Thus Far Our Treatment of Electrical Phenomena Has Been Confined to the Study of Charges in...

8/13/2019 Thus Far Our Treatment of Electrical Phenomena Has Been Confined to the Study of Charges in Equilibrium Situati

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Current And ResistanceThus far our treatment of electrical phenomena has been confined to the study of

charges in equilibrium situations, or electrostatics . We now consider situations involving

electric charges that are not in equilibrium. We use the term electric current, or simply

current, to describe the rate of flow of charge through some region of space. Most practical

applications of electricity deal with electric currents. For example, the battery in a flashlight

produces a current in the filament of the bulb when the switch is turned on. A variety of

home appliances operate on alternating current. In these common situations, current exists in

a conductor, such as a copper wire. It also is possible forcurrents to exist outside a

conductor. For instance, a beam of electrons in a television picture tube constitutes a

currentThis chapter begins with the definition of current. A microscopic description of

current is given, and some of the factors that contribute to the opposition to the flow of

charge in conductors are discussed. A classical model is used to describe electrical

conduction in metals, and some of the limitations of this model are cited. We also define

electrical resistance and introduce a new circuit element, the resistor. We conclude by

discussing the rate at which energy is transferred to a device in an electric circuit.24.1 Electric Current And Current Density

Whenever electric charges move, a current is said to exist. To define current more

precisely, consider a surface of area A through which charges are moving as in figure 24.1.

The current is the rate at which charge flows through this surface. If Q is the amount of

charge that passes through this area in a time interval t , the average current I is equal to

the charge that passes through A per unit time:

(24.1) t Q

I


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Figure 24.1 Charges in motion through an area A. The time rate of flow of charge through

the area is defined as the current I. The direction of the current is in the direction of the

current is in the direction of flow of positive charges

If the rate at which charge flows varies in time, then the current varies in time; we

define the instantaneous current I as the differential limit of average current:

24.2 dt dQ

I

The SI unit of current is the ampere (A),where:

sC A 1

That is, 1 A of current is equivalent to 1 C of charge passing through the surface area in 1s.

In practice, smaller units of current are often used, such as the milliampere (1 mA= 310 A)

and the microampere ( A A 6101 ).

The charges passing through the surface in Figure 24.1 can be positive or negative,

or both. It is conventional to choose the direction of the current to be in the direction as the

flow of positive charge. In electrical conductors, such as copper or aluminum, the current is

due to the motion of negatively charged electrons. Therefore, when we speak of current in

an ordinary conductor, the direction of the current is opposite the direction of flow of

electrons. However, if we are considering a beam of positively charged protons in an

accelerator, the current is in the direction of motion of the protons. In some cases such as

those involving gases and electrolytes, for instance the current is the result of the flow of

both positive and negative charges. This occurs, for example, in semiconductors, as

discussed in section 24.6. it is common to refer to a moving charge (whether it is positive or

negative) as a mobile charge carrier. For example, the charge carriers in a metal are

electrons.


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Figure 24.2 A section of a uniform conductor of cross-sectional area A. The mobile charge

carriers move with a speed vd , and the displacement they experience in the x direction in a

time interval t is t v x d . If we choose t to be the time interval during which the

charges are displaced, on the average, by the length of the cylinder, the number of carriers in

the section of length x is t nAv d , where n is the number of carriers per unit volume.We can relate current to the motion of the charge carriers by describing a microscopic

model of conduction in a metal. Consider the current in a conductor of cross-sectional area

A (Fig. 24.2). The volume of a section of the conductor of length x (the gray region shown

in Fig. 24.2) is A x . If n represents the number of mobile charge carriers per unit volume,

then the number of mobile charge carrierss in the volume element is give by nA x .

Therefore, the total charge Q in this elements is give by q xnAvQ d

where q is the charge on each carrier. If the carriers move with a speed vd , the displacement

they experience in the x direction in a time interval t is t v x d . Let us choose t to

be the time interval required for the charges in the cylinder to move through a displacement

whose magnitude is equal to the length of the cylinder. This time interval is also that

required for all of the charges in the cylinder to pass through the circular area at one end.With this choice, we can write Q in the form

qt nAvQ d

Using Eq.24.1 we see that the current in the conductor os given by

Anqvt Q

I d

Figure 24.3 A schematic representation of the zigzag motion of an electron in a conductor.

