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12/1/2016
AP PHYSICS 2
UNIT 3
Electrostatics:
electric force,
electric field, and
electric potential.
CHAPTER 14
ELECTRIC CHARGE,
FORCE, AND
ENERGY
EXPERIMENTS
The earliest account of
electric phenomena comes
from the Ancient Greeks.
In the 1700’s, an American scientist conducted
extensive research in these phenomena.
He sold many of his possessions to fund his
work.
They observed that
after amber tools were
cleaned with animal
fur, they would attract The word electricity comes
from the Greek word for
small bits of dust and
amber – electron.
feathers.
EXPERIMENT 1 – COMB AND BITS OF
PAPER
• Everyday observations show that uncharged lightweight
objects, such as small bits of paper that have not been
rubbed against anything, are attracted to charged objects.
EXPERIMENT 2 – TRANSPARENT TAPE
• Take two 9-inch-long pieces of transparent tape and
place them sticky side down on a plastic, glass, or
wooden tabletop. Now pull on one end of each tape to
remove them from the table. Bring the pieces of tape
near each other. They repel each other, as shown in the
figure. Can we explain the repulsion of the tapes as an
electric interaction?
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EXPERIMENT 3 – BALLOONS
If you rub two
balloons the same
way with a wool cloth,
Just like the experiments in class, Benjamin Franklin
recorded that when an amber rod was rubbed with fur, the
amber would attract the fur that it was rubbed with.
Is the electrical interaction just a magnetic
interaction?
However, he also discovered that two amber rods rubbed with
fur would repel one another.
• Perhaps the electric interaction is
He was able to achieve similar results with a glass rod rubbed
with silk, but with one major difference:
actually the magnetic interaction, just
they will attract the
The rubbed glass rod and
the rubbed amber rod repel
one another too !
wool but repel each
other.
EXPERIMENTS WITH MAGNETS
Ben Franklin’s Conclusions
described using different terminology.
Ben Franklin’s Conclusions
• The property acquired by materials due to
rubbing is called electric charge.
• Objects acquire positive or negative electric
charge (He could have named them anything else – the
names are arbitrary)
• Electrostatics is not the same phenomenon as
magnetism!
• These objects did not have two poles!
Charged rod
attracted south pole
Charged rod
attracted north pole
• Objects that interact with each other because
they are charged exert electric forces on each
other.
• When charged objects are at rest, the force is an
electrostatic force.
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Ben Franklin’s Conclusions
• Materials rubbed against each other acquire
electric charge.
• Two objects with the same type of charge repel
each other.
• Two objects with opposite types of charge attract
each other.
Charged objects attract uncharged objects
• Uncharged objects are attracted to
WHITEBOARD
• You pull your sweater and shirt off together, and then
pull them apart. You notice that they attract each other—
a phenomenon called "static" in everyday life. Explain
the mechanism behind this attraction and suggest an
experiment to test your explanation.
both positively and negatively charged
objects; uncharged metal objects are
attracted more strongly than nonmetal
• Two objects made of different materials rubbed
against each other acquire opposite charges.
HISTORICAL MODELS FOR ELECTRIC CHARGE
• Fluid models of electric charge
• Too much electric fluid in an object makes it
positively charged and too little makes it
negatively charged.
• Particle model of electric charge
• Electric charge is carried by microscopic
particles. Removing or adding these particles
to an object changes the charge of the object.
objects.
CONTEMPORARY MODEL FOR ELECTRIC CHARGE
• Two objects start as neutral—the
total electric charge of each is
zero.
• During rubbing, one object gains
electrons
and
becomes
negatively charged.
• The other object loses an equal
number of electrons and with this
deficiency of electrons becomes
positively charged.
What exactly is electric
charge, anyway??
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Early Theory of Electric Charge
Physicists like Benjamin Franklin and Charles Coulomb
believed that electric charge was a fluid. They
theorized that when two objects are rubbed together,
the fluid travels.
Their theory also said that objects that have lots of the
fluid are positively charged, and objects that have a
lack of the fluid are negatively charged.
Coulomb trying to
capture electric
fluid in a jar
(unsuccessfully)
Elementary particles have two key components
WHAT ACTUALLY IS ELECTRIC CHARGE?
Electric charge is a property of subatomic
particles (protons and electrons) that
cause them to feel a force when other
protons or electrons are present.
Since all matter is composed of protons,
electrons and neutrons, it is possible for any
object to become positively or negatively
charged!
(but very difficult to accomplish for certain
materials)
2) Electric Charge
Mass gives particles two fundamental properties.
Electric charge gives particles one fundamental
property.
b) Mass causes particles to attract all other particles that
have mass
This is the gravitational property of matter.
Protons and neutrons have roughly the same mass
as one another, and they have about 10,000 times
as much mass as electrons.
If an object is negatively charged, it
means that it has excess electrons.
If an object is positively charged, it
means that it has a lack of electrons.
If an object has an equal number of
protons as electrons, then it is
electrically neutral.
PHYSICAL QUANTITY:
ELECTRIC CHARGE
Elementary particles have two key components
1) Mass
a) Mass causes particles to resist being accelerated
This is called the property of inertia.
CHARGED OBJECTS
a) Electric charge causes protons and electrons
to create an electric field
This means that all other charged particles in their
vicinity will feel an electric force
Protons and electrons exactly opposite charges to one
another.
(This shows the complete independence of mass and charge)
Neutrons have an electric charge of 0.
• Symbol
• Units
q
Coulomb
• Units symbol
C
Electric charges are
small and they are
usually written using
prefixes of the metric
system
q = +3C
q = +3x10-6 C
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Protons and electrons are the fundamental building
block of all charged objects.
FUNDAMENTAL CHARGE
• Electron:
• Proton:
q = 1.6×10−19 C
q=
1.6×10−19
This is called the elementary charge, e
(e = 1.6 x 10-19 C)
C
• Ohhhhh yesss !!! electrons and protons have
the same magnitude of electric charge !!
• To differentiate the charge of an electron from
the charge of a proton we use the NEGATIVE
SIGN for the CHARGE OF AN ELECTRON.
q=
-1.6×10−19
No object could ever exist in nature that has a charge
of 1 x 10-20 C
The most physics-y way of stating this concept is that
electric charge is quantized.
Electric charge comes in indivisible “packages” of
magnitude e. Any charged object in nature must have
a charge that is a direct multiple of e.
C
+
+
+
--
• Metals
have
free
electrons that are free to
move inside.
+
+
+
+
+
+
+
+
+
+
+- +- ++- +- ++- +- +-
• Plastic, glass, and other
nonmetal materials do not
have free electrons or any
other charged particles that
are free to move inside.
• Both materials are composed of oppositely
charged particles with a total charge of zero.
• In conductors, some of
can move freely.
-
+- +- ++- +- ++- +- +-
Neutral atoms with charge
uniformly distributed.
Polarization of dielectrics
- +
-+
-
+- +- + -
When the positively charged
object is brought near the
dielectric material, its atoms
and molecules become
polarized.
A similar thing happens if
we bring a negatively
charged object near a nonconducting material.
the charged particles
• In dielectrics, charges can only be redistributed
slightly (set off), a process called polarization..
POLARIZATION OF A DIELECTRIC
Copper
wire
- - - - -
• Materials can be divided into two groups:
conductors and insulators (dielectrics).
Why?
DIELECTRICS vs. CONDUCTORS
Latex
Balloon
Electric properties of materials
-
++
++
++
+
+
- +
- ++
- ++
- +
- ++
- ++
- ++
- +
- ++
- ++
- ++
- +
-
+ -
Charges can only be
redistributed slightly (set off)
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Polarization of dielectrics
- +
-+
Conductors: Separation of charges
-
- +
-
++
++
++
+
+- +- + -
-
+
- +
- ++
- ++
- +
++
++
++
+
-
- ++
- ++
- ++
- +
-
+ -
- +
-
+- +- + -
+- -
+
+-
+
+-
+
+-
++
+
+-
+
+-
+
+-
+
+-
+
+
-
++
+
+-
-
Charged objects
+
+
+
+
+
+
+
+
+
+
Positively charged object:
Lack of negative charges
+
+
-
-
-
+
+-
-
+
+-
-
- ++
-
-
-
-
Negatively charged object:
Excess of negative charges
-
+- +- + -
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
-
+
+-
-
+
+-
-
- ++
-
-
-
-
+ -
Negative charges on metal cups are
free to move, and therefore repelled
by the negatively charged rod
When separated, the metal
cups are charged
CONDUCTORS
+
+
+
+
-+
-
-
+ -
When separated, the plastic
cups are uncharged
+
+
-+
-
Conductors: Separation of charges
FREE ELECTRONS ?
• In metals, some electrons
can move freely
throughout the metal.
– When we bring a
positively charged rod
next to a metal bar, the
free electrons move
closer to the positively
charged rod, leaving
the other side with a
deficiency of electrons.
© 2014 Pearson Education, Inc.
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CONDUCTORS
Conductors
Is the human body a conductor or a
dielectric?
As your hand approaches
a positively charged cup,
the free electrons on the
surface of your hand and
arm move toward it until
the net charge on the cup
is zero.
ELECTROPHORUS
Grounding
• If the cup has a positive charge, negative electrons
in the ground are attracted toward the cup and
travel from the ground to the cup, causing it to
become neutral.
Discharging by moisture in the air
• If you leave a negatively or
positively charged object
alone for a long time
interval, the object can
become neutral in part
because of the presence
of water vapor in the air.
The electroscope
• An electroscope consists of a metal ball attached to a
metal rod. A very lightweight needle-like metal rod is
connected on a pivot near the bottom of the larger rod.
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The electroscope
• The angle of deflection is related to the magnitude of the
electric charge of the electroscope.
Several years later, while Benjamin Franklin was
helping to build our nation, a French scientist
named Charles-Augustin de Coulomb ventured a
step further.
“The attractive force between
two oppositely charged
spheres is proportional to the
product of the quantities of
charge on the spheres and is
inversely proportional to the
square of the distance
between the spheres”
Coulomb's force law
• In 1785, Charles Coulomb determined the
relationship between distance and magnitude of
charges, and the force between charges.
• The experimental apparatus Coulomb used is
called a torsion balance.
(Originally in French)
Coulomb's data
Coulomb’s Law
Electric Force (FE )
Electrostatic interactions
• Sometimes more vigorous rubbing leads to a
greater force exerted by the rubbed objects on
each other.
q1
q2
d
• The magnitude of the force that the charged
objects exert on each other increases when the
distance between the objects decreases.
𝐹𝐸 =
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑑2
𝐹𝑞1𝑜𝑛𝑞2 =
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑑2
AP Equation sheet
AP Book
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Coulomb’s Law
𝐹𝑞1𝑜𝑛𝑞2
Coulomb’s Law Constant
𝑘 ∙ 𝑞1 ∙ 𝑞2
=
𝑑2
• q1 is the magnitude of the charge of object 1.
• q2 is the magnitude of the charge of object 2.
𝑘=
1
4 ∙ 𝜋 ∙ 𝜀0
0 is the vacuum permittivity
(constant number)
−12 2
𝜀0 =
𝑘=
• d is the distance between the charged objects.
8.85𝑥10 𝐶
𝑁 ∙ 𝑚2
1
8.85𝑥10−12 𝐶 2
4∙𝜋∙
𝑁 ∙ 𝑚2
• k is Coulomb’s Law Constant
𝑘=
What charge exerts a bigger force?
𝐹𝐸
q2
q1
d
𝐹𝑞1𝑜𝑛𝑞2 = 𝐹𝑞2𝑜𝑛𝑞1
Newton’s third Law !!!
Coulomb’s Law
𝑘 ∙ 𝑞1 ∙ 𝑞2
=
𝑑2
If the two objects have the
same type of charge, they
will repel one another.
