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4/7/2016
1
FOR SCIENTISTS AND ENGINEERS
physics
a strategic approachTHIRD EDITION
randall d. knight
© 2013 Pearson Education, Inc.
Chapter 28 Lecture
© 2013 Pearson Education, Inc.
Chapter 28 The Electric Potential
Chapter Goal: To calculate and use the electric potential and electric potential energy.
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Chapter 28 Preview
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Chapter 28 Preview
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Chapter 28 Preview
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Chapter 28 Preview
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Chapter 28 Preview
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Chapter 28 Preview
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Chapter 28 Reading Quiz
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The electric potential energy of a system of two point charges is proportional to
A. The distance between the two charges.
B. The square of the distance between the two charges.
C. The inverse of the distance between the two charges.
D. The inverse of the square of the distance
between the two charges.
Reading Question 28.1
Slide 28-10
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The electric potential energy of a system of two point charges is proportional to
A. The distance between the two charges.
B. The square of the distance between the two charges.
C. The inverse of the distance between the two charges.
D. The inverse of the square of the distance
between the two charges.
Reading Question 28.1
Slide 28-11
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What are the units of potential difference?
A. Amperes.
B. Potentiometers.
C. Farads.
D. Volts.
E. Henrys.
Reading Question 28.2
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What are the units of potential difference?
A. Amperes.
B. Potentiometers.
C. Farads.
D. Volts.
E. Henrys.
Reading Question 28.2
Slide 28-13
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New units of the electric field were introduced in this chapter. They are:
A. V/C.
B. N/C.
C. V/m.
D. J/m2.
E. Ω/m.
Reading Question 28.3
Slide 28-14
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New units of the electric field were introduced in this chapter. They are:
A. V/C.
B. N/C.
C. V/m.
D. J/m2.
E. Ω/m.
Reading Question 28.3
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Which of the following statements about equipotential surfaces is true?
A. Tangent lines to equipotential surfaces are always parallel to the electric field vectors.
B. Equipotential surfaces are surfaces that have the same value of potential energy at every point.
C. Equipotential surfaces are surfaces that have the same value of potential at every point.
D. Equipotential surfaces are always parallel planes.
E. Equipotential surfaces are real physical surfaces that exist in space.
Reading Question 28.4
Slide 28-16
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Which of the following statements about equipotential surfaces is true?
A. Tangent lines to equipotential surfaces are always parallel to the electric field vectors.
B. Equipotential surfaces are surfaces that have the same value of potential energy at every point.
C. Equipotential surfaces are surfaces that have the same value of potential at every point.
D. Equipotential surfaces are always parallel planes.
E. Equipotential surfaces are real physical surfaces that exist in space.
Reading Question 28.4
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The electric potential inside a capacitor
A. Is constant.
B. Increases linearly from the negative to the positive plate.
C. Decreases linearly from the negative to the positive plate.
D. Decreases inversely with distance from the negative plate.
E. Decreases inversely with the square of the distance from the negative plate.
Reading Question 28.5
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The electric potential inside a capacitor
A. Is constant.
B. Increases linearly from the negative to the positive plate.
C. Decreases linearly from the negative to the positive plate.
D. Decreases inversely with distance from the negative plate.
E. Decreases inversely with the square of the distance from the negative plate.
Reading Question 28.5
Slide 28-19
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Chapter 28 Content, Examples, and
QuickCheck Questions
Slide 28-20
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Energy
The kinetic energy of a system, K, is the sum of the kinetic energies Ki = 1/2mivi
2 of all the particles in the system.
The potential energy of a system, U, is the interaction energy of the system.
The change in potential energy, ∆U, is −1 times the work done by the interaction forces:
If all of the forces involved are conservative forces (such as gravity or the electric force) then the total energy K + U is conserved; it does not change with time.
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Work Done by a Constant Force
Recall that the work done by a constant force depends on the angle θ between the force F and the displacement ∆r.
If θ = 0°, then W = F∆r.
If θ = 90°, then W = 0.
