
Electric Field (E) and Electric Potential
(V)
Having finished
our discussion of DC circuits (Chapters 25 and 26), we now return
to the study of the details regarding the force between two charges,
the force experienced by an electric charge when placed in an electric
field, and the calculation of the electric field that surrounds
an electric charge (Chapter 21 sections 3, 4, and 5).
We then cover
an easier and more elegant way to calculate the electric field using
Gauss's Law (Chapter 22) and then the important topic of Electric
Potential (Chapter 23). The electric potential (given in units of
Volts) is smoothes reefed to as the voltage, and we will see that
we can also easily calculate the electric field from the electric
potential.
KEY TERMS:
point charge, Coulomb's Law, coulomb, principle of superposition
of forces, electric field, test charge, vector field, source point,
field point.

Coulomb's
Law  Section 21.3
For a clear picture
of Coulomb's Law go to an informative website with interactive pictures
that will quicky show you pictures that will help give you a better idea
of what Coulomb's Law is all about:
http://230nsc1.phyastr.gsu.edu/hbase/hframe.html
then click on "Electricity
and Magnetism" button,
then "Coulomb's
Law" button and scroll down to a description of conventional electric
current that we will discuss later.
You can click anyplace
on the interactive page to get an explanations of the many new ideas in
this chapter. You will find this very helpful to have a picture of some
of these abstract and complicated concepts.

(a)
A torsion balance of the type used by Coulomb to measure the electric
force between two charges. (b) Electric charges of opposite sign
attract each other. The forces obey Newton's third Law:
F 1 on 2 =  F 2 on 1.

Fig.
21.10 Torsion balance used by Coulomb to measure the electric force
between two charges.




By
careful laboratory experiments, Coulomb in 1784 deduced that
the magnitude of the electric force between two charges is directly
proportional to the product of the charges and inversely proportional
to the square of the distance between them:
F
= k q_{1} + q_{2} / r^{2} or
_{ }
Here
the carefully measured proportionality constant is (as far as
we are concerned):
k = 1^{ } / (4p eo) = 9 (10)^{+9} N m^{2}/
C^{2}^{}
( eo is called the permittivity
of free space, and we will discuss this further in our study of
capacitors and dielectrics in Chapter 24.)
The
direction of the force (vector) is always directed along the
line joining the two charges, and when the charges have the
same sign the force is repulsive and when the signs are opposite
the force is attractive.






Fig.
21.11 Total force exerted on q3 by the other charges is vector sum
of two forces.




What
is the total force exerted on the charge q_{3}
by the other two point charges?
a)
The three charges. (b)Free body diagram for q3.
Vetor
addition of electric forces on a line.
q3 = 1.0 nC, q2 = 3.0 nC, q3
= 5.0 nC (n = nano = 10^{9})
F1 on 3 = 112 uN, F 2 on 3 = 84 uN, and the total force on
q3 is (84  112) uN toward the
left
(u = micro = 10^{6}).





Since
there is symmetry about the xaxis the vertical (yaxis) components
of the two forces cancel and we only need to calculate the horizontal
forces.
F 1 on q
= k q1 Q / r^{2} =
0.29 N
and the horizontal
(xcomponent) =
0.29 N cos a
=
0.29 N (0.40/0.50) = 0.23 N.
The total force on Q is 2(0.23 N) = 0.46 N to the left (+xaxis).
F
1
on q is the force on Q
due to the upper charge q_{1}.

Fig.
21.14 Net force on Q is vector sum of forces from other charges.





We
know that there exists a Coulomb force of repulsion between the
two charged bodies at A and B, (having charges Q and qo respectively) whose magnitude is:
F = k Q qo /
r^{2}
=
qo [k Q /
r^{2} ] where
we have factored out the small charge qo.
We can write the force in terms of an (electric) field
having units of force per unit charge, or Newtons per Coulomb, so
that: F = qo E. Therefore the electric field
E = [ k Q / r^{2} ]
which has units of N/C.
(This
is very similar to Newton's gravitational force F = G mM / r^{2}
= m [G M / r^{2}] = m g where the gravitational field, g,
has units of force per mass, or Newtons per kilogram, which equals
mass per seconds squared, an acceleration of 9.8 m/s^{2}
near the surface of the Earth.)

Fig.
21.16 A charged body creates an electric
field around it.





The
force F_{0} is
exerted on a charge q_{0} by an electric field
E.
(a) If q_{0} is positive, then F_{0} and E are in the same direction.
(b) If q_{0} is negative, then F_{0} and E are in opposite directions.

Fig.
21.17 The force exerted on a charge q_{0}
by an electric field E.





Electric
field at three points, a, b, c, set up by charges q_{1} and
q_{1}, which form an electric dipole.
The electric dipole's charges are q1 = +12 nC and q2 = 12 nC
Calculate E_{1}, E_{2}, and total field E_{T} at points
a, b, c.
At point
"a"
E_{1}=3.0 (10)^{4}
N/C,
E_{2}=6.8 (10)^{4}
N/C,
E_{T}=9.8 (10)^{4}
N/C, all in +x direction.
At point
"b"
E_{1}=6.8 (10)^{4}
N/C,
E_{2}=.55 (10)^{4}
N/C,
E_{T}=6.2 (10)^{4}
N/C
all in +x direction.
At point
"c"
E_{1}=6.4 (10)^{4}
N/C,
E_{2}=6.4 (10)^{4}
N/C,
E_{T}=4.9 (10)^{4}
N/C,
E_{T} is in the +x direction.

Fig.
21.23 Electric field at three points a, b, and c, caused by two
charges (dipole).




Calculating
the electric field on the axis of a ring of charge.
In this figure, the charge is assumed to be positive.
The
ring has a total charge of Q distributed uniformly around it. Find
the electric field at a point P that lies on the axis of the ring
at a distance x from its center.


The magnitude
of dE = k dQ/ r^{2}
and the xcomponent is
dEx = dE cos a = {k dQ / x^{2}
+ a^{2} ] {x / [x^{2} + a^{2}]^{1/2}
}.
From symmetry we see that all the y and z components of E
cancel and only the xcomponents do not cancel. Now get the total
xcomponent of the field by adding up the contribution of all the
dEx's (i.e.,
by integrating):
Ex = ò k
x dQ /
[x^{2} + a^{2}]^{3/2}
where x is constant as we go from point to point in the integration,
so
Ex = k x Q /
[x^{2} + a^{2}]^{1/2}
Fig. 21.24 Electric
field on the axis of a ring of charge.



Finding
the electric field at a point P on the perpendicular bisector of
a line of charge with length 2a and total charge Q. In this
figure, the charge is assumed to be positive.

Fig.
21.25 Electric field at P caused by a line of charge.





The
net force on this electric dipole is zero, but there is a torque
directed into the page that tends to rotate the dipole clockwise.

Fig.
21.29 The net force on the electric DIPOLE is zero, but there is
a torque into the page.






Fig.
21.47 Calculate the electric field at q caused by Q.




Problem
21.89: Positive charge Q is distributed uniformly along the positive
xaxis between x = 0 and x = a. A positive point charge +q lies
on the positive xaxis, a distance x = a + b, a distance b to the
right of the end of Q. (a) Calculate the x and ycomponents of
the electric field produced by the charge distribution Q
at points on the positive xaxis where x > a. (b) Then calculate
the force (magnitude and direction) that the charge distribution
Q exerts on q.
Note
the tabulated integral:
ò dz
/ (cz)^{2} = 1 / (cz)






Fig.
21.48 Calculate the electric field at q caused by Q.




