chapter 21 current and direct current circuits. electric current electric current is the rate of...

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