+
+
_
_
𝐹𝑞1𝑜𝑛𝑞2 =
9𝑥109 𝑁 ∙ 𝑚2
𝐶2
With the math model you can get
you the magnitude of the
electric forces exerted on each
object, and you can use some
reasoning to find the direction.
If the two objects have
different types of charge,
they will attract one
another.
+
𝐹𝑞1𝑜𝑛𝑞2 =
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑑2
1
𝑞1 ∙ 𝑞2
4 ∙ 𝜋 ∙ 𝜀0
𝑑2
WHITEBOARD
Find magnitude and direction of
• FqA on qB
• FqB on qA
qB = -9C
qA = +3C
+
d = 0.05 m
_
The unit of electric charge (q) is the
Coulomb [C], in honor of Charles
Coulomb.
• FqA on qB = - 97.2 N
• FqB on qA = + 97.2 N
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WHITEBOARD
Four electric charges are fixed on a grid.
Find the 4 missing quantities.
qA = -4C
-
FqAonqB = ?
d = 0.04 m
+
• d = 0.08 m
d=?
FqBonqD = +14.0625 N
• qD = -2C
-
d = 0.04 m qD = ?
FqConqD = -90 N
WHITEBOARD
• Three identical metal spheres A, B, and C are on
separate insulating stands. You charge sphere
A with charge +q. Then you touch sphere B to
sphere A, and separate them by distance d.
1. Write an expression for the electric force that the
spheres exert on each other.
• You now take sphere C, touch it to sphere A, and
then remove sphere C. You then separate
spheres A and B by a distance d/2.
1. Write an expression for the magnitude of the
force that spheres A and B exert on each other.
• FqA on qB = -112.5 N
qB = +5C
+
d=?
FqAonqC = +45 N
qC = +8C
WHITEBOARD - SOLUTION
• FqA on qD = 9 N
FqAonqD = ?
𝐹𝑞𝐴 𝑜𝑛 𝑞𝐵
𝑘 ∙ 𝑞2
=
4𝑑2
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Whiteboard Reasoning with Coulomb’s Law
F
+q
F
+q
𝐹𝑞 =
d
𝐹𝑞𝐴 𝑜𝑛 𝑞𝐵
𝑘 ∙ 𝑞2
=
2𝑑2
If one of the spheres is replaced with a sphere of charge
+2q, and they are brought to half the distance apart, what
will be the resulting force exerted on each of the spheres?
a
+2q
b
+2q
+3q
d
c
+q
𝐹𝑞 =
𝟖 ∙ 𝑘 ∙ 𝑞2
𝑑2
a>b>c
WHITEBOARD
(RANKINGS)
(RANKINGS)
(RANKINGS)
+q
c
+q
d
a=b=c
+q
a
-q
d
+q
b
+q
d
+q
d
WHITEBOARD
b
+q
d
WHITEBOARD
a
d
(RANKINGS)
+2q
d/2
-q
𝑘∙
𝑑2
Two charged spheres are held near one another, as shown
above.
+q
-q
WHITEBOARD
𝑞2
+q
c
+q
2d
a>b>c
+q
a
+q
3d
+q
b
+q
d
+q
c
+q
2d
+q
+q
½d
c>a>b
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WHITEBOARD
WHITEBOARD
Both spheres are
positive
(RANKINGS)
a
b
+q
+q
+2q
d
c
+q
2d
+3q
+q
1/3 d
c>a>b
WHITEBOARD
Electrostatic Conceptual Question
Two isolated charges, +q and -2q, are 2 centimeters
apart. If F is the magnitude of the force exerted on
charge -2q, what are the magnitude and direction of
the force exerted on charge +q?
Magnitude
(A)
(B)
(C)
(D)
(E)
F/2
2F
F
F
2F
Direction
Toward charge -2q
Away from charge -2q
Toward charge -2q
Away from charge -2q
Toward charge -2q
Newton’s Third Law!
A positively charged sphere
with charge q and mass m
hangs at the end of a string.
