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Transcript
OpenStax Staging4 module: m10495
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Electric Field Lines
∗
Words Numbers
This work is produced by OpenStax Staging4 and licensed under the
Creative Commons Attribution License 4.0†
Abstract
By the end of this section, you will be able to:
• Explain the purpose of an electric eld diagram
• Describe the relationship between a vector diagram and a eld line diagram
• Explain the rules for creating a eld diagram and why these rules make physical sense
• Sketch the eld of an arbitrary source charge
Now that we have some experience calculating electric elds, let's try to gain some insight into the
geometry of electric elds.
As mentioned earlier, our model is that the charge on an object (the source
charge) alters space in the region around it in such a way that when another charged object (the test charge)
eld
lines, and of electric eld line diagrams, enables us to visualize the way in which the space is altered, allowing
is placed in that region of space, that test charge experiences an electric force. The concept of electric
us to visualize the eld. The purpose of this section is to enable you to create sketches of this geometry, so
we will list the specic steps and rules involved in creating an accurate and useful sketch of an electric eld.
It is important to remember that electric elds are three-dimensional. Although in this book we include
some pseudo-three-dimensional images, several of the diagrams that you'll see (both here, and in subsequent
chapters) will be two-dimensional projections, or cross-sections. Always keep in mind that in fact, you're
looking at a three-dimensional phenomenon.
Our starting point is the physical fact that the electric eld of the source charge causes a test charge in
that eld to experience a force. By denition, electric eld vectors point in the same direction as the electric
force that a (hypothetical) positive test charge would experience, if placed in the eld (Figure 1)
∗ Version
1.5: Sep 13, 2016 3:03 am -0500
† http://creativecommons.org/licenses/by/4.0/
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Figure 1: The electric eld of a positive point charge. A large number of eld vectors are shown. Like
all vector arrows, the length of each vector is proportional to the magnitude of the eld at each point.
(a) Field in two dimensions; (b) eld in three dimensions.
We've plotted many eld vectors in the gure, which are distributed uniformly around the source charge.
Since the electric eld is a vector, the arrows that we draw correspond at every point in space to both the
magnitude and the direction of the eld at that point.
As always, the length of the arrow that we draw
corresponds to the magnitude of the eld vector at that point. For a point source charge, the length decreases
by the square of the distance from the source charge. In addition, the direction of the eld vector is radially
away from the source charge, because the direction of the electric eld is dened by the direction of the
force that a positive test charge would experience in that eld. (Again, keep in mind that the actual eld is
three-dimensional; there are also eld lines pointing out of and into the page.)
This diagram is correct, but it becomes less useful as the source charge distribution becomes more
complicated. For example, consider the vector eld diagram of a dipole (Figure 2).
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Figure 2: The vector eld of a dipole. Even with just two identical charges, the vector eld diagram
becomes dicult to understand.
There is a more useful way to present the same information. Rather than drawing a large number of
increasingly smaller vector arrows, we instead connect all of them together, forming continuous lines and
curves, as shown in Figure 3.
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Figure 3: (a) The electric eld line diagram of a positive point charge. (b) The eld line diagram of
a dipole. In both diagrams, the magnitude of the eld is indicated by the eld line density. The eld
vectors (not shown here) are everywhere tangent to the eld lines.
Although it may not be obvious at rst glance, these eld diagrams convey the same information about
the electric eld as do the vector diagrams.
First, the direction of the eld at every point is simply the
direction of the eld vector at that same point. In other words, at any point in space, the eld vector at
each point is tangent to the eld line at that same point. The arrowhead placed on a eld line indicates its
direction.
As for the magnitude of the eld, that is indicated by the
eld line densitythat is, the number of eld
lines per unit area passing through a small cross-sectional area perpendicular to the electric eld. This eld
line density is drawn to be proportional to the magnitude of the eld at that cross-section. As a result, if the
eld lines are close together (that is, the eld line density is greater), this indicates that the magnitude of the
eld is large at that point. If the eld lines are far apart at the cross-section, this indicates the magnitude
of the eld is small. Figure 4 shows the idea.
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Figure 4: Electric eld lines passing through imaginary areas. Since the number of lines passing
through each area is the same, but the areas themselves are dierent, the eld line density is dierent.
This indicates dierent magnitudes of the electric eld at these points.
S
In Figure 4, the same number of eld lines passes through both surfaces (
is larger than surface
the location of
S0,
S
0
.
and
S0,
but the surface
S
Therefore, the density of eld lines (number of lines per unit area) is larger at
indicating that the electric eld is stronger at the location of
S0
than at
S. The rules for
creating an electric eld diagram are as follows.
note:
1.Electric eld lines either originate on positive charges or come in from innity, and either
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terminate on negative charges or extend out to innity.
2.The number of eld lines originating or terminating at a charge is proportional to the magnitude of that charge. A charge of 2
q will have twice as many lines as a charge of q.
3.At every point in space, the eld vector at that point is tangent to the eld line at that same
point.
