Chapter 21

Current

and

Direct Current Circuits

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21.1 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

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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

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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

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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

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Model of current in a Conductor

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 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

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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

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Current and Drift Speed 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

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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

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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

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21.2 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

SI units of resistance are ohms (Ω) 1 Ω = 1 V / A

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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

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

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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

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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

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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

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Resistance and Resistivity, Summary 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

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 0, called the residual resistivity

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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 electrons and the vibrations of the metal atoms 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

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

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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

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Resistor Values

Values of resistors are commonly marked by colored bands

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21.3 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

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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

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21.4 A Model of Electrical Conduction 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 of an electron is random The electrons undergo many collisions The average velocity of the electrons is zero

There is zero current in the conductor

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Conduction Model, 2 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 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 Assumptions:

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 The force experienced by an electron is

From Newton’s Second Law, the acceleration is

The equation of motion between two collisions

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

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Conduction Model, 5 Let be the average time interval

between two 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

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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 collision time is also related to the free mean path lavg and average speed vavg

= lavg / vavg

2em

ne

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21.5 Electrical Power 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

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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

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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

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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

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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

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Electric Power Transmission Real power lines have

resistance Power companies

transmit electricity at high voltages and low currents to minimize power losses

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