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Chapter 16 Electric Field
Main Points of Chapter 16
• Electric field
• Superposition
• Electric dipole
• Electric field lines
• Field of a continuous distribution of charge
• Motion of a charge in a field
• Electric dipole in external electric field
16-1 Electric Field
1. Electric Field
Early：an “action-at-a-distance” force
Later: Faraday introduced “Electric Field ”
E field
Q1
Charge
Q2
Electric field
charge
2. Definition of Electric field
Field Point
Test charge
Size is small enough
Charge is very small

F
q0
Q
source charge
Test charge must be small enough that it does not affect field
The electric field is defined:
SI
N/C (or V/m)
Some values of Electric fields:
atmosphere at earth’s surface in clear weather
100-200 N/C
value sufficient to cause electrical breakdown in
dry air
For a point charge
Therefore, the electric field of a point charge is:
The field points outwards from a positive charge and
inwards to a negative one:
Comparison of Electric Force with Electric Field
• Electric Force (F) - the actual force felt by a charge
at some location.
• Electric Field (E) - found for a location only – tells
what the electric force would be if a charge were
located there:
A useful concept
3. Principle of Superposition of Electric Fields
Since electric forces add by superposition, the electric
field does as well.
For a group of point charges:
q1
q4
q0
q2
q3
the unit vector directed from the position of the kth
charge to the point where the field is evaluated
 Act What is the electric field at the origin due to
this set of charges?
y
Solution
a
+q
a
Notice that the fields from the topright and bottom left cancel at the
origin
a
+q
a
x
a
+q
4. Electric Dipoles and Their Fields
An electric dipole is defined as equal and
opposite charges a distance l apart:
electric dipole moment is defined as
and points from the negative charge towards
the positive one.
Electric dipoles are often found in nature. Charged
objects can induce electric dipoles (left); molecules may
have permanent electric dipole moments due to their
structure (right).
Example: Find the electric field of the electric dipole
at point a and point b.
b
Solution
r
O
a
x
Similarly, for point b,
b
r
At any point far from a dipole in any direction
ACT The electric field of a dipole at distance large
compared to the charge separation ________
A) decreases linearly with increasing distance.
B) remains constant as the distance increases.
C) decreases inversely with the cube of the distance.
D) decreases inversely with the square of the distance.
E) cannot be determined.
16-2 Electric Field Lines
Electric field lines are a useful aid to visualizing the
electric field. There are two rules to drawing these
lines:
1. The electric field is tangent to the field line at
every point.
2. The density of electric field lines is an indicator of
relative field strength.
The next slide shows field lines for a point charge,
including the decrease in density as one moves farther
from the charge.
Field lines of a
positive point
charge:
Left: two equal, same-sign charges
Right: an electric dipole
Important note: We always draw only a few sample
field lines; otherwise the sketch would be solid color.
Field Lines of two equal, same-sign charges
• There is a zero halfway between the two charges
• r >> a: looks like the field of point charge (+2q)
Electric field lines have certain properties which
should be carefully noted:
• Lines leave (+) charges and return to (-) charges,
never discontinue in empty space
• Tangent of line = direction of E
• Local density of field lines  local magnitude of E
• No two field lines ever cross, even when multiple
charges are present.
ACT What are the signs of the charges whose
electric fields are shown at right?
1)
2)
3)
4)
5) no way to tell
Electric field lines originate on positive
charges and terminate on negative charges.
Which of the charges has the greater magnitude?
The red one
16-3 The Field of a Continuous Distribution
To find the field of a continuous distribution of
charge, treat it as a collection of near-point charges:
Summing over the
infinitesimal fields:
Finally, making the charges infinitesimally small and
integrating rather than summing:
 : linear charge density
 : surface charge density
 : volume charge density
 Example An infinitely long wire is uniformly charged.
The charge density is  .Find the Electric field at point
P on the x-axis at x=x0
Solution
y
dy
r
P
x0
x
The Electric Field produced by an infinite
line of charge is:
- everywhere perpendicular to the line
- is proportional to the charge density
1
- decreases as
r
+
as seen from above
next lecture: Gauss’ Law makes this trivial!!
ACT A long line of charge with charge per unit
length λ1 is located on the x-axis and another long line
of charge with charge per unit length λ2 is located on
the y-axis with their centers crossing at the origin. In
what direction is the electric field at point z = a on the
positive z-axis if λ1 and λ2 are positive?
A) the positive z-direction
B) halfway between the x-direction and the ydirection
C) the negative z-direction
D) all directions are possible parallel to the x-y plane
E) cannot be determined
Act The figure here shows three nonconducting rods,
one circular and two straight. Each has a uniform
charge of magnitude Q along its top half and another
along its bottom half. For each rod, what is the
direction of the net electric field at point P?
 Example A uniformly charged circular arc has
radius R and subtends an angle 20.The total charge
is q. Calculate the electric field at point P .
Solution
y
dq