Another positively charged
sphere of a different charge
5q is secured at the top end
of the string to a wooden
support.
+
Draw a force
diagram for the
hanging sphere
(qh)
Represent
mathematically
using Newton’s
2nd Law
+
Draw Newton’s
3rd Law forces
that the
spheres exert
on each other.
FT on qh
Fq2 on qh
+ qh
FE on qh
FE on qh
- q2
+ q2
+ qh
Fq2 on qh
Fqh on q2
Calculate the ratio of the strength of the electric force that
each particle exerts on the other to the strength of the
gravitational force that particle exerts on the other.
q1q2
r2
Fgravitational = G
Fgravitational
=
kq1q2
Gm1m2
Fq2 on qh
qh +
Felectric = k
Felectric
FT on qh
+ qh
Fqh on q2
A hydrogen atom is composed of a proton and an electron.
mproton = 1.67 x 10-27 kg
melectron = 9.1 x 10-31 kg
G = 6.67 x 10-11 Nm2/kg2
Fq2 on qh
FT on qh = Fq2 on qh FT on qh + Fq2 on qh
+ FE on qh
= FE on qh
Whiteboard: The Hydrogen Atom
qelectron = -1.6 x 10-19 C
qproton = 1.6 x 10-19 C
k = 9 x 109 Nm2/C2
Hanging sphere is
positive, top sphere
is negative
m1m2
r2
= 2.27 x 1039 times as strong!!!
The electrical force between the particles is much,
much stronger than the gravitational force!
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The Principle of Superposition
The net force exerted on an object is the vector sum
of the individual forces exerted on it. Treat each force
separately using Coulomb’s Law, and then add them
as vectors!
Whiteboard: Superimpose!
Whiteboard: Solution
Determine the magnitude and direction of the net force
exerted on q1 in the configuration shown below.
Fnet = 3.6 N
60°
+
60°
q1 = 6 x 10-7 C
60°
+
3 cm
+
_
q2 = -6 x 10-7 C
+
_
3 cm
3 cm
+
q3 = 6 x 10-7 C
+
Whiteboard
60°
+
The y-components of the forces
negate one another, and the xcomponents add to one another!
Whiteboard
Whiteboard: solution
Fq1onq3
60°
X-component
Y-component
-0.00102
0.001357
L
+q
qq
+
L
+
+q
d
• Determine the signs of the charges q1 and q2.
• Calculate the magnitude of the net electrostatic
force exerted on particle 3.
• Where should another positively charged particle
could be fixed in place so that the electrostatic force
on particle 3 is zero?
Fq2onq3
-0.0018
-0.001354
F
-0.00282
0
Two objects of mass m and charge +q hang at
the bottom of a string of length L.
• Draw a Force diagram for any +q
• Write an expression for any known force (or
components)
• Write an expression for the electric charge
that keeps the system in equilibrium.
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FT
q
+q
Fq
𝐹𝑞 =
𝑘 ∙ 𝑞2
𝑑2
+
FTx
FTy
𝐹𝑇𝑦 = 𝐹𝑇 ∙ 𝑐𝑜𝑠 𝜃
𝐹𝑇𝑦 = 𝑔𝑚
𝐹𝑇𝑥 = 𝐹𝑇 ∙ 𝑠𝑖𝑛 𝜃
FG
Σ𝐹𝑦 = 0
Σ𝐹𝑥 = 0
𝐹𝐺 = 𝑔𝑚
Whiteboard
𝐹𝐺 + 𝐹𝑇𝑦 = 0
𝐹𝑞 + 𝐹𝑇𝑥 = 0
𝑔𝑚 + 𝐹𝑇 ∙ 𝑐𝑜𝑠 𝜃 = 0
𝑘𝑞 2
+ 𝐹𝑇 ∙ 𝑠𝑖𝑛 𝜃 = 0
2
𝑑
𝑔𝑚
𝑘𝑞2
𝐹𝑇 = −
𝐹𝑇 = − 2
𝑐𝑜𝑠 𝜃
𝑑 ∙ 𝑠𝑖𝑛 𝜃
𝑑2
𝑘𝑞2
𝑔𝑚
=
∙ 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃
𝑞=𝑑
𝑔𝑚 ∙ 𝑡𝑎𝑛 𝜃
𝑘
Suppose in the scenario with the hanging charged spheres, one of
the spheres had a charge +q, and the other had a charge +2q.
