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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)
1 A = 1 C / s The symbol for electric current is I
Average Electric Current Assume charges are
moving perpendicular to a surface of area A
If Q is the amount of charge that passes through A in time t, the average current is
avg
QI
t
Instantaneous Electric Current If the rate at which the charge flows
varies with time, the instantaneous current, I, can be found
0limt
Q dQI
t 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 cross-sectional 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 ΔQ = (n A Δ x) q
The drift speed, vd, is the speed at which the carriers move 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 = E The constant of proportionality, , 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
VR
I
Resistance, cont SI units of resistance are ohms (Ω)
1 Ω = 1 V / 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:
is called the resistivity of the material The inverse of the resistivity is the
conductivity: = 1 / and R = l / A
Resistivity has SI units of ohm-meters ( . m)
RA
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 wire-wound
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
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
[1 ( )]o oT T
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
[1 ( )]o oR R T T
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
From Newton’s Second Law, the acceleration is
Applying a motion equation
Since the initial velocities are random, their average value is zero
eF E
e
e e e
e
m m m
FF Ea
o oe
et t
m
Ev v a v
Conduction Model, 5 Let be the average time interval
between successive collisions The average value of the final velocity
is the drift velocity
This is also related to the current: I = n e vd A = (n e2 E / me) A
de
e
m
E
v
Conduction Model, final Using Ohm’s Law, an expression for the
resistivity of a conductor can be found:
Note, the resistivity does not depend on the strength of the field
The average time is also related to the
free mean path: = l avg/vavg
2em
ne
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 QV
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:
Applying Ohm’s Law, alternative expressions can be found:
Units: I is in A, R is in , V is in V, and P is in W
I V
22 V
I V I RR
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, V = - 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 = IV =I This power is delivered to the external
resistor (I2 R) and to the internal resistor (I2r)
P = II2 R + I2 r The current depends on the internal and
external resistances
IR r
Resistors in Series When two or more resistors are connected end-to-
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
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
321eq R
1
R
1
R
1
R
1
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 and 4.0 resistors
are in series and can be replaced with their equivalent, 12.0
The 6.0 and 3.0 resistors are in parallel and can be replaced with their equivalent, 2.0
These equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0
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:0
closedloop
V
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
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
Solve the equations simultaneously for the unknown quantities
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 = C(1 – e-t/RC) =
Q(1 – e-t/RC) is the time constant
=RC
The current can be found
( ) t RCI t eR
Time Constant, Charging The time constant represents the time
required for the charge to increase from zero to 63.2% of its maximum
has units of time The energy stored in the charged
capacitor is ½ Q = ½ C2
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 = = 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
Both charge and current decay exponentially at a rate characterized by = RC
,t RCo o
dq QI t I e I
dt 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 This is spread out over the entire globe
The average fair-weather charge density is 2 x 10-12 A / m2
During the lightning stoke, J ~ 105 A/m2
Fair-weather current is in the opposite direction from the lightning current