The changes in direction are the result of collisions between the electron and atoms in the

conductor. Note that the net motion of the electron is opposite the direction of the electric


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field. Because of the acceleration of the charge carriers due to the electric force, the paths

are actually parabolic. However, the drift speed is much smaller than the average speed, so

the parabolic shape is not visible on this scale.

The velocity of the charge carriers vd is an average velocity called the drift velocity.

To understand the meaning of drift speed, consider a conductor in which the charge carriers

are free electrons. If the conductor is isolated that is, the potential difference across it is zero

then these electrons undergo random motion that is analogous to the motion of gas

molecules. As we discussed earlier, when a potential difference is applied across the

conductor (for example, by means of a battery), an electric field is set up in the conductor;

this field exerts an electric force on the electrons, producing a current. However, the

electrons do not move in straight lines along the conductor. Instead, they collide repeatedly

with the metal atoms, and their resultant motion is complicated and zigzag motion (Fig.

24.3). Despite the collisions, the electrons move slowly along the conductor (in a direction

opposite that of E) at the drift velocity v d . As we shall see in an example that follows, drift

velocities are much smaller than the average speed between collisions. We shall discuss this

model in more detail in section 24.3 . One can think of the collisions of the electrons within

a conductor as being an effective internal friction (or drag force), similar to that experienced by the molecules of a liquid flowing through a pipe sruffed with steel wool

We can think of the atom electron collisions in a conductor as an effective internal

friction (or drag force) similar to that experienced by the molecules of a liquid flowing

through a pipe stuffed with steel wool. The energy transferred from the electrons to the

metal atoms during collisions causes an increase in the vibrational energy of the atoms and a

corresponding increase in the temperature of the conductor.


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Example 24.1

The 12-gauge copper wire in a typical residential building has a cross-

sectional area of 3.31 x 10 -6 m 2. If it carries a current of 10.0 A, what is

the drift speed of the electrons? Assume that each copper atomcontributes one free electron to the current. The density of copper is

8.95 g/cm 3.

SolutionFrom the periodic table of the elements in Appendix C, we find that the

molar mass of copper is 63.5 g/mol. Recall that 1 mol of any substance

contains Avogadros number of atoms (6.02 x 10 23). Knowing the

density of copper, we can calculate the volume occupied by 63.5 g (=1

mol) of copper:

3

3

09.795.8

5.63cm

cm g

g mV

Because each copper atom contributes one free electron to the current,

we have

3

283

36

3

23

1049.811000.1

09.71062.6

melectron x

mcm x

cmelectron x

n

where q is the absolute value of the charge on each electron. Thus,

sm

x

m xC xm x s

c

nqAvd

4

2619328

1022.2

1031.31060.11049.8

101


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Resistance and ohms Law Charges move in a conductor to produce a current under the action of an electric

field inside the conductor. An electric field can exist in the conductor in this case since we

are dealing with charges in motion, a nonelectrostatic situation. This in contrast with the

situation in which a conductor in electrostatic equilibrium (where the charges are a rest) can

have no electric field inside.

Consider a conductor of cross-sectional area A carrying a current I . The current

density J in the conductor is defined as the current per unit area. Because the current

Anqv I d the current density is give by

(24.5) d nqv A

I J

where J has SI units of 2m A . This expression is valid only if the current density is uniform

and only if the surface of cross-sectional area A is perpendicular to the direction of the

current. In general, current density is a vector quantity:

(24.6) d nqv J

From this equation, we see that current density is in the direction of charge motion for

positive charge carriers and opposite the direction of motion for negative charge carriers.

A current density J and an electric field E are established in a conductor whenever a

potential difference is maintained across the conductor. In some materials, the current

density is proportional to the electric field:

lawsohm' (24.7) E J

where the constant of proportionality is called the conductivity of the conductor.1

Materials that obey Equation 27.7 are said to follow Ohms law, named after Georg e Simon

Ohm (1789 1854). More specifically, Ohms law states that


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Materials that obey Ohms law and hence demonstrate this simple relationship between E

and J are said to be ohmic. Experimentally, however, it is found that not all materials have

this property. Materials and devices that do not obey Ohms law are said to be nonohmic.

Ohms law is not a fundamental law of nature but rather an empirical relationship valid only

for certain materials.

Figure 24.4 A uniform conductor of length l and cross-sectional area A. A potential

difference ab vv maintained across the conductor sets up an electric field E ,

and this field produces a current I that is proportional to the potential difference.

A form of ohms law that is more directly useful in practical applications can be

obtained by considering a segment of a straight wire of cross-sectional area A and length l

as in figure 24.4. A potential difference ab vv is maintained across the wire, creating an

electric field in the wire and current. If the electric field in the wire is assumed and current.

If the electric field in the wire is assumed to be uniform, the potential difference ab vvv

is related to the electric field through the relationship

El V

Therefore, we can express the magnitude of the current density in the wire as

l V

E J

Because A I J , we can write the potential difference as

for many materials (including most metals), the ratio of the

current density to the electric field is a constant that is

independent of the electric field producing the current.


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I A

J l

V

1

The quantity A

l

is called the resistance of the conductor. We can define the resistance as

the ratio of the potential difference across a conductor to the current in

the conductor:

conductoaof resistance (24.8) I V

Al

R

We will use this equation over and over again when studying electric circuits. From this

result we see that resistance has SI units of volts per ampere. One volt per ampere is

defined to be one ohm )(

AV 11

This expression shows that if a potential difference of 1 V across a conductor causes a

current of 1 A, the resistance of the conductor is 1 . For example, if an electrical appliance

connected to a 120-V source of potential difference carries a current of 6 A,

its resistance is 20

The inverse of conductivity is resistivity

yresistivit 1

where has the units ohm-meters )( m . Because Al

R

, we can express theresistance

of a uniform block of material along the length l as

conductor uniformaof resistance Al R

From table 24.1 gives the resistivities of a variety of materials at 20 0C


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Equation 24.10 shows that the resistance of a given cylindrical conductor such as a

wire is proportional to its length and inversely proportional to its cross-sectional area. If the

length of a wire is doubled, then its resistance doubles. If its cross-sectional area is doubled,

then its resistance decreases by one half. The situation is analogous to the flow of a liquid

through a pipe. As the pipes length is increased, the resistance to flow increases. As the

pipes cross -sectional area is increased, more liquid crosses a given cross section of the pipe

per unit time interval. Thus, more liquid flows for the same pressure differential applied to

the pipe, and the resistance to flow decreases.

Ohmic materials and devices have a linear current potential difference relationship

over a broad range of applied potential differences (Fig. 24.5a). The slope of the I -versus V

curve in the linear region yields a value for 1/ R. Nonohmic materials have a nonlinear

current potential difference relationship. One common semiconducting device that has

nonlinear I -versus- V characteristics is the junction diode (Fig. 24.5b). The resistance of this

device is low for currents in one direction (positive V ) and high for currents in the reverse

direction (negative V ). In fact, most modern electronic devices, such as transistors, have

nonlinear current potential difference relationships; their proper operation depends on the particular way in which they violate Ohms law .


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Figure 24.5 (a) The current potential difference curve for an ohmic material. The

curve is linear, and the slope is equal to the inverse of the resistance of the conductor.

(b) A nonlinear current potential difference curve for a junction diode. This device

does not obey Ohms law.


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Example 24.2

(A)Calculate the resistance per unit length of a 22-gauge Nichrome wire, which has a

radius of 0.321 mm.

Solution The cross-sectional area of this wire is

27232 1024.310321.0 m xm xr A

The resistivity of Nichrome is m x 105.1 6 (see Table 24.1). Thus, we can use

Equation 24.10 to find the resistance per unit length:

mm xm x

Al R

63.41024.3 105.1

27

6

(B) If a potential difference of 10 V is maintained across a 1.0-m length of the

Nichrome wire, what is the current in the wire?