Problem
21.90: Positive charge Q is distributed uniformly along the positive
yaxis between y = 0 and y = a. A negative point charge q lies
on the positive xaxis, a distance x from the origin. (a) Calculate
the x and ycomponents of the electric field produced by the charge
distribution Q at points on the positive xaxis. (b) Calculate
the x and ycomponents of the force that the charge distribution
Q exerts on q.
Tabulated
integral:
ò
dz / (z^{2}
+ a^{2})^{3/2} = z / [a^{2} (z^{2}
+ a^{2)½} ]








Now
that we know how to calculate the electric field from a distribution
of electric charges, we can look at another way to do the calculation
which is much more elegant and much more simple  the integrals
will be trivial to perform. The method is called GAUSS'S LAW
and is discussed in Chapter 22.
GAUSS'S LAW (Chapter 22)
Using
Gauss's Law to calculate the electric field caused by a line
of charge or other distribution of charges will be much easier
than dealing with the messy integration we used in the previous
chapter. In fact, the integrals will be so trivial we can literally
do them in our heads! Here's how we use Gauss's Law: Surround
a given distribution of electric charges with an imaginary surface
that totally surrounds the charge. Gauss's Law gives the relationship
between the electric field at all points on the imaginary surface
and the electric charge enclosed by the surface. In practice,
this is very easy to use to determine the electric field caused
by the charges inside the surface.
Check
the Summary and these "Key Terms" at the end of the chapter:
closed surface, electric flux, surface integral, Gauss's Law,
Gaussian surface.
Section 22.1 Charge and Electric Flux is worth reading slowly and
carefully. It gives the answer to the question "If the electric
field pattern is known in a given region, what can we determine
about the charge distribution in that region?" Knowing this
we will use the law to calculate (easily!) the electric field caused
by a given distribution of electric charges.
For a clear
picture of Gauss's Law go to an informative website with interactive
pictures that will quicky show you pictures that will help give
you a better idea of what Coulomb's Law is all about:
http://230nsc1.phyastr.gsu.edu/hbase/hframe.html
then click on
"Electricity and Magnetism" button,
then "Gauss's
Law" button and then Gauss's Law Applications." Check
out the several applications using the buttons at the bottom of
the page for calculations you may see on a test later.
You can click
anyplace on the interactive page to get an explanations of the many
new ideas in this chapter. You will find this very helpful to have
a picture of some of these abstract and complicated concepts.




(a) A box containing
an unknown amount of charge.
(b) The charge inside the box can be probed (or studied, or determined)
by using a test charge q_{0} to measure the electric field outside the box.


By
using a test charge to probe the electric field outside a box containing
electric charge one can map the field
(E
= F / q_{0}). In Figure 22.1(b) the electric field is the
same as the electric field caused by a point charge (see Fig. 22.15(a)
in the text). In fact, we only need to measure the field E
on the surface of the box.


Fig.
22.1 A charge inside a box can be probed using
a test charge qo.



The
electric field on the surface of boxes containing
(a) a single positive point charge,
(b) two positive point charges, (c) a single negative point charge,
(d) two negative point charges.

Fig.
22.2 The electric field on the surface of boxes containing charges.




The
Figure (22.2) shows that the pattern of the field on the surfaces
of the box enclosing the charges is different depending on what
charges are enclosed. For example, when the charges are positive
the field is outward, when the charges are negative the field is
inward, and it is clear that if there were no charges the field
on the surface of the box would be zero. If the amount of charge
inside the box increases, the field pattern on the surface of the
box changes accordingly.
The
electric field vectors surrounding electric charges are considered
to be a vector field, similar to the vector field of
the wind patterns we sometimes see in a weather forecast. We
can visualize the flow of the wind by looking at the wind velocity
(field) vectors. Although the electric field does not flow
in any physical sense, we can think of the electric field
vectors to be analogous to the velocity (flow) vectors on a
weather map, and we call the "flow" of the electric
field vectors the electric flux (from the Latin word
flux which means flow). This concept of electric
flux gives us a convenient way to visualize the electric field
vectors.
Looking
at Figure 22.2 we can now call the outward pointing electric field
vectors an outward electric flux through the box's surface, and
the inward pointing vectors an inward electric flux through the
box's surface. If we had one positive and one negative charge in
a box the net flux through the surface of the box would be zero
because the outward electric field vectors would cancel the inward
pointing vectors. The zero flux correlates with the zero net electric
charge in the box.
If
we had a charge outside the box it would contribute a
zero net flux through the box surfaces because every electric
field line from the charge that entered the box would have to
exit through the other side of the box. This is an important
result  we need only consider charges inside our imaginary
box when calculating the electric field vectors on the surface
of the box.
Since
the magnitude of an electric field vector is directly proportional
to the magnitude of the electric charge, the electric flux through
the box surfaces will be directly proportional to the magnitude
of the strength of the charge inside the box.
To
summarize:
 The sign
of the enclosed charge determines the direction (in or out)
of the electric flux through the surface of the box.
 Charges
outside the box do not contribute to the net electric flux
through the box.
 The net
electric flux through the box surface is proportional to the
magnitude of the electric charge inside the box.





