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Physics For Scientists And Engineers 6E - part 209

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:

(27.2)

The SI unit of current is the

ampere (A):

(27.3)

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

The charges passing through the surface in Figure 27.1 can be positive or negative,

or both.

It is conventional to assign to the current the same 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.

If the ends of a conducting wire are connected to form a loop, all points on the

loop are at the same electric potential, and hence the electric field is zero within and
at the surface of the conductor. Because the electric field is zero, there is no net
transport of charge through the wire, and therefore there is no current. However, if
the ends of the conducting wire are connected to a battery, all points on the loop are
not at the same potential. The battery sets up a potential difference between the ends
of the loop, creating an electric field within the wire. The electric field exerts forces on
the conduction electrons in the wire, causing them to move in the wire, thus creating a
current.

It is common to refer to a moving charge (positive or negative) as a mobile

charge

carrier. For example, the mobile charge carriers in a metal are electrons.

Microscopic Model of Current

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. 27.2). The volume of a section of the conductor of length !x (the gray
region shown in Fig. 27.2) is A !x. If n represents the number of mobile charge
carriers per unit volume (in other words, the charge carrier density), the number of
carriers in the gray section is nA !x. Therefore, the total charge !Q in this section is

Q " number of carriers in section # charge per carrier " (nA !x)q

where q is the charge on each carrier. If the carriers move with a speed v

, the displace-

ment they experience in the x direction in a time interval !t is !x " v

t. 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

Q " (nAv

t)q

If we divide both sides of this equation by !t, we see that the average current in the
conductor is

(27.4)

∆Q

∆t

nqv

1 A "

1 C

1 s

SECTION 27.1 • Electric Current

833

▲

PITFALL PREVENTION

27.1 “Current Flow” Is

Redundant

The  phrase  current  flow is  com-
monly used, although it is strictly
incorrect,  because  current  is a
flow  (of  charge).  This  is  similar
to  the  phrase  heat  transfer,  which
is also redundant because heat is
a  transfer  (of  energy).  We  will
avoid  this  phrase  and  speak  of
flow of charge or charge flow.

Electric current

∆x

∆t

Figure 27.2 A section of a uniform

conductor of cross-sectional area A.

The mobile charge carriers move

with a speed v

, and the displace-

ment they experience in the x

direction in a time interval !t is
!

x " v

t. 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 nAv

where n is the number of carriers

per unit volume.

Current in a conductor in terms

of microscopic quantities

The speed of the charge carriers v

is an average speed called the

drift speed. To

understand the meaning of drift speed, consider a conductor in which the charge car-
riers 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 (Fig. 27.3). Despite the collisions, the electrons move slowly along the conduc-
tor (in a direction opposite that of

E) at the drift velocity v

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.

834

CHAPTE R 27 • Current and Resistance

Figure 27.3 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 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.

Quick Quiz 27.1

Consider positive and negative charges moving horizon-

tally through the four regions shown in Figure 27.4. Rank the current in these four
regions, from lowest to highest.

Quick Quiz 27.2

Electric charge is conserved. As a consequence, when current

arrives at a junction of wires, the charges can take either of two paths out of the junction
and the numerical sum of the currents in the two paths equals the current that entered
the junction. Thus, current is (a) a vector (b) a scalar (c) neither a vector nor a scalar.

–

Figure 27.4 (Quick Quiz 27.1) Charges move through four regions.

(a)

–

(b)

(c)

(d)

Example 27.1 Drift Speed in a Copper Wire

The 12-gauge copper wire in a typical residential building
has a cross-sectional area of 3.31 # 10

. If it carries a

current of 10.0 A, what is the drift speed of the electrons?
Assume that each copper atom contributes one free
electron to the current. The density of copper is 8.95 g/cm

Solution From  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
Avogadro’s  number  of  atoms  (6.02 # 10

). Knowing the

density of copper, we can calculate the volume occupied by
63.5 g (" 1 mol) of copper:

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

V "

63.5 g

8.95 g/cm

7.09 cm

From Equation 27.4, we find that the drift speed is

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

2.22 # 10

m/s

10.0 C/s

(8.49 # 10

)(1.60 # 10

C)(3.31 # 10

)

nqA

" 8.49 # 10

electrons/m

n "

6.02 # 10

electrons

7.09 cm

1.00 # 10

1 m

Example 27.1 shows that typical drift speeds are very low. For instance, electrons

traveling with a speed of 2.22 # 10

m/s would take about 75 min to travel 1 m! In

view of this, you might wonder why a light turns on almost instantaneously when a
switch  is  thrown.  In  a  conductor,  changes  in  the  electric  field  that  drives  the  free
electrons  travel  through  the  conductor  with  a  speed  close  to  that  of  light.  Thus,
when  you flip  on  a  light  switch,  electrons  already  in  the  filament  of  the  lightbulb
experience  electric  forces  and  begin  moving  after  a  time  interval  on  the  order  of
nanoseconds.

27.2 Resistance

In Chapter 24 we found that the electric field inside a conductor is zero. However, this
statement is true only if the conductor is in static equilibrium. The purpose of this
section is to describe what happens when the charges in the conductor are not in equi-
librium, in which case there is an electric field in the conductor.

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
I " nqv

A, the current density is

(27.5)

where J has SI units of A/m

. This expression is valid only if the current density is uni-

form 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:

(27.6)

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:

J " &E

(27.7)

where the constant of proportionality & is called the

conductivity of the conductor.

Materials that obey Equation 27.7 are said to follow

Ohm’s law, named after Georg

Simon Ohm (1789–1854). More specifically, Ohm’s law states that

J " nq

nqv

SECTION 27.2 • Resistance

835

▲

PITFALL PREVENTION

27.2 Electrons Are

Available Everywhere

Electrons  do  not  have  to  travel  from
the  light  switch  to  the  light  in  order
for  the  light  to  operate.  Electrons
already  in  the  filament  of  the
lightbulb move in response to the
electric field set up by the battery.
Notice  also  that  a  battery  does
not  provide  electrons  to  the
circuit.  It  establishes  the  electric
field  that  exerts  a  force  on
electrons already in the wires and
elements of the circuit.

Current density

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.

Materials that obey Ohm’s 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 Ohm’s law are said
to be nonohmic. Ohm’s law is not a fundamental law of nature but rather an empirical
relationship valid only for certain materials.

We can obtain an equation useful in practical applications by considering a

segment of straight wire of uniform cross-sectional area A and length !, as shown in

Georg Simon Ohm

German physicist (1789–1854)

Ohm, a high school teacher and

later a professor at the University

of Munich, formulated the

concept of resistance and

discovered the proportionalities

expressed in Equations 27.7 and

Do not confuse conductivity & with surface charge density, for which the same symbol is used.

Figure 27.5. A potential difference !V " V

is maintained across the wire, creat-

ing in the wire an electric field and a current. If the field is assumed to be uniform, the
potential difference is related to the field through the relationship

V " E!

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

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

The quantity R " !/&A 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:

(27.8)

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 ('):

(27.9)

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

(27.10)

where % has the units ohm-meters (' ( m). Because R " !/&A, we can express the
resistance of a uniform block of material along the length ! as

(27.11)

R " %

% "

1 '

1 V
1 A

∆V

∆V "

J "

I " R

J " &

E " &

∆V

836

CHAPTE R 27 • Current and Resistance

Resistivity is the inverse of

conductivity

Resistance of a uniform

material along the length !

Figure 27.5 A uniform conductor of length ! and cross-sectional area A. A potential

difference !V " V

maintained across the conductor sets up an electric field E,

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

This result follows from the definition of potential difference:

Do not confuse resistivity % with mass density or charge density, for which the same symbol is used.

" $

E(d

s " E

dx " E

▲

PITFALL PREVENTION

27.3 We’ve Seen

Something Like
Equation 27.8 Before

In  Chapter  5,  we  introduced
Newton’s  second  law,  )F " ma,
for  a  net  force  on  an  object  of
mass m. This can be written as

In  that  chapter,  we  defined  mass
as resistance to a change in motion in
response to an external force. Mass as
resistance to changes in motion is
analogous  to  electrical  resistance
to charge flow, and Equation 27.8
is  analogous  to  the  form  of
Newton’s second law shown here.

m "

▲

PITFALL PREVENTION

27.4 Equation 27.8 Is Not

Ohm’s Law

Many  individuals  call  Equation
27.8 Ohm’s law, but this is incor-
rect.  This  equation  is  simply  the
definition of resistance, and pro-
vides  an  important  relationship
between  voltage,  current,  and
resistance.  Ohm’s  law  is  related
to a linear relationship between J
and E (Eq. 27.7) or, equivalently,
between  I and  !V,  which,  from
Equation  27.8,  indicates  that  the
resistance  is  constant,  indepen-
dent of the applied voltage.

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