Solution Because a 1.0-m length of this wire has a resistance

of 4.63 Equation 24.8 gives

AV

R

V I 16.2

63.4

10

C. calculate the current density and electric field in the wire assuming that the current

it carries is 26.16 A.

26

27 1067.61024.316.2

m A

xm x

A A I

J

Since E J and

1

the electric field in the wire is give by

C N

m A

xm x E

J J

E

0.101067.6105.1 266

Note the copper hasa resistivity of 1.7 x 10 -8m ,about 10 -2 that of nichrome.

Therefore a copper wire of the same gaughe would have a resistance per unit length

of only 0.05 m . A1-m length of such copper wire would carry the same current

(2.16 A) with an applied voltage of only 0.11 V.


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24.3The Resistivity of different conductors

The restivity of a conductor depends on a number of factors, one of which istemperature. For most metals, resistivity increases temperature. The resistivity of a

conductor varies approximately linear fashion with temperature over a limited temperature

range according to the expression

eraturewith tempof variation(24.11) 10 aT T

where is the resistivity at some temperature T (in degrees Celsius), 0 is the resistivity at

some reference temperature T 0 (usually taken to be 20C), and is the temperaturecoefficient of resistivity . From Equation 24.11, we see that the temperature coefficient of

resistivity can be expressed as

T p

0

1

where 0 is the change in resistivity in the temperature interval 0T T T

The temperature coefficients of resistivity for various materials are given in Table 24.1. Note that the unit for is degrees Celsius -1 [(C) -1]. Because resistance is proportional to

resistivity (Eq. 24.10), we can write the variation of resistance as

(24.13) 10 aT T R R


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For metals like copper, resistivity is nearly proportional to temperature, as shown in

Figure 24.6. However, a nonlinear region always exists at very low temperatures, and the

resistivity usually reaches some finite value as the temperature approaches absolute zero.

This residual resistivity near absolute zero is caused primarily by the collision of electrons

with impurities and imperfections in the metal. In contrast, hightemperature resistivity (the

linear region) is predominantly characterized by collisions between electrons and metal

atoms.

Example 24.3 A Platinum Resistance Thermometer

A resistance thermometer, which measures temperature by measuring

the change in resistance of a conductor, is made from platinum and has a

resistance of 50.0 at 20.0C. When immersed in a vessel containing

melting indium, its resistance increases to 80.7 . Calculate the melting point of the indium. Note that 1031092.3 C x for platinum.

Solution Solving Equation 24.13 for T and using the

value for platinum given in Table 24.1, we obtain

0

1030

0 137

0.501092.3

0.507.80C

C x R

R RT

Since 0T T T and C T 0

0 20 , we find that T=1570C


14/21

Figure 27.6 Resistivity versus temperature for a metal such as

copper. The curve is linear over a wide range of temperatures, and increases with

increasing temperature. As T approaches absolute zero (inset), the resistivity approaches a

finite value 0 .

Notice that three of the values in Table 24.1 are negative; this indicates that the

resistivity of these materials decreases with increasing temperature (Fig. 24.7), which is

indicative of a class of materials called semiconductors . This behavior is due to an increase

in the density of charge carriers at higher temperatures.

Figure 24.7 Resistivity versus temperature for a pure

semiconductor, such as silicon or germanium.

Because the charge carriers in a semiconductor are often associated with impurity

atoms, the resistivity of these materials is very sensitive to the type and concentration of

such impurities.

24.2 Electrical energy and Power

If a battery is used to establish an electric current in a conductor, there is a continuous

transformation of chemical energy in the battery to kinetic energy of the electrons to internal

energy in the conductor,resulting in an increase in the temperature of the conductor.

In typical electric circuits, energy is transferred from a source such as a battery, to

some device, such as a lightbulb or a radio receiver. Let us determine an expression that will

allow us to calculate the rate of this energy transfer. First, consider the simple circuit in

Figure 24.9, where we imagine energy is being delivered to a resistor. (Resistors are

designated by the circuit symbol . ) Because the connecting wires also have

resistance, some energy is delivered to the wires and some energy to the resistor. Unless


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noted otherwise, we shall assume that the resistance of the wires is so small compared to the

resistance of the circuit element that we ignore the energy delivered to the wires

Imagine following a positive quantity of charge Q that is moving clockwise around the

circuit in Figure 24.9 from point a through the battery and resistor back to point a . We

identify the entire circuit as our system. As the charge moves from a to b through the

battery, the electric potential energy of the system increases by an amount Q V while the

chemical potential energy in the battery decreases by the same amount. However, as the

charge moves from c to d through the resistor, the system loses this electric potential energy

during collisions of electrons with atoms in the resistor. In this process, the energy is

transformed to internal energy corresponding to increased vibrational motion of the atoms in

the resistor. Because we

have neglected the resistance of the interconnecting wires, no energy transformation occurs

for paths bc and da. When the charge returns to point a , the net result is that some of the

chemical energy in the battery has been delivered to the resistor and resides in the resistor as

internal energy associated with molecular vibration.

Active Figure 24.9 A circuit consisting of a resistor ofresistance R and a battery having a potential difference V across

its terminals. Positive charge flows in the clockwise direction.

The resistor is normally in contact with air, so its increased temperature will result in a

transfer of energy by heat into the air. In addition, the resistor emits thermal radiation,

representing another means of escape for the energy. After some time interval has passed,

the resistor reaches a constant temperature, at which time the input of energy from the

battery is balanced by the output of energy by heat and radiation. Some electrical devices


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include heat sinks 4 connected to parts of the circuit to prevent these parts from reaching

dangerously high temperatures. These are pieces of metal with many fins. The high thermal

conductivity of the metal provides a rapid transfer of energy by heat away from the hot

component, while the large number of fins provides a large surface area in contact with the

air, so that energy can transfer by radiation and into the air by heat at a high rate.

Let us consider now the rate at which the system loses electric potential energy as the

charge Q passes through the resistor:

IV V t Q

t U

where I is the current in the circuit. The system regains this potential energy when the

charge passes through the battery, at the expense of chemical energy in the battery. 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. Thus, the power p,

representing the rate at which energy is delivered to the resistor, is

(24.14) IV P

We derived this result by considering a battery delivering energy to a resistor. However,

Equation 24.14 can be used to calculate the power delivered by a voltage source to any

device carrying a current I and having a potential difference V between its terminals.

Using Equation 24.14 and the fact that V = IR for a resistor, we can express the power

delivered to the resistor in the alternative forms

conductor ainloss power(24.15) 2

2

RV

R I P

A battery any other device that provides electrical energy is called a seat of

electromotive force, usually referred to as emf. (the phrase electromotive force is anunfortunate one, since it does not really describe a force but actually refers to a potential

diference between in volts). Neglecting the internal resistance of the battery, the potential

difference between points a and b is equal to the emf of the battery. That is,

ab V V V ,and the current in the circuit is given by R RV I . Since equals the

power lost in the resistor R I 2 .


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27.3 A Model for Electrical ConductionIn this section we describe a classical model of electrical conduction in metals that

was first proposed by Paul Drude (1863 1906) in 1900. This model leads to Ohms law and

shows that resistivity can be related to the motion of electrons in metals. Although the

Drude model described here does have limitations, it nevertheless introduces concepts that

are still applied in more elaborate treatments.

Consider a conductor as a regular array of atoms plus a collection of free electrons,

which are sometimes called conduction electrons. The conduction electrons, although bound

to their respective atoms when the atoms are not part of a solid, gain mobility when the free

atoms condense into a solid. In the absence of an electric field, the conduction electrons

move in random directions through the conductor with average speeds on the order of 106

m/s. The situation is similar to the motion of gas molecules confined in a vessel. In fact,some scientists refer to conduction electrons in a metal as an electron gas. There is no

current in the conductor in the absence of an electric field because the drift velocity of the

Example 24.4

An electric heater is constructed by applying a potential

difference of 110 V to a Nichrome wire that has a total

resistance of 8.00 . Find the current carried by the wire

and the power rating of the heater.