The
volume (V) flow rate (dV/dt) of fluid through the wire rectangle
(a) is vA when the area of the rectangle is perpendicular to the
velocity vector v and (b) is VA cos f when
the rectangle is tilted at an angle f.
We
will next replace the the fluid velocity flow vector v with
the electric field vector E to get the concept of electric
flux F_{E}.

Fig.
22.5 Volume flow rate through the wire rectangle.





(a) The electric flux through the surface equals EA.
(b) When the area vector makes and angle f with
the vector E, the area projected onto a plane oriented perpendicular
to the flow is A cos f.
The flux is zero when f = 90
deg. because the rectangle lies in a plane parallel
to the flow and no fluid flows through the rectangle.

Fig.
22.6 A flat surface in a uniform electric field.




Let's
define an area vector A to have a magnitude equal
to the area of a surface and a direction perpendicular to
the area of the surface. Then we can write the term VA cos f
more concisely as a dot product of two vectors: v .
A = VA cos f.
If we now replace the fluid velocity flow vector v with the
electric field vector E we get an expression for the electric
flux: F_{E} = E . dA = VA
cosf.
If
the electric field vector E varies from place to place
over the area we simply divide the area into small pieces dA
which is a vector with magnitude dA and direction perpendicular
to the surface ( and direction outward if the surface is curved)
to get:
F_{E}
=ò E.dA = ò E
dA cos f
a
general definition of the electric flux. This is a surface integral
in which the value of E at each point on the surface is multiplied
by the surface area element dA at that point and integrated
(or "added").







Consider
Gauss's Law and the electric field caused by a point charge q.
(We know from last chapter that the magnitude of the field vector
is
E = (4pe_{o}) q /R^{^2}).
Place the positive charge q at the center of an imaginary spherical
surface whose radius is R.
F_{E}
= ò E.dA = ò E
dA cos f =
ò E
dA = E ò dA
=
E (4pR^2) = (4pe_{o}) q /R^{^2})(4pR^2) = q / e_{o}. So we have the
electric flux F_{E}
= q / e_{o}.
And if we have more than one charge inside the sphere we simply
add them up because each charge will contribute to the vector field
E. Therefore the total charge enclosed by the imaginary
sphere is Qencl = q_{1}+ q_{}_{2} + q_{}_{3}
+
...
Now we can write Gauss's Law:
F_{E}
= = ò E.dA = = ò E
dA cos f =
Qencl/ e_{o}

Fig.
22.9 Electric FLUX through a sphere centered on a point charge.





The
projection of an element of area dA of a sphere of radius R UP onto
a concentric sphere of radius 2R. The projection multiplies each
linear dimension by two, so the area element on the larger sphere
is 4 dA The same number of line of flux pass through each area element.

Fig.
22.11 Flux from a point charge q.





(a)
The outward normal to the surface makes an angle f
with the
direction
of E.
(b) The projection of the area element dA onto the spherical surface
is
dA cosf.