4.The eld line density at any point in space is proportional to (and therefore is representative
of ) the magnitude of the eld at that point in space.
5.Field lines can never cross. Since a eld line represents the direction of the eld at a given
point, if two eld lines crossed at some point, that would imply that the electric eld was
pointing in two dierent directions at a single point. This in turn would suggest that the (net)
force on a test charge placed at that point would point in two dierent directions. Since this
is obviously impossible, it follows that eld lines must never cross.
Always keep in mind that eld lines serve only as a convenient way to visualize the electric eld; they are
not physical entities. Although the direction and relative intensity of the electric eld can be deduced from a
set of eld lines, the lines can also be misleading. For example, the eld lines drawn to represent the electric
eld in a region must, by necessity, be discrete. However, the actual electric eld in that region exists at
every point in space.
Field lines for three groups of discrete charges are shown in Figure 5. Since the charges in parts (a) and
(b) have the same magnitude, the same number of eld lines are shown starting from or terminating on each
charge. In (c), however, we draw three times as many eld lines leaving the +3q charge as entering the
The eld lines that do not terminate at
−q
−q .
emanate outward from the charge conguration, to innity.
Figure 5: Three typical electric eld diagrams. (a) A dipole. (b) Two identical charges. (c) Two charges
with opposite signs and dierent magnitudes. Can you tell from the diagram which charge has the larger
magnitude?
The ability to construct an accurate electric eld diagram is an important, useful skill; it makes it much
easier to estimate, predict, and therefore calculate the electric eld of a source charge.
The best way to
develop this skill is with software that allows you to place source charges and then will draw the net eld
upon request. We strongly urge you to search the Internet for a program. Once you've found one you like,
run several simulations to get the essential ideas of eld diagram construction. Then practice drawing eld
diagrams, and checking your predictions with the computer-drawn diagrams.
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note:
One example of a eld-line drawing program
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1
is from the PhET Charges and Fields
simulation.
1 Summary
ˆ
ˆ
ˆ
Electric eld diagrams assist in visualizing the eld of a source charge.
The magnitude of the eld is proportional to the eld line density.
Field vectors are everywhere tangent to eld lines.
2 Conceptual Questions
Exercise 1
(Solution on p. 10.)
If a point charge is released from rest in a uniform electric eld, will it follow a eld line? Will it
do so if the electric eld is not uniform?
Exercise 2
Under what conditions, if any, will the trajectory of a charged particle not follow a eld line?
Exercise 3
(Solution on p. 10.)
How would you experimentally distinguish an electric eld from a gravitational eld?
Exercise 4
A representation of an electric eld shows 10 eld lines perpendicular to a square plate. How many
eld lines should pass perpendicularly through the plate to depict a eld with twice the magnitude?
Exercise 5
What is the ratio of the number of electric eld lines leaving a charge 10
(Solution on p. 10.)
q and a charge q?
3 Problems
Exercise 6
Which of the following electric eld lines are incorrect for point charges? Explain why.
1 https://openstax.org/l/21eldlindrapr
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Exercise 7
(Solution on p. 10.)
In this exercise, you will practice drawing electric eld lines. Make sure you represent both the
magnitude and direction of the electric eld adequately. Note that the number of lines into or out
of charges is proportional to the charges.
(a) Draw the electric eld lines map for two charges +20
−20 µC
µC
and
µC
and +20 µC situated 5 cm from
µC
and
situated 5 cm from
each other.
(b) Draw the electric eld lines map for two charges +20
each other.
(c) Draw the electric eld lines map for two charges +20
−30 µC
situated 5 cm from
each other.
Exercise 8
Draw the electric eld for a system of three particles of charges +1 µC,+2 µC, and
−3 µC
xed at
the corners of an equilateral triangle of side 2 cm.
Exercise 9
(Solution on p. 11.)
Two charges of equal magnitude but opposite sign make up an electric dipole.
A quadrupole
consists of two electric dipoles are placed anti-parallel at two edges of a square as shown.
Draw the electric eld of
the charge distribution.
Exercise 10
Suppose the electric eld of an isolated point charge decreased with distance as
as
1/r2 .
1/r2+δ
rather than
Show that it is then impossible to draw continous eld lines so that their number per unit
area is proportional to
E.
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Solutions to Exercises in this Module
Solution to Exercise (p. 7)
yes; no
Solution to Exercise (p. 7)
At the surface of Earth, the gravitational eld is always directed in toward Earth's center. An electric eld
could move a charged particle in a dierent direction than toward the center of Earth. This would indicate
an electric eld is present.
Solution to Exercise (p. 7)
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Solution to Exercise (p. 9)
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Solution to Exercise (p. 9)
Glossary
Denition 1: eld line
smooth, usually curved line that indicates the direction of the electric eld
Denition 2: eld line density
number of eld lines per square meter passing through an imaginary area; its purpose is to indicate
the eld strength at dierent points in space
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