If θ = 180°, then W = F∆r.Slide 28-22
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Work
If the force is not
constant or the displacement is not along a linear
path, we can calculate the work by dividing the path into many small segments.
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Gravitational Potential Energy
Every conservative force is associated with a potential energy.
In the case of gravity, the work done is:
Wgrav = mgyi − mgyf
The change in gravitational potential energy is:
∆Ugrav = − Wgrav
where
Ugrav = U0 + mgy
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Two rocks have equal mass. Which has more gravitational
potential energy?
QuickCheck 28.1
A. Rock A.
B. Rock B.
C. They have the same potential energy.
D. Both have zero potential energy.
Slide 28-25
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Two rocks have equal mass. Which has more gravitational
potential energy?
QuickCheck 28.1
A. Rock A.
B. Rock B.
C. They have the same potential energy.
D. Both have zero potential energy.
Slide 28-26
Increasing PE
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Electric Potential Energy in a Uniform Field
A positive charge q inside a capacitor speeds up as it “falls” toward the negative plate.
There is a constant force F = qE in the direction of the displacement.
The work done is:
Welec = qEsi − qEsf
The change in electric potential energy is:
∆Uelec = −Welec
where
Uelec = U0 + qEsSlide 28-27
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Two positive charges are equal. Which has more electric potential energy?
QuickCheck 28.2
A. Charge A.
B. Charge B.
C. They have the same potential energy.
D. Both have zero potential energy.
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Two positive charges are equal. Which has more electric potential energy?
QuickCheck 28.2
A. Charge A.
B. Charge B.
C. They have the same potential energy.
D. Both have zero potential energy.
Slide 28-29
Increasing PE
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A positively charged particle gains kinetic energy as it moves in the direction of
decreasing potential energy.
Electric Potential Energy in a Uniform Field
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A negatively charged particle gains kinetic energy as it moves in the direction of
decreasing potential energy.
Electric Potential Energy in a Uniform Field
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The figure shows the energy diagram for a positively charged
particle in a uniform electric field.
The potential energy increases linearly with
distance, but the total mechanical energy Emech is fixed.
Electric Potential Energy in a Uniform Field
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Two negative charges are equal. Which has more electric potential energy?
QuickCheck 28.3
A. Charge A.
B. Charge B.
C. They have the same potential energy.
D. Both have zero potential energy.
Slide 28-33
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Two negative charges are equal. Which has more electric potential energy?
QuickCheck 28.3
A. Charge A.
B. Charge B.
C. They have the same potential energy.
D. Both have zero potential energy.
Slide 28-34
Increasing
PE for
negative
charge
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A positive charge moves as shown. Its kinetic energy
QuickCheck 28.4
A. Increases.
B. Remains constant.
C. Decreases.
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A positive charge moves as shown. Its kinetic energy
QuickCheck 28.4
A. Increases.
B. Remains constant.
C. Decreases.
Slide 28-36
Increasing PE
Decreasing KE
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The Potential Energy of Two Point Charges
The change in electric potential energy of the system is ∆Uelec = −Welec if:
Consider two like charges q1 and q2.
The electric field of q1
pushes q2 as it moves from xi to xf.
The work done is:
Slide 28-37
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The Potential Energy of Point ChargesConsider two point charges, q1 and q2, separated by a distance r. The electric potential energy is
This is explicitly the energy of the system, not the energy of just q1 or q2.Note that the potential energy of two charged particles approaches zero as r → ∞.
Slide 28-38
The Potential Energy of Two Point Charges
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Two like charges approach each other.
Their total energy is Emech > 0.
They gradually slow down until the distance separating them is rmin.
This is the distance of
closest approach.
The Potential Energy of Two Point Charges
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Two opposite charges are shot apart from one another with equal and opposite momenta.
Their total energy is Emech < 0.
They gradually slow down until the distance separating them is rmax.
This is their maximum
separation.