The spheres have the same mass. Which of the following shows
the equilibrium position that will be reached by the spheres?
(A)
(B)
(C)
(D) None of the above are correct.
WHITEBOARD
Metal spheres on insulating stands have the
F2q on q
Fq on 2q
following electric charges: qA = +2.0 nC,
qB = +2.0 nC, and qC = –4.0 nC. The spheres are
𝐹𝑁𝐸𝑇 = 124.7 𝑛𝑁
placed at the corners of an equilateral triangle
The magnitude of the electric force
exerted on each particle is the same.
Newton’s 3rd Law is a beautiful
thing!
whose sides have length d = 1.0 m, with qC at the
top of the triangle. What is the magnitude of the total
(net) electric force that spheres A and B exert on
sphere C?
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Whiteboard
Whiteboard: solution
X-component
• Calculate the magnitude and direction of the net electric
force exerted on the object of charge q1.
• If the mass of each of the particles is 30 grams, and the
particles are released from rest in the configuration
shown above, what will be the magnitude and direction
of the instantaneous acceleration of the object of charge
q1 at the moment of release?
Whiteboard: solution
Y-component
Fq2onq1
0
0.02025
Fq3onq1
0.011691
-0.00675
F
Whiteboard
0.011691
Fq2onq1
𝐹𝑁𝐸𝑇 = 0.0179 𝑁
𝜃 = 49.1
Fq3onq1
0.0135
Whiteboard
𝑎=
Σ𝐹
𝑚
𝑎=
0.0179 𝑁
0.03 𝑘𝑔
𝑎=
0.595 𝑚
𝑠2
Whiteboard
• On the axes below, qualitatively sketch a graph of the
acceleration a of the object of mass m2 versus the
distance d between the objects after the string has been
cut.
• Two small objects, each with a charge of -4.0 nC,
are held together by a 0.020 m length of insulating
string as shown in the diagram above. The objects
are initially at rest on a horizontal, nonconducting
frictionless surface. The gravitational effect on each
object due to the other is negligible.
𝐹𝑞1 𝑜𝑛 𝑞2 =
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑑2
𝑎=
𝐹𝑞1 𝑜𝑛 𝑞2 = +0.00036 𝑁
𝑎=
+0.00036
+0.006 𝑚
𝑎=
0.06 𝑘𝑔
𝑠2
𝐹𝑞2 𝑜𝑛 𝑞1 = −0.00036 𝑁
𝑎=
−0.012 𝑚
−0.00036
𝑎=
𝑠2
0.03 𝑘𝑔
• The masses of the objects are m1 = 0.030 kg and
m2 = 0.060 kg. The string is now cut.
• Calculate the magnitude of the initial acceleration of
each object.
Σ𝐹
𝑚
• Describe qualitatively what happens to the speeds of the
objects as time increases, assuming that the objects
remain on the horizontal, nonconducting frictionless
surface.
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1
Felectric a 2
d
&
a a Fnet

1
aa 2
d
Whiteboard
Bonus Follow-Up Question!
What would the velocity vs time graph for m2 look like after the string was cut?
v
A positively charged cannonball is held near another fixed
positively charged object in the barrel of the cannon.
Build an Energy bar chart for the initial and final states.
After the string is cut, m2 will continue to speed up
forever, but with decreasing acceleration.
m2 will never slow down, because it is always
being repelled by m1!
0.006 m/s2
However, it will speed up by less and less each
second as it gets further away.
Electric potential energy: A qualitative
analysis
• Some type of energy must decrease if gravitational
and kinetic energies increase in this process.
0.07
0.06
This shows a decreasing acceleration
(slope)
t
m2 is always being repelled by m1, so it will continue to speed
up forever. However, as they get further apart, Felectric decreases.
This causes m2 to speed up with a decreasing acceleration.
0.02 m
0.08
Velocity keeps increasing, but by less and
less each second.
Cannonball at a
some elevation
Moving
Cannonball
Whiteboard
Consider two oppositely charged blocks, one of which can
slide without friction..
Build an Energy bar chart for the initial and final states.
Electric potential energy: A qualitative
analysis
• When the negatively charged block is released and
moves nearer the nut, the kinetic energy of the
system increases.
8.00E+00
6.00E+00
0.05
4.00E+00
0.04
2.00E+00
KE cant be
negative
0.00E+00
0.03
-2.00E+00
0.02
-4.00E+00
0.01
Uqi Ugi Kei
UqF UgF KeF
-6.00E+00
0
Uqi Ugi Kei
UqF UgF KeF
-8.00E+00
-1.00E+01
-1.20E+01
Cannonball at a
some elevation
Cannonball at a
higher elevation
The electric potential
energy of oppositely
charged objects is negative
Uq becomes more
negative as the charged
object gets closer
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analyze a situation where only the
electric potential energy changes when
work is done.
Distance decreases, electric
force increases.
It is harder to push !
Electric potential energy
𝑊 = 𝐹𝑞 ∙ ∆𝑟
𝑊=
𝑘 ∙ 𝑞1 ∙ 𝑞2
∙ ∆𝑟
𝑟2
𝑈𝑞 =
Calculus !
𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑊=
−
𝑟𝐹
𝑟𝑖
𝑊 = 𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑟
• Sign of the charge does
matter !
1 1
−
𝑟𝐹 𝑟𝑖
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Tip
Skills for analyzing processes involving
electric force and electric potential energy
ELECTRIC POTENTIAL ENERGY
In conjunction with your problem-solving strategy:
• Decide whether you can consider the charged
objects to be point-like.
PRACTICE PROBLEMS
• If you are using the work-energy principle,
construct an energy bar chart. Decide where the
zeros for potential energies are.
WHITEBOARD: Graph electric potential
energy versus distance
• Because of the 1/r dependence, the electric potential
energy approaches positive infinity when the separation
approaches zero, and it becomes less positive and
approaches zero as like charges are moved far apart.
WHITEBOARD 1 / 6
• Two oppositely charged objects (with positive charge +q
and negative charge –q) are separated by distance ri.
Will the electric potential energy of the system decrease
or increase if you pull the objects farther apart?
• Explain (Use a bar chart)
• Graph Uq vs r
WHITEBOARD SOLUTION
4
3
2
1
0
-1
Uqi
w
UqF
-2
-3
-4
-5
-6
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WHITEBOARD 2 / 6
• Suppose that a radon atom in the air in a home is
inhaled into the lungs. While in the lungs, the
nucleus of the radon atom undergoes radioactive
decay, emitting an α (alpha) particle, which is
composed of two protons and two neutrons.
During this process, the radon nucleus turns into
a polonium nucleus with charge +84e and mass
3.6 x 10–25 kg. The α particle has charge +2e
and mass 6.6 x 10–27 kg. Suppose the two
particles are initially separated by 1.0 x 10–15 m
and are at rest. How fast is the α particle moving
when it is very far from the polonium nucleus?
WHITEBOARD 3/6
(PROBLEM 24)
𝑈𝑞𝑖 = 𝑈𝑞𝑓 + 𝐾𝐸𝑓
𝑈𝑞𝑖 = 𝐾𝐸𝑓
Uq = 0
R=
𝑣=
𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑚 ∙ 𝑣
=
𝑑
2
2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑚∙𝑑
𝑣=
𝑣=
108,302,606 m
𝑠
2
2 ∙ 𝑘 ∙ 84𝑒 ∙ 2𝑒
6.6𝑥10−27 ∙ 1𝑥10−15
• Determine the change in electric potential energy
of a system of two charged objects when a -1.5 C
charged object and a -4.0 C charged object
move from an initial separation of 500 km to a
final separation of 100 km.