Fig.
22.12 Flux through an irregular surface.




Section 22.4 Applications of Gauss's Law


Fig.
22.17 Under electrostatic conditions, any excess charge resides
entirely on the surface of a solid conductor.




If
the conductor is isolated (not connected to a battery, etc.)
all motion of excess charges will stop after the charging process
has taken place. By excess charges we mean those charges other
than the ions and free electrons that the neutral conductor
is made of. So if the situation is electrostatic there can
be no electric field inside the conductor. If there were a
field it would exert a force on the charges and they would move.
If
we now draw our imaginary Gaussian surface completely inside
the conductor as shown in Figure 22.17 there can be no electric
field on the imaginary surface because it is inside the conductor.
By Gauss's Law we must have Qencl
= 0. We conclude that all the excess charges must reside on the
outer surface of the solid conductor.
This
is a very important result that we will use repeatedly in our calculations
of electric fields by Gauss's Law: There are no excess charges
in the interior of a solid conductor, i.e., Qencl = 0.
Use
the following recipe for Gauss's Law types of problems:
1.
Carefully draw a figure showing the location of all charges, the
direction of the resulting electric field vectors E, and
the dimensions given.
2.
Draw an imaginary closed Gaussian surface so that the value
of the magnitude of the electric field is constant on the
surface (or a portion of the surface) and the surface
contains the point at which you want to calculate the field.
This is very important because you will not be able to do the
integral if the magnitude of E cannot be taken outside
the integral in Gauss's Law.
3.
Write Gauss's Law and perform the dot product E . dA on
the vectors to get rid of the vectors (vector calculus is difficult).
4.
Since you drew the surface in such a way that the magnitude of the
E is constant on the surface, you can factor the E
out of the integral and then easily perform the integral.
5.
Determine the value of Qencl
from your figure and insert it into Gauss's equation.
6.
Solve the equation for the magnitude of E.





Under
electrostatic conditions the electric field inside a solid conducting
sphere is zero. Outside the sphere the electric field drops off
as 1 / r^2, as though all of the excess charge on the sphere were
concentrated at its center.

Fig.
22.18 Under electrostatic conditions the electric field inside a
solid conducting sphere is zero.




If
a conducting sphere of radius R has a positive charge +q, we can
calculate the field inside and outside the sphere by using Gauss's
Law.
1.
Inside the conducting sphere the electric field is zero.
(We have already shown this.)
2.
To calculate the field outside the sphere draw the figure showing
all the charges, indicate the direction of E at the point
(r) you want to calculate E, and let r > R.
3.
Do the dot product in Gauss's Law:
_{ }ò_{ }E.dA
= ò EdA
cos f =
Qencl
/e_{o}.
Evaluate cos f = cos 0 deg = 1
4.
Factor E out of the integral and integrate
E ò dA
= E (4p R^^{2}) = Qenc
/e_{o}).
5.
Insert Qencl = +q (see Gaussian surface) and solve for E to
get:
E = +q / e_{o}(4p R^^{2})






Fig.
22.19 A coaxial cylindrical Gaussian surface is used to find the
electric field outside an infinitely long charged wire.




Field
of a Line Charge 
Electric charge is distributed uniformly along an infinitely long,
thin wire. The charge per unit length is +l
(in Coulombs per meter). Find the electric field at a distance
r perpendicular to the wire from the center of the line (see figure).
1.
Draw the figure showing all the charges, indicate the direction
of E at the point (r) you want to calculate E, and let
r > R. The symmetry of the charge distribution indicates
that the resultant electric field vector will be in a plane
perpendicular to the axis and radially outward.
2.
The geometry of the charge distribution suggests that we draw
a cylindrical closed surface containing the point at r, the
distance from the charges. The direction of E will be
perpendicular to the curved portion of the cylinder and outward.
3.
Write the integral over the closed surface in three parts  an integral
over each flat end and an integral over the curved surface. Write
Gauss's Law with the three integrals and do the dot product
in each:
_{ }ò_{ } E1.dA1
+ ò_{ } E2.dA2
+ ò_{ } E3.dA3
= _{
}ò_{ } E1
dA1 cos f1 + ò_{ } E2
dA2 cos f2 + ò_{ } E3
dA3 cos f3 =
0 + 0 + _{
}ò
E3 dA3
The zeros result because the angle between the E vectors
and the area dA vectors of the end caps of the cylindrical
surface is 90 deg. and cos 90^{ }^{deg.}
= 0. The angle between the E vectors and the area vector
of the curved surface is 0^{ deg.} , and
cos 0^{ deg.} = 1. (Let's discuss this in class!)
4.
So Gauss's Law becomes ò_{ } E.dA
=ò_{} EdA
= E ò dA
=
E (2prl) = Qencl
/ e_{o}, where (2prl) is the area of the curved
Gaussian surface.
5.
Now evaluate by noting that the "linear charge density"
is l = charge / length or charge = l (length) or Qencl
= l l where l is the length of our cylinder.
6.
Solving for E we get E = l l / e_{o}
(2prl) and the l's cancel, so
E = l / e_{o}
(2pr).






Fig.
22.20 A cylindrical Gaussian surface is used to find the electric
field of an infinite plane sheet of charge.




Field
of an infinite plane sheet of charge  Find the electric field caused by a thin,
flat, infinite sheet on which there is a uniform positive charge
per unit area s.
The
nature of the charge distribution suggests that our Gaussian surface
will be a cylinder with its axis perpendicular to the plane of charge.
We need to write the integral in Gauss's Law in terms of three integrals
over the three parts of our closed cylindrical surface just as we
did in the problem above for the long charged wire. When we evaluate
the dot products in the three integrals we will find that one of
them goes to zero because the cos 90^{ deg.} = 0. When
we evaluate the term Qencl we use the definition of the "surface charge
density" s =
charge / unit area = Qencl
/ A
where A is the area of the end caps of our cylinder. We get Qencl
= sA and
the rest is easy. (Let's discuss this in class!) The answer
is E = s / 2e_{o}.





A
parallel plate capacitor.

Fig.
22.21 Electric field between two oppositely charged parallel plates.
The field between the plates is uniform except near the edges, which
are ignored.




There
is an electric field between oppositely charged parallel conducting
plates, i.e., a capacitor  the subject of Chapter 24.
Two
large plane parallel conducting plates are given charges of equal
magnitude and opposite sign; the charge per unit area is +s for one and s for the other. Find the electric field in the
region between the plates.
The
nature of the charge distribution suggests that our Gaussian surface
will be a cylinder with its axis perpendicular to the plane of charge.
We need to write the integral in Gauss's Law in terms of three integrals
over the three parts of our closed cylindrical surface just as we
did in the problem above for the infinite plane sheet of charge
density s.
We will wisely place one of the end caps of our Gaussian cylinder
inside the conducting plates. When we evaluate the surface integral
for this surface we can make use of the fact that we know the electric
field inside a conductor is zero and the integral will nicely go
to zero!
(Let's discuss this in class!)





The
field outside the sphere is the same as that for a solid conducting
sphere with the same amount of total charge but spread over its
surface. The field inside the sphere increases linearly with r,
i.e., the field gets larger as we go further away from the
center and our Gaussian spherical surface encloses more and more
charge. We need to characterize the charge distribution causing
a "volume charge density" r =
charge / unit volume.
We will omit this problem!

Fig.
22.22 The magnitude of the electric field of a uniformly charged
INSULATING sphere. Compare this with the earlier case for the charged
conducting sphere.




(a) Charge on
a solid conductor resides entirely on its outer surface.
(b) If there
is no charge inside the conductor's cavity, the net charge on the
surface of the CAVITY is zero.
(c) If there is a charge +q inside the cavity, the total charge
on the cavity surface is q.
The solution
of this problem lies in the fact that the field inside a conductor
is zero and if we place our Gaussian surface inside the conductor
(the field is zero), the charge enclosed must be zero, i.e.,
+q and q = 0 charge enclosed by the Gaussian surface.


Fig.
22.23 The electric charge on a solid conductor resides entirely
on its outer surface.





Sensitive
instruments are often enclosed in a "Faraday cage" to
shield them from unwanted radio frequencies (electromagnetic waves).
The field pushes electrons toward the left, leaving a net negative
charge on the left side and a net positive charge on the right side.
The result is that the net electric field inside the box is zero.
A Gaussian surface drawn inside the conducting material of which
the box is made must have zero electric field on it (field inside
a conductor is zero). If the Gaussian surface has zero field on
it, the charge enclosed must be zero according to Gauss's Law.

Fig.
22.28 A conducting box
(a Faraday cage) in an electric field.




To see where we are
in the course and where we are going, take a look at the website:
http://230nsc1.phyastr.gsu.edu/hbase/hframe.html
then click on the
"Electricity and Magnetism" button,
then "Maxwell's
Equations" button,
then "Integral
form with no magnetic media" button to see the four Maxwell's Equations.
The first is Gauss's Law which we have just convered, and next we will
cover Faraday's and Ampere's Laws.
But first we cover the important concept of Electric Potential.
ELECTRIC POTENTIAL (Chapter 23)
KEY
TERMS: electric potential energy, electric potential, volt,
voltage,
electron volt, equipotential surface, gradient, cathode ray tube.

For a clear picture
of Electric Potential go to an informative website with interactive pictures
that will quicky show you pictures that will help give you a better idea
of what Electric Potential (voltage) is all about:
http://230nsc1.phyastr.gsu.edu/hbase/hframe.html
then click on "Electricity
and Magnetism" button,
then "Electric
Potential Energy" in the
index on the righthand side of the page.

(a)
When a positive charge moves in the direction of an electric field,
the field does positive work and the potential energy decreases.
Work = qo E d
(b)
When
a positive charge moves in a direction opposite to an electric field,
the field does negative work and the potential energy increases.

Fig.
23.3 A POSITIVE charge moving in an electric field.





When
a NEGATIVE charge moves in the direction of an electric field, the
field does negative work and the potential energy increases.

Fig.
23.4 A NEGATIVE charge moving in an electric field.





The
charge qo moves along a straight line extending
radially from charge q. As it moves from a to b, the distance varies
from ra to rb.

Fig.
23.5 The charge qo moves along a straight line extending radially
from charge q.
As it moves from a to b,
the distance varies from ra to rb.





Since
cos f dl = dr the path the charge takes
does not affect the work done on it by the field. Since the work
is path independent, only the end points are needed to determine
the work done. So we can describe the work done in terms of the
prtential energy at th ened points. (It's the work  energy theorem.)

Fig.
23.6 The work done on charge qo by the electric field produced by
charge q depends only on the
distances ra and rb.





The
potential energy associated with the charge q0:
U = k q0 (q1/r1 + q2/r2 + q3/r3)

Fig.
23.8 The potential energy associated with a charge qo at point a
depends on charges q1, q2, and q3 and on their distances r1, r2,
and r3 from point a.





In
both cases (poinitive or negative piont charges), if you move in
the direction of E, electric potential V decreases; if you
move in the direction opposite E, V increases.

23.12
(a) A positive point charge. (b) A negative point charge.





Note:
The electric field is the negative derivative of the electric
potential. 
23.17
Electric field magnitude E and potential V at points inside and
outside a positively charged spherical conductor.





Find
the potential V
at point P
on the axis
of the ring.

Fig.
23.21 All the charge in the ring of charge Q is the same distance
r from a point P on the ring axis.





Find
the potential V
at point P a distance x away
from the line of charge.

Fig.
23.22 Finding the electric potential on the perpendicular bisector
of a
uniformly charged rod of length 2a.




Go
to Capacitors page next
© 2009 J. F. Becker


