices are
inoperative
Equivalent Resistance –
Series – An Example

Two resistors are replaced with their
equivalent resistance
Resistors in Parallel


The potential difference across each resistor
is the same because each is connected
directly across the battery terminals
The current, I, that enters a point must be
equal to the total current leaving that point



I = I1 + I2
The currents are generally not the same
Consequence of Conservation of Charge
Equivalent Resistance –
Parallel

Equivalent Resistance
1
1
1
1




Req R1 R2 R3

The inverse of the
equivalent resistance of
two or more resistors
connected in parallel is
the algebraic sum of the
inverses of the
individual resistance

The equivalent is always
less than the smallest
resistor in the group
Equivalent Resistance –
Parallel, Examples


Equivalent resistance replaces the two original
resistances
Household circuits are wired so the electrical devices
are connected in parallel

Circuit breakers may be used in series with other circuit
elements for safety purposes
Resistors in Parallel, Final



In parallel, each device operates
independently of the others so that if one is
switched off, the others remain on
In parallel, all of the devices operate on the
same voltage
The current takes all the paths


The lower resistance will have higher currents
Even very high resistances will have some current
Circuit Reduction

A circuit consisting of resistors can often be
reduced to a simple circuit containing only
one resistor




Examine the original circuit and replace any
resistors in series with their equivalents and any
resistors in parallel with their equivalents
Sketch the new circuit
Examine it and replace any new series or parallel
combinations with their equivalents
Continue until a single equivalent resistance is
found
Circuit Reduction – Example



The 8.0 W and 4.0 W resistors
are in series and can be
replaced with their equivalent,
12.0 W
The 6.0 W and 3.0 W resistors
are in parallel and can be
replaced with their equivalent,
2.0 W
These equivalent resistances
are in series and can be
replaced with their equivalent
resistance, 14.0 W
Kirchhoff’s Rules


There are ways in which resistors can
be connected so that the circuits formed
cannot be reduced to a single
equivalent resistor
Two rules, called Kirchhoff’s Rules, can
be used instead
Statement of Kirchhoff’s Rules

Junction Rule

At any junction, the sum of the currents
must equal zero


A statement of Conservation of Charge
Loop Rule

The sum of the potential differences across
all the elements around any closed circuit
loop must be zero

A statement of Conservation of Energy
Mathematical Statement of
Kirchhoff’s Rules


Junction Rule:
Iin = Iout
Loop Rule:
 DV  0
closed
loop
More About the Junction Rule

I1 - I2 - I3 = 0




Use +I for currents
entering a junction
Use –I for currents
leaving a junction
From Conservation
of Charge
Diagram b shows a
mechanical analog
More About the Loop Rule



Traveling around the loop
from a to b
In a, the resistor is
transversed in the direction of
the current, the potential
across the resistor is –IR
In b, the resistor is
transversed in the direction
opposite of the current, the
potential across the resistor is
is +IR
Loop Rule, final


In c, the source of emf
is transversed in the
direction of the emf
(from – to +), the
change in the electric
potential is +ε
In d, the source of emf
is transversed in the
direction opposite of the
emf (from + to -), the
change in the electric
potential is -ε
Junction Equations
from Kirchhoff’s Rules

Use the junction rule as often as
needed, so long as, each time you write
an equation, you include in it a current
that has not been used in a previous
junction rule equation

In general, the number of times the
junction rule can be used is one fewer than
the number of junction points in the circuit
Loop Equations
from Kirchhoff’s Rules


The loop rule can be used as often as
needed so long as a new circuit element
(resistor or battery) or a new current
appears in each new equation
You need as many independent
equations as you have unknowns
Kirchhoff’s Rules’
Equations, final

In order to solve a particular circuit
problem, the number of independent
equations you need to obtain from
the two rules equals the number of
unknown currents
Problem-Solving Hints –
Kirchhoff’s Rules

Conceptualize




Study the circuit diagram
Identify all the elements in the circuit
Identify the polarity of all the batteries and imagine
the directions in which the current would exist
through the batteries
Categorize


Determine if the circuit can be reduced by
combining series and parallel resistors
If not, continue with the application of Kirchhoff’s
rules
Problem-Solving Hints –
Kirchhoff’s Rules, cont

Analyze

Draw the circuit diagram and assign labels and
symbols to all known and unknown quantities.
Assign directions to the currents


The direction is arbitrary, but you must adhere to the
assigned directions when applying Kirchhoff’s Rules
Apply the junction rule to any junction in the circuit
that provides new relationships among the various
currents
Problem-Solving Hints, cont

Analyze, cont

Apply the loop rule to as many loops as are
needed to solve for the unknowns


Solve the equations simultaneously for the
unknown quantities


To apply the loop rule, you must choose a direction to
travel around the loop and correctly identify the potential
difference as you cross various elements
If a current turns out to be negative, the magnitude will
be correct and the direction is opposite to that which you
assigned
Finalize

Check your answers for consistency
RC Circuits




A direct current circuit may contain capacitors
and resistors, the current will vary with time
When the circuit is completed, the capacitor
starts to charge
The capacitor continues to charge until it
reaches its maximum charge (Q = Cε)
Once the capacitor is fully charged, the
current in the circuit is zero
Charging an RC Circuit



As the plates are being charged, the potential
difference across the capacitor increases
At the instant the switch is closed, the charge
on the capacitor is zero
Once the maximum charge is reached, the
current in the circuit is zero

The potential difference across the capacitor
matches that supplied by the battery
Charging Capacitor
in an RC Circuit

The charge on the
capacitor varies with
time


q = Ce(1 – e-t/RC) =
Q(1 – e-t/RC)
t is the time constant


t=RC
The current can be
found
e t RC
I (t ) 
R
e
Time Constant, Charging



The time constant represents the time
required for the charge to increase from
zero to 63.2% of its maximum
t has units of time
The energy stored in the charged
capacitor is ½ Qe = ½ Ce2
Discharging Capacitor
in an RC Circuit

When a charged
capacitor is placed
in the circuit, it can
be discharged


q = Qe-t/RC
The charge
decreases
exponentially
Discharging Capacitor

At t = t = RC, the charge decreases to 0.368
Qmax


In other words, in one time constant, the capacitor
loses 63.2% of its initial charge
The current can be found
dq
Q
t RC
I t  
 Io e
, Io 
dt
RC

Both charge and current decay exponentially
at a rate characterized by t = RC
Atmosphere as a Conductor

Lightning and sparks are examples of
currents existing in air


Earlier examples of the air as an insulator
were a simplification model
Whenever a strong electric field exists
in air, it is possible for the air to undergo
electrical breakdown in which the
effective resistivity of the air drops and
the air becomes a conductor
Creating a Spark

(a) A molecule is ionized as
a result of a random event


Cosmic rays and other events
produce the ionized molecules
(b) The ion accelerates
slowly and the electron
accelerates rapidly due to
the force from the electric
field


This is if there is a strong
electric field
In a weak field, they both
accelerate slowly and
eventually neutralize as they
recombine
Creating a Spark, cont


(c) The accelerated
electron approaches
another molecule at
a high speed
(d) If the field is
strong enough, the
electron may have
enough energy to
ionize the molecule
during the collision
Creating a Spark, final

(e) There are now two
electrons to be
accelerated by the field


Each of these electrons
can strike another
molecule and repeat the
process
The result is a very
rapid increase in the
number of charge
carriers available in the
air and a corresponding
decrease in the
resistance of the air
Lightning


Lightning occurs when a large current in the
air neutralizes the charges that established
the initial potential difference
Typical currents during lightning can be very
high


Stepped leader current is in the range of 200 –
300 A
Peak currents are about 5 x 104 A

Power is in the billions of watts range
Fair Weather Currents

The average fair-weather current in the
atmosphere is about 1000 A


The average fair-weather charge
density is 2 x 10-12 A / m2


This is spread out over the entire globe
During the lightning stoke, J ~ 105 A/m2
Fair-weather current is in the opposite
direction from the lightning current