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Chapter 21 – Electric Charge and Electric Field
I.
Intro – bring together electricity and magnetism
A.
E & M considered independent until 1820 – Hans Christian Oersted
B.
Interrelationship formulated in 1865 by James Clerk Maxwell – Maxwell’s Equations
1.
Gauss’ Law for Electricity:


 E  dA 
2.
Q encl
o
Gauss’ Law for Magnetism:


 B  dA  0
3.
Faraday’s Law:

4.
 

E  dl  
t

 
B  dA
Ampere’s Law:

 
  
B  dl   o i c   o  o
E  dA
t
II.
Model of the Atom
A.
Size and Planetary Model
B.
What holds the atom together?

21-1
C
Characteristics of the Particles that make up the Atom
nucleons
D.
proton (p)
neutron (n)
electron (e)
mass, m(kg)
1.67 x 10-27
1.67 x 10-27
9.11 x 10-31
mass, m(u)
1 u = 1 atomic mass unit = 1.6606 x 10-27 kg
1.007276
1.008665
0.000549
mass relative to me
1835me
1837me
me
charge, q (Coulombs)
+1.60 x 10-19
0
-1.60 x 10-19
charge relative to fundamental unit of charge,
e = 1.60 x 10-19 C
+1e
0
-1e
Ionized atoms – ions
Electrically neutral atom:
amount of positive charge = amount of negative charge
number of protons = number of electrons
e.g., Oxygen (8):
Aluminum (13):
Positive ion: amount of positive charge > amount of negative charge
number of protons > number of electrons because the atom lost one or more electrons
e.g., Sodium (11):
Potassium (19):
Negative ion: amount of positive charge < amount of negative charge
number of protons < number of electrons because the atom gained one or more electrons
e.g., Chlorine (17):
E.
Iodine (53):
Observation of the types of interactions that occur between charges
1.
positive charge vs positive charge
2.
negative charge vs negative charge
3.
positive charge vs negative charge
Law of Electrostatics:
21-2
F.
G.
H.
Types of materials and their electrical properties;
1.
Conductors (Al, Ni, Cu, Ag, Au)
2.
Insulators (glass, rubber, plastic)
3.
Semiconductors (silicon, germanium, gallium)
How to Charge an object:
1.
charging by friction
2.
charging by direct transfer of charge
3.
charging by induction
Attractive force on an uncharged object
1.
conductor
2..
insulator
21-3
III. Coulomb’s Law – quantitative statement of the law of electrostatics
A.
Given two point charges with charges q1 and q2 , separated by a distance r:
q1
q2
r
The force is an inverse square law force, that is, F 
Fk
q 1q 2
r2
q 1q 2
r2
k = 9 x 109
N  m2
C2
=
1
4  o
 o = permittivity of free space = 8.85 x 10 -12
C2
N  m2
So, Coulomb’s Law can also be written as:
F
B.
1 q1q2
4 o r 2
Examples
1.
Two point charges, q1 = +2 C and q2 = -5 C, are placed 4 cm apart. Find the magnitude and
direction of the force q1 exerts on q2 , and find the same for the force q2 exerts on q1 .
21-4
2.
Three point charges are positioned as shown. Find the magnitude and direction of the force on
q3 . Take q1 = +2 C, q2 = -5 C, and q3 = -1 C.
q1
5 cm
q3
4 cm
q2
3.
Two point charges, q1 = +2 C and q2 = -5 C, are placed 4 cm apart. Find a point (or points
other than infinity) where the force on a third charge, q3, is zero.
21-5
4.
Two small identical steel spheres are placed so that their centers are 4 cm apart and are given
charges q1 = +2 C and q2 = -5 C.
a.
Digression: How many electrons were removed from the first steel sphere so that the
charge on the sphere was +2 C?
b.
Find the magnitude and direction of the force one sphere exerts on the other. If the spheres
were not small, then is the force different? Why?
c.
The spheres are momentarily placed in contact (or a wire is attached from one sphere to the
other). They are then separated again to a distance of 4 cm. What is the magnitude and
direction of the force on the spheres?
IV. The Electric Field
A.
The interaction between charges was analyzed as a charge exerting a force on another charge
(Coulomb’s Law). Another way to analyze the interaction is to say that one charge is interacting
with the “electric field” produced by the other charge.
That is, a charge placed at a particular position in space alters the space around itself. This alteration
of the space is referred to as its “field.” A second charge place in this field then interacts with the
field and feels a force as a result of being in the field.
B.
Quantify the idea of the electric field by using a test charge, +q, that is positive and placed at the
position at which you want to know the strength of the field.
Suppose we want to find the strength and direction of the electric field at a point P due to a
positively charged sphere, as shown.
P
+
21-6

 F
Operational Definition: E 
q
Units:
What is the difference if the charged sphere is negatively charged?
P
_
A more precise definition of the electric field:


F
E  lim
q 0 q
What happens if the test charge, q, is not small?
C.

