Physics For Scientists And Engineers 6E - part 191

72.

A slab of insulating material has a nonuniform positive
charge density 1 # Cx

, where x is measured from the

center  of  the  slab  as  shown  in  Figure  P24.71,  and  C is  a
constant.  The  slab  is  infinite  in  the  y and  z directions.
Derive expressions for the electric field in (a) the exterior
regions  and  (b)  the  interior  region  of  the  slab
('d/2 , x , d/2).

73.

(a) Using the mathematical similarity between Coulomb’s
law and Newton’s law of universal gravitation, show that
Gauss’s law for gravitation can be written as

where m

is the net mass inside the gaussian surface and

g # F

/m represents the gravitational field at any point on

g"dA # '4(Gm

the gaussian surface. (b) Determine the gravitational field
at a distance r from the center of the Earth where r , R

assuming that the Earth’s mass density is uniform.

Answers to Quick Quizzes

24.1 (e). The same number of field lines pass through a

sphere of any size. Because points on the surface of the
sphere are closer to the charge, the field is stronger.

24.2 (d). All field lines that enter the container also leave the

container so that the total flux is zero, regardless of the
nature of the field or the container.

24.3 (b) and (d). Statement (a) is not necessarily true because

an equal number of positive and negative charges could
be present inside the surface. Statement (c) is not neces-
sarily  true,  as  can  be  seen  from  Figure  24.8:  a  nonzero
electric  field  exists  everywhere  on  the  surface,  but  the
charge  is  not  enclosed  within  the  surface;  thus,  the  net
flux is zero.

24.4 (c). The charges q

and q

are outside the surface and

contribute zero net flux through S*.

24.5 (d). We don’t need the surfaces to realize that any given

point in space will experience an electric field due to all
local source charges.

24.6 (a). Charges added to the metal cylinder by your brother

will reside on the outer surface of the conducting cylin-
der.  If  you  are  on  the  inside,  these  charges  cannot
transfer  to  you  from  the  inner  surface.  For  this  same
reason,  you  are  safe  in  a  metal  automobile  during  a
lightning storm.

Answers to Quick Quizzes

761

Figure P24.71 Problems 71 and 72.

Electric Potential

C H A P T E R O U T L I N E

25.1 Potential Difference and

Electric Potential

25.2 Potential Differences in a

Uniform Electric Field

25.3 Electric Potential and

Potential Energy Due to Point
Charges

25.4 Obtaining the Value of the

Electric Field from the
Electric Potential

25.5 Electric Potential Due to

Continuous Charge
Distributions

25.6 Electric Potential Due to a

Charged Conductor

25.7 The Millikan Oil-Drop

Experiment

25.8 Applications of Electrostatics

▲

Processes occurring during thunderstorms cause large differences in electric potential

between a thundercloud and the ground. The result of this potential difference is an electrical
discharge that we call lightning, such as this display over Tucson, Arizona. (© Keith Kent/
Photo Researchers, Inc.)

Chapter 25

762

763

he concept of potential energy was introduced in Chapter 8 in connection with such

conservative forces as the gravitational force and the elastic force exerted by a spring. By
using the law of conservation of energy, we were able to avoid working directly with
forces when solving various problems in mechanics. The concept of potential energy is
also of great value in the study of electricity. Because the electrostatic force is conserva-
tive, electrostatic phenomena can be conveniently described in terms of an electric
potential energy. This idea enables us to define a scalar quantity known as electric potential.
Because the electric potential at any point in an electric field is a scalar quantity, we can
use it to describe electrostatic phenomena more simply than if we were to rely only on
the electric field and electric forces. The concept of electric potential is of great practical
value in the operation of electric circuits and devices we will study in later chapters.

25.1 Potential Difference and Electric Potential

When a test charge q

is placed in an electric field

E created by some source charge

distribution, the electric force acting on the test charge is q

E. The force q

E is

conservative because the force between charges described by Coulomb’s law is conserv-
ative. When the test charge is moved in the field by some external agent, the work
done by the field on the charge is equal to the negative of the work done by the exter-
nal agent causing the displacement. This is analogous to the situation of lifting an
object with mass in a gravitational field—the work done by the external agent is mgh
and the work done by the gravitational force is !mgh.

When analyzing electric and magnetic fields, it is common practice to use the

notation d

s to represent an infinitesimal displacement vector that is oriented tangent

to a path through space. This path may be straight or curved, and an integral
performed along this path is called either a path integral or a line integral (the two terms
are synonymous).

For an infinitesimal displacement d

s of a charge, the work done by the electric

field on the charge is

F " ds # q

E " ds. As this amount of work is done by the field, the

potential energy of the charge–field system is changed by an amount dU # ! q

E " ds.

For a finite displacement of the charge from point A to point B, the change in poten-
tial energy of the system $U # U

(25.1)

The integration is performed along the path that q

follows as it moves from A to B.

Because the force q

E is conservative, this line integral does not depend on the

path taken from A to B.

For a given position of the test charge in the field, the charge–field system has a

potential energy U relative to the configuration of the system that is defined as U # 0.
Dividing the potential energy by the test charge gives a physical quantity that depends
only on the source charge distribution. The potential energy per unit charge U/q

U # !q

E"d

Change in electric potential

energy of a system

764

C H A P T E R 2 5 • Electric Potential

independent of the value of q

and has a value at every point in an electric field. This

quantity U/q

is called the

electric potential (or simply the potential) V. Thus, the

electric potential at any point in an electric field is

(25.2)

The fact that potential energy is a scalar quantity means that electric potential also is a
scalar quantity.

As described by Equation 25.1, if the test charge is moved between two positions A

and B in an electric field, the charge–field system experiences a change in potential
energy. The

potential difference $V # V

between two points A and B in an

electric field is defined as the change in potential energy of the system when a test
charge is moved between the points divided by the test charge q

(25.3)

Just as with potential energy, only differences in electric potential are meaningful. To

avoid having to work with potential differences, however, we often take the value of the
electric potential to be zero at some convenient point in an electric field.

Potential difference should not be confused with difference in potential energy.

The potential difference between A and B depends only on the source charge distribu-
tion (consider points A and B without the presence of the test charge), while the differ-
ence in potential energy exists only if a test charge is moved between the points.
Electric potential is a scalar characteristic of an electric field, independent of
any charges that may be placed in the field.

If an external agent moves a test charge from A to B without changing the kinetic

energy of the test charge, the agent performs work which changes the potential energy
of the system: W # $U. The test charge q

is used as a mental device to define the elec-

tric potential. Imagine an arbitrary charge q located in an electric field. From Equation
25.3, the work done by an external agent in moving a charge q through an electric
field at constant velocity is

(25.4)

Because electric potential is a measure of potential energy per unit charge, the SI

unit of both electric potential and potential difference is joules per coulomb, which is
defined as a

volt (V):

That is, 1 J of work must be done to move a 1-C charge through a potential difference
of 1 V.

Equation 25.3 shows that potential difference also has units of electric field times

distance. From this, it follows that the SI unit of electric field (N/C) can also be
expressed in volts per meter:

Therefore,

we can interpret the electric field as a measure of the rate of change

with position of the electric potential.

A unit of energy commonly used in atomic and nuclear physics is the

electron volt

(eV), which is defined as

the energy a charge–field system gains or loses when

a charge of magnitude e (that is, an electron or a proton) is moved through a
potential difference of 1 V. Because 1 V # 1 J/C and because the fundamental
charge is 1.60 % 10

C, the electron volt is related to the joule as follows:

(25.5)

1 e

V # 1.60 % 10

C"V # 1.60 % 10

1 V

" 1

W # q $V

# !

E"d

V #

▲

PITFALL PREVENTION

25.1 Potential and

Potential Energy

The  potential  is  characteristic  of
the field  only,  independent  of  a
charged  test  particle  that  may  be
placed in the field. Potential energy
is  characteristic  of  the  charge–field
system due  to  an  interaction
between  the  field  and  a  charged
particle placed in the field.

▲

PITFALL PREVENTION

25.2 Voltage

A  variety  of  phrases  are  used  to
describe  the  potential  difference
between  two  points,  the  most
common  being

voltage, arising

from  the  unit  for  potential.  A
voltage  applied to  a  device,  such
as a television, or across a device is
the  same  as  the  potential  differ-
ence  across  the  device.  If  we  say
that  the  voltage  applied  to  a
lightbulb  is  120  volts,  we  mean
that  the  potential  difference
between  the  two  electrical
contacts  on  the  lightbulb  is
120 volts.

Potential difference between

two points

The electron volt

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