P
R
x
Act: Find the electric field at point P on the axis
of a uniformly charged ring of total charge q. The
radius of the ring is R, the distance from P to the center
of the ring is x.
x
Solution
P
r 
R
dq
 when x = 0（P is at
the center of the ring)
 when x>>R
O
 Act: Find the electric field at a distance x along the
axis of a uniformly charged circular disk of radius R
and charge Q
Solution
x
P
r
O
R
x
P
If x>>R
r
O
R
Field of a point charge
If R>>x
Field of an infinite uniformly charged plane
Field of an infinite uniformly charged plane
Divide the plane into narrow

straight strips
q1
x
q1
x
x
y
dy
x
as seen from above
From the electric field due to a uniform sheet of
charge, we can calculate what would happen if we put
two oppositely-charged sheets next to each other:
The individual fields:
The superposition:
The
result:
Summary of Electric Field Lines
Point Charge
~ 1/r2
Dipole
~ 1/r3
Infinite
Line of Charge
infinite uniformly
charged plane
~ 1/r
~ 1/r0
 Act A large flat has a uniform charge density  . A small
circular hole of radius R has been cut in the middle of the
surface. Calculate the electric field at point P(x)
Solution
Think that the configuration is composed
of one uniformly charged plane with
charge density + and a uniformly
charged circular disk -
R
x
Use superposition
.
P x
 Example: Find the electric field at point P on the
central axis of the solid cone. The total charge is q
Solution
z
P
r dz
H
z
R
O
x
y
16-4 Motion of a Charge in a Field
Deflection of Moving Charged Particles( electrons)
q
We can control the motion of a beam of charged particles
 ACT An electron beam moving horizontally at a
speed V enters a region between two horizontally
oriented plates of length L1. When the electrons reach
a fluorescent screen located at a distance L2 past these
plates, they have been deflected a vertical distance y
from their original direction. If the speed of the
electrons is doubled what is the new value of the
deflection?
A) y/4
B) 4y
C) 2y
D) y
E) y/2
 ACT A ring of negative, uniform charge density is
placed on the xz-plane with the center of the ring at the
origin. A positive charge moves along the y axis toward
the center of the ring. At the moment the charge passes
through the center of the ring ________
A) its velocity and its acceleration reach their
maximum values.
B) its velocity is maximum and its acceleration is zero.
C) its velocity and its acceleration have non-zero values
but neither is at its maximum.
D) its velocity and its acceleration are both equal to
zero.
E) its velocity is zero and its acceleration is maximum.
16-5 The Electric Dipole in an External Electric Field
1. The motion of a dipole in an uniform external
electric field

p

p
stable equilibrium

p

p
unstable equilibrium
A dipole in an uniform external electric field only rotates
2.The energy of a dipole in an external electric field
We choose
ACT The potential energies associated with four
orientations of an electric dipole in an electric field
are (1) -5U0, (2) -7U0, (3) 3U0, and (4) 5U0, where U0 is
positive. Rank the orientations according to (a) the
angle between the electric dipole moment
and the
electric field , and (b) the magnitude of the torque
on the electric dipole, greatest first.
(a) 4, 3, 1, 2
(b) 3, the 1 and 4 tie, then 2
 ACT An electric dipole of dipole moment
is
placed in an electric field
The dipole is then
slightly rotated a small angle θ away from its initial
direction and released. When released the dipole
will ________
A) stay in its new position, and angle θ away from
its
original position.

B) continue to rotate in the direction of θ until its
dipole moment is perpendicular to
C) oscillate back an forth around the new position, an
angle θ away from its original position.
D) continue to rotate until its dipole moment is parallel
to the
and then oscillate around that position.
E) go back to its original position.
i
3. The motion of a dipole in an nonuniform external
electric field
A dipole placed in an nonuniform external
electric field experiences not only a net force
but also a torque.
The motion is a combination of linear
acceleration and rotation
We can explain the phenomena that a charged rod
can attracts small pieces of paper.
_
+
-- -- -
 Act A neutral water molecule (H2O) in its vapor state
has an electric dipole moment of magnitude
(a) How far apart are the molecule's centers of
positive and negative charge
(b) If the molecule is placed in an electric field
of
, what maximum torque can the field
exert on it? (Such a field can easily be set up in the
laboratory.)
(c) How much work must an external agent do to
turn this molecule end for end in this field, starting
from its fully aligned position, for which  = 0?
Solution
Summary of Chapter 16
• Electric field is defined as the force per unit charge:
• Force on a point charge q’
• Electric field lines are very useful for visualizing the
electric field, as long as their limitations are taken into
account.
Summary of Chapter 16, cont.
• Electric field of a point charge:
• Electric fields obey the superposition principle.
• Electric dipole: equal and opposite charges separated by
a distance L. The electric field is proportional to the
dipole moment, which is:
Summary of Chapter16, cont.
• An electric dipole in an external electric field feels a
torque, and has potential energy:
• Electric field due to a continuous charge
distribution: