21. Coulomb’s Law
• Quantifies the electric force between
two charges.
ba
ba
ba
ba
ba
ba
ba r
r
QQ
kr
r
QQ
F ˆˆ
4
1
22
0
229
212
/10988.8
/10854.8
CNmk
and
NmC
where
0
41. Exercise 1
A point charge q = -8.0 nC is located at the
origin. Find the electric field vector at the point
x = 1.2 m, y = -1.6 m
m2.1
-
m6.1mr 0.2
45. Exercise 2
An electric dipole consists of a positive
charge q and negative charge –q
separated by a distance 2a, as shown in
the figure below. Find the electric field due
to these charges along the axis at the
point P, which is the distance y from the
origin. Assume that y>>a.
79. Electrostatics
Electric charge
Conservation of charge
Insulators & conductors
Charging objects
Electroscopes
Lightning
Van de Graff generators
Equilibrium problems
Grounding
Static electricity
Coulomb’s law
Systems of charges
80. Electric Charge
• Just as most particles have an attribute known as mass,
many possess another attribute called charge. Charge and
mass are intrinsic properties, defining properties that particles
possess by their very nature.
• Unlike mass, there are two different kinds of charge: positive
and negative.
• Particles with a unlike charges attract, while those with like
charges repel.
• Most everyday objects are comprised of billions of charged,
but usually there are about the same number of positive
charges as negative, leaving the object as a whole neutral.
• A charged object is an object that has an excess of one type
of charge, e.g., more positive than negative. The amount of
excess charge is the charge we assign to that object.
81. Conservation of Charge
Charged particles can be transferred from one object to another, but
the total amount of charge is conserved. Experiments have shown
that whenever subatomic particles are transferred between objects or
interact to produce other subatomic particles, the total charge before
and after is the same (along with the total energy and momentum).
Example: An object with 5 excess units of positive charge and
another with 2 units of excess negative charge are released from rest
and attract each other. (By Newton’s 3rd law, the forces are equal
strength, opposite directions, but their accelerations depend on their
masses too.) Since there is no net force on the system, their center of
mass does not accelerate, and they collide there. As they “fall” toward
each other, electric potential energy is converted to kinetic energy.
When contact is made charge may be exchanged but they total
amount before and after must be the same. After the collision the total
momentum must still be zero.
+5 -2 +1.5 +1.5
Before After
Total charge: +3 Total charge: +3
82. Conservation of Charge: β-decay
• The stability of the nucleus of an atom depends on its size
and its proton-neutron ratio. This instability sometimes results
in a radioactive process known as β-decay.
• A neutron can turn into a proton, but in the process an
electron (beta particle) is ejected at high speed from the
nucleus to conserve charge.
• A proton can turn into a neutron. In this case the beta
particle is an positron (an antielectron: same mass as an
electron but a positive charge) to make up for the loss of
positive charge of the proton.
• In either case, charge, momentum, and energy are
conserved.
83. SI unit of Charge: the Coulomb
• Just as we have an SI unit for mass, the kilogram, we
have one for charge as well. It’s called the coulomb, and
its symbol is C.
• It’s named after a French physicist, Charles Coulomb,
who did research on charges in the mid and late 1700’s.
• A coulomb is a fairly large amount of charge, so
sometimes we measure small amounts of charge in μC
(mircocoloumbs).
• An electron has a charge of -1.6 10-19 C.
• A proton has a charge of +1.6 10-19 C.
• In a wire, if one coulomb of charge flows past a point in
84. Elementary Charge
• Charges come in small, discrete bundles. Another way to
say this is that charge is quantized. This means an object
can possess charge in incremental, rather than continuous,
amounts.
• Imagine the graph of a linear function buy when you zoom
in very close you see that it really is a step function with very
small steps.
• The smallest amount of charge that can be added or
removed from an object is the elementary charge, e = 1.6
10-19 C.
• The charge of a proton is +e, an electron -e.
• The charge of an object, Q, is always a multiple of this
elementary charge: Q = Ne, where N is an integer.
85. Insulators vs. Conductors
• A conductor is a material in which excess charge freely
flows. Metals are typically excellent conductors because the
valence (outer shell) electrons in metal atoms are not
confined to any one atom. Rather, they roam freely about a
metal object. Metal are excellent conductors of electricity (and
heat) for this reason.
• An insulator is a material in which excess charge, for the
most part, resides where it is deposited. That is, once placed,
it does not move. Most nonmetallic material are good
insulators. Valence electrons are much more tightly bound to
the atoms and are not free to roam about. Insulators are
useful for studying electrostatics (the study of charge that can
be localized and contained).
• Semi-conductors, like silicon used in computer chips, have
electrical conductivity between that of conductors and
Details on Conductors, Semiconductors, and Insulators
86. Electrons and Chemical Bonds
All chemical bonding is due to forces between electrostatic
charges.
Covalent bonding: A pair of electrons is shared between two
nonmetal atoms, allowing each atom to have access to enough
electrons to fill its outer shell. Except for hydrogen, this usually
means 8 electrons in the outer shell (octet rule).
Ionic bonding: One or more valence electrons of a metal atom
are “stolen” by a nonmetal atom, leaving a positive metal ion
and a negative nonmetal ion, which then attract one another.
Metallic bonding: Valence electrons of metals flow freely
throughout a metal object. These delocalized electrons are
attracted to the nuclei of the atoms through which they are
moving about. This produces a strong binding force that holds
the atoms together. In an iron bar, for example, there is no
covalent or ionic bonding. Metallic bonding hold the metal
87. Charging up Objects
Charging up an object does not mean creating new charges.
Charging implies either adding electrons to an object, removing
electrons from an object, or separating out positive and
negative charges within an object. This can be accomplish in 3
different ways:
• Friction: Rubbing two materials together can rub electrons off
of one and onto the other.
• Conduction: Touching an object to a charged object could
lead to a flow of charge between them.
• Induction: If a charged object is brought near (but not
touching) a second object, the charged object could attract or
repel electrons (depending on its charge) in the second object.
This yields a separation charge in the second object, an
induced charge separation.
88. Electroscopes
An electroscope is an apparatus comprised of a metal
sphere and very light metal leaves. A metal rod connects the
leaves to the sphere. The leaves are enclosed in an
insulating, transparent container. When the electroscope is
uncharged the leaves hang vertically. The scope is charged
by placing a charged rod near the sphere. The rod is charged
by friction. If a rubber rod is rubbed
with fur, electrons will be rubbed off the fur
and
onto the rubber rod, leaving the rod
negatively
Electroscopes
uncharge
charged. If a glass rod is rubbed with silk,
electrons will be rubbed off the rod onto the
silk, leaving the glass rod positively charged.
Either rod, if brought near, will charge the
scope by induction. Also, either rod, if contact
is made with the sphere, will charge the
scope by conduction.
continued…
89. Electroscopes (cont.)
+ + + + + + + + + + + + + +
When a positively charged rod is placed near but not touching
the metal sphere, some of the valence electrons in the metal
leaves are drawn up into the sphere, leaving the sphere
negatively charged and the leaves positively charged. Thus,
the rod has induced a chargeseparation in the scope. The light,
positive leaves repel each other
and separate. The electroscope as
a whole is still electrically neutral,
but it has undergone a charge
separation. As soon as the rod is
removed from the vicinity, the
charge separation will cease to
exist and the leaves the drop.
Note: Only the electron are
mobile; the positives on the leaves
represent missing electrons.
+
+
+ +
+
+
--- -
-
-
-
continued…
90. Electroscopes (cont.)
- - - - - - - - - - - - - - - - - - -
When a negatively charged rod is placed near but not touching
the metal sphere, some of the valence electrons in the sphere
are repelled down into the metal leaves, leaving the sphere
positively charged and the leaves negatively charged. The rod
has again induced a chargeseparation in the scope. The light,
negative leaves repel each other
as before. Again, the electroscope
as a whole is electrically neutral,
but the charge separation will
remain so long as the rod remains
nearby. Note that this situation is
indistinguishable from the situation
with the positive rod. Since the
effects are the same, how do we
know that the rods really do have
different charges?
-
-
- -
-
-
+ +
++
+ +
continued…
91. Electroscopes (cont.)
- - - - - - - - - - - - - - -
Now let’s touch the negative rod to the sphere. Some of the
electrons can actually hop onto the sphere and spread
throughout the scope. This is charging by conduction since,
instead of rearranging charges in the scope, new charges have
been added; the scope is no longer neutral. The extra electrons
force the leaves apart, even when the rod is removed. If the
negative rod returns, it charges the leaves further, but this time
by induction (by driving some ofelectrons on the sphere
down to the leaves). This
causes an increased
separation of the leaves.
When the rod is removed,
the scope will return to the
state on the left.
-
-
- -
-
-
- -
--
- -
Continued…
extra e- ’s added
-
-
- -
-
-
--
--
- -
leaf spread
92. Electroscopes (cont.)
The pic on the left shows a scope that has acquired extra
electrons from a negative rod that has since been removed.
Now we bring a positive rod nearby. This has the opposite
effect of bringing the negative rod near. This time some of the
extra electrons in the leaves head to the sphere and the spread
of the leaves diminishes. Note: the scope is still negatively
charged overall, but the presence of thepositive rod means more of
the excess negative charge
will reside in the sphere and
less in the leaves. When the
rod is removed, the scope
return to the state on the
left.
-
-
- -
-
-
- -
--
- -
Continued…
extra e- ’s added leaf spread
-
-
- -
-
-
- -
--
- -
+ + + + + + + + + +
93. Grounding an Electroscope
Whether a scope has charged by conduction, either positively
or negatively, the quickest way to “uncharge” it is by grounding
it. To do this we simply touch the sphere. When a negatively
charged scope is grounded by your hand, the excess electrons
from the scope travel into your body and, from there, into your
surroundings. When apositively charged scope is
grounded, electrons from
your body flow into the
scope until it is neutral. Your
surroundings will replace
the electrons you’ve
donated to the scope. As
always, it’s only the
electrons that move around.
-
-
- -
-
-
- -
--
- -
+
+
+ +
+
+
+ +
++
+ +
-
-
- -
-
-
94. Electroscope Practice Problem
For the following scenario, try to predict what would happen
after each step. Explain each in terms of electrons and
charging.
1. A rod is rubbed with a material that has a greater affinity
for electrons than the rod does.
2. This rod is brought near a neutral electroscope.
3. This rod touches the electroscope and is removed.
4. A positive rod is alternately brought near and removed.
5. A negative rod is alternately brought near and removed.
6. Finally, you touch the scope with your finger.
95. Redistributing Charge on Conducting
Spheres
A B
-Q- - -
B
Two neutral spheres, A & B, are placed side by side, touching. A
negatively charged rod is brought near A, which induces a charge
separation in the “A-B system.” Some of the valence e-’s in A migrate
to B. When the rod is re-moved and A & B are separated, A is +, B is -,
but the system is still neutral.
A
+Q
A is now brought near neutral sphere C, inducing a charge separation
on it. Valence e-’s in C migrate toward A, but since C is being touched
on the positive side, e-’s from the hand will move into C. Interestingly,
C retains a net negative charge after A and the hand are removed
even though no charged object ever made contact with it.
A
+Q
C C
-
96. Static Electricity: Shocks
If you walk around on carpeting in your stocking feet, especially
in the winter when the air is dry, and then touch something
metal, you may feel a shock. As you walk you can become
negatively charged by friction. When you make contact with a
metal door knob, you discharge rapidly into the metal and feel
a shock at the point of contact. A similar effect occurs in the
winter when you exit a car: if you slide out of your seat and
touch then touch the car door, you might feel a shock.
The reason the effect most often occurs in winter is because
the air is typically drier then. Humidity in the air can rather
quickly rob excess charges from a charged body, thereby
neutralizing it before a rapid, localized discharge (and resulting
shock) can take place.
Care must be taken to prevent static discharges where
sensitive electronics are in use or where volatile substances
97. Static Electricity: Balloons
Pic #1: If you rub a balloon on your hair,
electrons will be rubbed off your hair onto
the balloon (charging by friction).
Pic #2: If you then place the negatively
charged balloon near a neutral wall, the
balloon will repel some of the electrons
near it in the wall. This is inducing a
charge separation in the wall. Now the
wall, while still neutral, has a positive
charge near the balloon. Thus, the balloon
sticks to the wall.
Pick #3: Your hair now might stand up.
This is because it has been left positively
charged. As with the leaves of a charged
# 1
- -
-
-
- -
-
-
+-
+-
+-
+-
+-
+-
+-
+-
-+
-+
-+
-+
+-
+-
+-
+-
+-
+-
# 2
# 3
98. #1
#
2
#
3
You hang two balloons from the
ceiling and rub them on your hair.
When you move out of the way, the negatively charged
balloons repel each other. On each balloon there are three
forces: tension in the string, gravity, and the electric force.
The angle of separation will grow until equilibrium is
achieved (zero net force).
If you move your head close to
either of the balloons, it will move
toward you since your hair remains
positively charged.
Hanging Balloons
99. Polarization of a Cloud
Detailed Lightning
One mechanism incorporates friction: when moist, warm air rises, it cools
and water droplets form. These droplets collide with ice crystals and
water droplets in a cloud. Electrons are torn off the rising water droplets
by the ice crystals. The positive droplets rise to the top of the cloud, while
the negative ice crystals remain at the bottom.
A second mechanism involves the freezing process: experiments have
shown that when water vapor freezes the central ice crystal becomes
negatively charged, while the water surrounding it becomes positive. If
rising air tears the surrounding water from the ice, the cloud becomes
polarized.
There are other theories as well.
Lightning is the discharge of static electricity
on a
massive scale. Before a strike the bottom part
of a
cloud becomes negatively charged and the top
part
positively charged. The exact mechanism by
which this polarization (charge separation)
takes place is uncertain, but this is the
precursor to a lightning strike from cloud to
cloud or cloud to ground.
100. Lightning Strikes
The negative bottom part of the cloud
induces
a charge separation in the ground below. Air
is normally a very good insulator, but if the
charge separation is big enough, the air
between the cloud and ground can become
ionized (a plasma). This allows some of the
electrons in the cloud to begin to migrate
into the ionized air below. This is called a
“leader.” Positive ions from the ground
migrate up to meet the leader. This is called
a “streamer.” As soon as the leader and
streamer meet, a fully conductive path
exists between the cloud and ground and a
lightning strike occurs. Billions of trillions of
electrons flow into the ground in less than a
millisecond. The strike can be hotter than
the surface of the sun. The heat expands
the surrounding air; which then claps as
+
+ + + + +
+ + +
- - - - - - - -
-
+
+
+ +
+
++ + +
101. Lightning Rods and Grounding
Discovered by Ben Franklin, a lightning rod is a long, pointed,
metal pole attached to a building. It may seem crazy to attract
lightning close to a susceptible structure, but a lightning rod
can afford some protection. When positive charges
accumulate beneath a cloud, the accumulation is extremely
high near the tip of the rod. As a result, an electric field is
produced that is much greater surrounding the tip than around
the building. (We’ll study electric fields in the next unit.) This
strong electric field ionizes the air around the tip of the rod and
“encourages” a strike to occur there.