Solution since IRV ,we have

AV RV I 8.138

110

We can find the power rating using R I P 2

kW A R I P 15888.13 22

If we were to double the applied voltage, the current would

double but the power would quadruple.


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free electrons is zero. That is, on the average, just as many electrons move in one direction

as in the opposite direction, and so there is no net flow of charge.

This situation changes when an electric field is applied. Now, in addition to

undergoing the random motion just described, the free electrons drift slowly in a direction

opposite that of the electric field, with an average drift speed vd that is much smaller

(typically 10- 4 m/s) than their average speed between collisions (typically 10 6 m/s).

Figure 24.11 (a) A schematic diagram of the random motion of two charge carriers in a

conductor in the absence of an electric field. The drift velocity is zero. (b) The motion of the

charge carriers in a conductor in the presence of an electric field. Note that the random

motion is modified by the field, and the charge carriers

have a drift velocity.

Figure 24.11 provides a crude description of the motion of free electrons in a

conductor. In the absence of an electric field, there is no net displacement after many

collisions (Fig. 24.11a). An electric field E modifies the random motion and causes the

electrons to drift in a direction opposite that of E (Fig. 24.11b).

In our model, we assume that the motion of an electron after a collision is independent

of its motion before the collision. We also assume that the excess energy acquired by the

electrons in the electric field is lost to the atoms of the conductor when the electrons and

atoms collide. The energy given up to the atoms increases their vibrational energy, and this

causes the temperature of the conductor to increase. The temperature increase of a conductor

due to resistance is utilized in electric toasters and other familiar appliances.

We are now in a position to derive an expression for the drift velocity. When a free

electron of mass me and charge q (= - e) is subjected to an electric field E, it experiences a


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force F = qE. Because this force is related to the acceleration of the electron through

Newtons second law, F =mea, we conclude that the acceleration of the electron is

mqE

a

This acceleration, which occurs for only a short time interval between collisions, enables the

electron to acquire a small drift velocity. If 0v is the electrons initial velocity the instant

after a collision (which occurs at a time that we define as t = 0), then the velocity of the

electron at time t (at which the next collision occurs) is

t mqE

vat vv 00

We now take the average value of 0v over all possible collision times t and all possible

values of 0v . If we assume that the initial velocities are randomly distributed over all

possible values, we see that the average value of 0v is zero and we get

(24.16) t mqE

v

If the average time between collisions is , then drift velocity, we have

(24.17) mqE

vd

We can relate this expression for drift velocity to the current in the conductor. Substituting

Equation 24.17 into Equation 24.6, we find that the magnitude of the current density is

densitycurrent(24.18) 2

m

E nqnqv J d

where n is the number of charge carriers per unit volume. Comparing this expression with

Ohms law, E J ,we obtain the following relationships for conductivity and

resistivity of a conductor

restivity (24.20) 1

tyconductivi (24.19)

2

2

nq

mm

nq


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The average time interval between collisions is related to the average distance between

collisions l (that is, the mean free path ) and the average v speed through the expression

(24.21) 1v

According to this classical model, conductivity and resistivity do not depend on the strength

of the electric field. This feature is characteristic of a conductor obeying Ohms law .

Example 24.6

(A) Using the data and results from Example 24.1 and the classical model of

electron conduction, estimate the average time interval between collisions for

electrons in household copper wiring 20 0C.

Solution From Equation 24.20, we see that

2

nq

m

Where m x 8107.1 (m for copper and the carrier density is

3281084.18

melectrons xn for the wire described in Example 27.1.

Substitution of these values into the expression above gives

s xm xC m x

kg x 148219-328

31

105.2107.11.6x101048.8

1011.9

(B) Assuming that the average speed for free electrons in copper is 1.6 x10 6 m/s

and using the result from part (A),calculate the mean free path for electrons in

copper

m x s x sm xvr l 8146 100.4105.2106.1 which is equivalent to 40 nm (compared with atomic spacings of about 0.2 nm).

Thus, although the time interval between collisions is very short, an electron inthe wire travels about 200 atomic spacings between collisions .


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Thus Far Our Treatment of Electrical Phenomena Has Been Confined to the Study of Charges in...

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