• Sketch
• Energy bar chart
• Find Uq
𝑈𝑞𝑖 + 𝑊 = 𝑈𝑞𝑓
𝑊 = ∆𝑈𝑞 =
𝑘 ∙ 𝑞1 ∙ 𝑞2 𝑘 ∙ 𝑞1 ∙ 𝑞2
−
𝑟𝐹
𝑟𝑖
∆𝑈𝑞 = 𝑘 ∙ 𝑞1 ∙ 𝑞2
∆𝑈𝑞 = 𝑘 ∙ −1.5 𝐶 ∙ −4 𝐶 ∙
1 1
−
𝑟𝐹 𝑟𝑖
1
1
−
100000 500000
∆𝑈𝑞 = 432,000 𝐽
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WHITEBOARD SOLUTION
WHITEBOARD 4/6
(PROBLEM 33)
∆𝑈𝑞 = 𝑘 ∙ 𝑞1 ∙ 𝑞2
600000
540000
500000
432000
400000
300000
200000
108000
100000
0
Uqi
W
UqF
WHITEBOARD SOLUTION
600
• A stationary block has a charge of +6.0x10-4 C. A
0.80 kg cart with a charge of 4.0x10-4 C is initially
at rest and separated 4.0 m from the block. The
cart is released and moves along a frictionless
surface to a distance of 10.0 m from the block.
•
•
•
•
Sketch
Find the change in electric potential energy.
Find the speed of the cart.
Energy bar chart
∆𝑈𝑞 = 𝑘 ∙ 6𝑥10−4 ∙ 4𝑥10−4 ∙
540
∆𝑈𝑞 = −∆𝐾𝐸
324
∆𝑈𝑞 = −
200
100
0
Uqi
KEF
UqF
• Sketch
• How fast is the cart moving when very far
(infinity) from the fixed charge?
• How fast is the cart moving when 2.0 m from the
fixed charge?
𝑚𝑣 2
2
𝑣=
−2 ∙ ∆𝑈𝑞
𝑚
28.46 𝑚
𝑠
∆𝑈𝑞 = −∆𝐾𝐸
𝑘 ∙ 𝑞1 ∙ 𝑞2
300
216
𝑣=
Think: Conservation
of energy
• A 0.4 kg cart with charge +4.0x10-5 C starts at
rest on a horizontal frictionless surface 0.50 m
from a fixed object with charge +2.0x10-4 C.
When the cart is released, it moves away from
the fixed object.
400
1 1
−
10 4
∆𝑈𝑞 = −324 𝐽
WHITEBOARD 5/6
(PROBLEM 46)
500
1 1
−
𝑟𝐹 𝑟𝑖
𝑣=
𝑟𝐹 = ∞
1 1
𝑚𝑣 2
−
=−
𝑟𝐹 𝑟𝑖
2
−2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2
1 1
−
𝑟𝐹 𝑟𝑖
𝑚
1
=0
𝑟𝐹
𝑣=
26.83 𝑚
𝑠
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12/1/2016
∆𝑈𝑞 = −∆𝐾𝐸
𝑘 ∙ 𝑞1 ∙ 𝑞2
𝑣=
𝑟𝐹 = 2 𝑚
Van de Graaff generator
1 1
𝑚𝑣 2
−
=−
𝑟𝐹 𝑟𝑖
2
−2 ∙ 𝑘 ∙ 𝑞1 ∙ 𝑞2
1 1
−
𝑟𝐹 𝑟𝑖
𝑚
𝑣=
Wimshurst machine
• The Wimshurst,
invented in the 1880s,
consists of two plastic
disks that rotate in
opposite directions. It
can produce large
charge separations.
23.24 𝑚
𝑠
© 2014 Pearson Education, Inc.
A photocopier
• In a copy machine, a drum gets electrically
charged with an image of the page being copied.
Dark toner particles stick to the places where the
drum is charged.
21