If a strike does occur, the electricity travels down the rod into a
copper cable that connects the lightning rod to a grounding rod
buried in the earth. There the excess charge is grounded, i.e.,
the electrons are dissipated throughout the landscape. By
taking this route, rather than through a building and its wiring,
102. A Van de Graaff generator consists of a large metal dome attached to a tube,
within which a long rubber belt is turning on rollers. As the belt turns friction
between it and the bottom roller cause the e-’s to move from the belt to the
roller. A metal brush then drains these e-’s away and grounds them. So, as the
belt passes the bottom roller it acquires a positive charge, which is transported
to the top of the device (inside the dome). Here another metal brush facilitates
the transfer of electrons from the dome to the belt, leaving the dome positively
charged.
In short, the belt transports electrons from a metal dome to the ground,
producing a very positively charged dome. No outside source of charge is
required, and the generator could even be powered by a hand crank. A person
touching the dome will have some of her e-’s drained out. So, her lightweight,
positive hair will repel itself. Coming close to the charge dome will produce
sparks when electrons jump from a person to the dome.
Van de Graaff
Generator
Internal workings Detailed explanation
103. Coulomb’s Law
K = 9 109 Nm2 /C2
Coulomb's Law DetailedCharges in Motion
F =
K q1 q2
r 2
There is an inverse square formula, called Coulomb’s law, for
finding the force on one point charge due to another:
This formula is just like Newton’s law of uniform gravitation with
charges replacing masses and K replacing G. It states that the
electric force on each of the point charges is directly proportional to
each charge and inversely proportional to the square of the distance
between them. The easiest way to use the formula to ignore signs
when entering charges, since we already know that like charges
repel and opposites attract. K is the constant of proportionality. Its
units serve to reduce all units on the right to nothing but newtons.
Forces are equal but opposite.
+ -
q1 q2
rF F
104. Electric Force vs. Gravitational Force
K = 9 109 N m2 / C2
FE =
K q1 q2
r 2
G = 6.67 10-11 N m2 / kg2
FG =
G m1 m2
r 2
Gravity is the dominant force when it comes to shaping galaxies
and the like, but notice that K is about 20 orders of magnitude
greater than G. Technically, they can’t be directly compared, since
they have different units. The point is, though, that a whole lot of
mass is required to produce a significant force, but a relatively
small amount of charge can overcome this, explaining how the
electric force on a balloon can easily match the balloon’s weight.
When dealing with high-charge, low-mass objects, such as protons
105. Electric Force Example
+ +15 μm
A proton and an electron are separated by 15 μm. They are released
from rest. Our goal is to find the acceleration each undergoes at the
instant of release.
1. Find the electric force on each particle.
2. Find the gravitational force on each particle. A proton’s
mass is 1.67 10-27 kg, and an electron’s mass is 9.11
10-31 kg.
3. Find the net force on each and round appropriately. Note
that the gravitational force is inconsequential here.
4. Find the acceleration on each particle.
5. Why couldn’t we use kinematics to find the time it would
take the particles to collide?
1.024 10-18 N
4.51 10-58 N
1.024 10-18 N
e-: 1.124 1012 m/s2, p+: 6.13 × 108
m/s2
r changes, so F changes, so a
changes.
106. System of 3 Charges
17
cm
14 cm
115
º
+3
μC
-5 μC
+2 μC
A
C
B
In a system of three point charges, each charge exerts a forces
on the other two. So, here we’ve got a vector net force problem.
Find the net force on charge B. Steps:
1. Find the distance in meters between A and B
using the law of cosines.
2. Find angle B in the triangle using the law of
sines.
3. Find FBA (the magnitude of the force on
charge
B due to charge A).
4. Find FBC.
5. Break up the forces on B into components
and find the net horiz. & vertical forces.
0.261947 m
36.027932 º
0.786981 N
4.591836 N
3.78 N (right) , 1.25 N (up)
3.98 N at
18.3 º N of E
107. System of 4 Charges
-16 µC
+9
µC
-7 µC
+25 µC
3
cm
4 cm
A
B
C
D
Here four fixed charges are arranged in a rectangle.
Find Fnet on charge D.
Solution:
Link
767.2 N at 59.6 º N of W
108. Hanging Charge Problem
q, m q, m
L L
mg
T
FE
Two objects of equal charge and mass are
hung from the same point on a ceiling
with equally long strings. They repel each
other forming an angle between the strings.
Find q as a function of m, L and .
Solution: Draw a f.b.d. on one of the
objects, break T into components, and
write net vertical and horiz. equations:
T sin( /2) = FE , T cos( /2) = mg.
Dividing equations and using Coulomb’s law yields:
mg tan( /2) = FE = Kq2 / r2, where r = 2Lsin( /2). Thus,
q = 4L2 mg tan( /2) sin2( /2)
K
109. Point of Equilibrium
+2q
d
x = ?
+q
Clearly, half way between two equal charges is a point of
equilibrium, P, as shown on the left. (This means there is zero
net force on any charge placed at P.) At no other point in space,
even points equidistant between the two charges, will
equilibrium occur.
Depicted on the right are two positive point charges, one with
twice the charge of the other, separated by a distance d. In this
case, P must be closer to q than 2q since in order for their
forces to be the same, we must be closer to the smaller charge.
Since Coulomb’s formula is nonlinear, we can’t assume that P is
twice as close to the smaller charge. We’ll call this distance x
and calculate it in terms of d.
+q+q P P
Continued…
110. Point of Equilibrium (cont.)
+2q
d
x
+q P
Since P is the equilibrium point, no
matter what charge is placed at P, there
should be zero electric on it. Thus an
arbitrary “test charge” q0 (any size any
sign) at P will feel a force due to q and
an equal force due to
2q. We compute each of these forces
via Coulomb’s law:
K q q0
x2
K (2q)q0
(d - x)2
=
The K’s, q’s, and q0’s cancel, the
latter showing that the location of P
is independent of the charge placed
there. Cross multiplying we obtain:
(d - x)2 = 2x2 d2 - 2xd + x2 = 2x2
x2 + 2xd - d2 = 0.
111. Point of Equilibrium (cont.)
From x2 + 2xd - d2 = 0,
the quadratic formula
yields:
+2q
d
x
+q P
x =
-2d (2d)2 - 4(1)(-d2)
2(1)
-2d 8d2
2
=
= -d d 2 Since x is a distance, we choose the positive
root:
x = d( 2 - 1) 0.41d. Note that x < 0.5d, as predicted.
Note that if the two charges had been the same, we would
have started with (d - x)2 = x2 d2 - 2xd + x2 = x2
d2 - 2xd = 0 d(d - 2x) = 0 x = d/2, as
predicted. This serves as a check on our reasoning.
112. Equilibrium with Several Charges
Several equal point charges are to be arranged in a plane so that another
point charge with non-negligible mass can be suspended above the plane.
How might this be done?
Answer: Arrange the charges in a circle, spaced evenly, and fix them in
place. Place another charge of the same sign above the center of the
circle. If placed at the right distance above the plane, the charge could
hover. This arrangement works because of symmetry. The electric force
vectors on the hovering charge are shown. Each vector is the same
magnitude and they lie in a cone. Each vector has a vertical component and
a component in the plane. The planar components cancel out, but the
vertical components add to negate
the weight vector. Continued…
113. Equilibrium with Several Charges (cont.)
Note that the charges in the plane are fixed. That is, they are attached
somehow in the plane. They could, for example, be attached to an
insulating ring, which is then set on a table. Regardless, how could the
arrangement of charges in the plane be modified so as to maintain
equilibrium of the hovering charge but allow it to hover at a different height?
Answer: If the charges in the plane are arranged in a circle with a large
radius, the electric force vectors would be more horizontal, thereby working
together less and canceling each other more. The hovering charge would
lower. Since its weight doesn’t change, it must be closer to the plane in
order to increase the forces to compensate for their partial cancellation. If
the charges in the plane were arranged in a small circle, the vectors would
be more vertical, thereby working together more and canceling each other
less. The hovering charge would rise and the vectors would decrease in
magnitude. To maximize the height of the hovering charge, all the charges
in the plane should be brought to a single point. Continued…