If the charge q is of such a size that it does not affect the existing electric field E , then the force on the
charge is just


F  qE .
D.
Calculations of Electric Fields.
1.
E-field of a point charge, +Q
P
r
+Q
21-7
2.
E-field for many point charges: q1 , q2 , q3 , etc.
q1
r1
P
q2
r2
r3
q3
a.
Find the magnitude and direction of the electric field at point P.
q1 = +2 C
5 cm
P
4 cm
q2 = -5 C
a’.
Place a charge q3 = -1. C at point P. Find the magnitude and direction of the force on the
charge
b.
Find a position or positions where the electric field is zero, E = 0.
-4Q
+Q
d
21-8
b’.
Find the strength of the electric field in the three regions, and graph the strength of the
electric fields versus its position on the x-axis.
-4Q
+Q
d
21-9
c.
An electric dipole (like a water molecule) is placed at the origin. Find the magnitude and
direction of the electric field a distance x from the dipole. The magnitude of the charges on
the dipole is q and the distance between the charges is l.
+q
P
l
x
x
-q
d.
3.
What is the strength of the electric field a distance y from the origin on the y-axis? What
happens when y >> l? (Try this.)
The electric field due to charges that are distributed along a line, over an area, or throughout a
volume. (continuous charge distributions)
Ei
P
ri
qi
21-10
What is dq?
Charge density
a.
1-D
Linear charge
density, 
2-D
Surface charge
density, 
3-D
Volume charge
density, 
Diagram
Def
dq
=
dq
ds
=
dq
dA
=
dq
dV
ds
dq
dA
dq
dV
Units
dq
For constant
charge
density
C
m
dq = ds
=
Q
L
dq = dA
=
Q
A
dq = dV
=
Q
V
C
m
2
C
m
3
Find the x and y components of the electric field at point P a perpendicular distance a from
the wire. The coordinates of the ends of the wire are x1 and x2. The wire has a uniform
charge density   = constant).
y
P
a
x
x2
x1
21-11
What if the wire is infinitely long?
What if the wire is of length L and point P is on the perpendicular bisector of the wire?
What if P is on the perpendicular bisector and a >> L?
21-12
b.
A ring of radius R lies in the y-z plane and has a charge Q uniformly distributed around its
circumference. Find the magnitude and direction of the electric field at a distance x from
the center of the ring.
y
x
x
R
z
What do you think the electric field strength is when x >> R?
c.
A single layer of charge is uniformly distributed over a large, flat surface. The surface
charge density is   constant). Find the magnitude and direction of the electric field a
distance z above the surface of charge.
What kind of one-dimensional
z
charge distributions can I
combine together to get a large,
flat sheet of charge?
y
x
21-13
c’.
z
y
x
c’’. How about multivariable calculus?
z
y
x
21-14
d.
More sheets of charge (edge-on view of large, flat sheets). Remember that the magnitude

of the electric field for a large, flat sheet of charge is E 
 2k , and that its direction
2 o
is perpendicular to the sheet.
E-field for a layer of negative charge
E-field for two layers of positive charge
E-field for one positive and one negative layer
of charge
Flat sheet of conducting material with charge
placed on the sheet
e.
What is the strength of the electric field inside all conductors in the electrostatic case? Why?
21-15
V.
Electric Field Lines of Force
A.
Method of picturing what the electric field looks like
B.
C.
-
path taken by a small positive charge moved quasistatically
-
direction of motion of positive charge gives the direction of the field
Meaning of the lines of force
-
density of lines is proportional to the strength of the electric field
-
direction of lines gives the direction of the electric field
-
lines of force are perpendicular to the surface of conductors (why?)
-
E = 0 inside of conductors in the electrostatic situation (why?)
Examples
positive point charge
Two point charges – one positive and one
negative
+
+
very large, flat sheet of positively charged
21-16
-
two very large flat sheets oppositely charged
VI. Motion of Charge Particles in a Uniform Electric Field (E = constant)

Suppose a positive charge, q, is placed in a uniform electric field, E , directed in the vertically
upward direction. What are the forces and the accelerations of the charge in the horizontal and
vertical directions?
Example: An electron is fired with a velocity of 5 x 10 6 m/s into a uniform electric field that is
directed vertically downward. The strength of the electric field is 2000 N/C. How far does the
electron travel vertically as it travels 2 cm horizontally?
y
E = 2000
N/C
vo = 5 x 106 m/s
x
-
2 cm
21-17
VII. The Electric Dipole examined
A.
Electric dipole moment
+q
l

l
-q
B.
Torques on a dipole in a uniform electric field
C.
Potential Energy of a dipole in an electric field
21-18