The Potential Energy of Two Point Charges
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A positive and a negative charge are released from rest in vacuum. They move toward each other. As they do:
QuickCheck 28.5
A. A positive potential energy becomes more positive.
B. A positive potential energy becomes less positive.
C. A negative potential energy becomes more negative.
D. A negative potential energy becomes less negative.
E. A positive potential energy becomes a negative potential energy.
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A positive and a negative charge are released from rest in vacuum. They move toward each other. As they do:
QuickCheck 28.5
A. A positive potential energy becomes more positive.
B. A positive potential energy becomes less positive.
C. A negative potential energy becomes more negative.
D. A negative potential energy becomes less negative.
E. A positive potential energy becomes a negative potential energy.
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The Electric Force Is a Conservative Force
Any path away from q1
can be approximated using circular arcs and radial lines.
All the work is done along the radial line segments, which is equivalent to a straight line from i to f.
Therefore the work done by the electric force depends only on initial and final position, not the path followed.
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Example 28.2 Approaching a Charged Sphere
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Example 28.2 Approaching a Charged Sphere
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Example 28.2 Approaching a Charged Sphere
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Example 28.3 Escape Velocity
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Example 28.3 Escape Velocity
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Example 28.3 Escape Velocity
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Example 28.3 Escape Velocity
Slide 28-50
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The Potential Energy of Multiple Point
ChargesConsider more than two point charges, the potential energy is the sum of the potential energies due to all pairs of charges:
where rij is the distance between qi and qj.The summation contains the i < j restriction to ensure that each pair of charges is counted only once.
Slide 28-51
The Potential Energy of Multiple Point Charges
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Example 28.4 Launching an Electron
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Example 28.4 Launching an Electron
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Example 28.4 Launching an Electron
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Example 28.4 Launching an Electron
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The Potential Energy of a Dipole
The change in electric potential energy of the system is ⊗Uelec = −Welec if:
Consider a dipole in a uniform electric field.
The forces F+ and F
exert a torque on the dipole.
The work done is:
Slide 28-56
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The potential energy of a dipole is φ = 0° minimum at where the dipole is aligned with the electric field.
A frictionless dipole with mechanical energy Emech will oscillate back and forth between turning points on either side of φ = 0°.
The Potential Energy of a Dipole
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Example 28.5 Rotating a Molecule
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Example 28.5 Rotating a Molecule
Slide 28-59
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The Electric Potential
We define the electric potential V (or, for brevity, just the potential) as
The unit of electric potential is the joule per coulomb, which is called the volt V:
Slide 28-60
This battery is a source of electric potential. The electric potential difference between the + and − sides is 1.5 V.
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Test charge q is used as a probe to determine the electric potential, but the value of V is independent of q.
The electric potential, like the electric field, is a
property of the source charges.
The Electric Potential
Slide 28-61
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Using the Electric Potential
As a charged particle moves through a changing electric potential, energy is conserved:
Slide 28-62
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A proton is released from rest at the dot. Afterward, the proton
QuickCheck 28.6
A. Remains at the dot.
B. Moves upward with steady speed.
C. Moves upward with an increasing speed.
D. Moves downward with a steady speed.
E. Moves downward with an increasing speed.
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A proton is released from rest at the dot. Afterward, the proton
QuickCheck 28.6
A. Remains at the dot.
B. Moves upward with steady speed.
C. Moves upward with an increasing speed.
D. Moves downward with a steady speed.
E. Moves downward with an increasing speed.
Slide 28-64
Decreasing PE
Increasing KE
© 2013 Pearson Education, Inc.
If a positive charge is released from rest, it
moves in the direction of
QuickCheck 28.7
A. A stronger electric field.
B. A weaker electric field.
C. Higher electric potential.
D. Lower electric potential.
E. Both B and D.
Slide 28-65
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If a positive charge is released from rest, it
moves in the direction of
QuickCheck 28.7
A. A stronger electric field.
B. A weaker electric field.
C. Higher electric potential.
D. Lower electric potential.
E. Both B and D.
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Problem-Solving Strategy: Conservation of
Energy in Charge Interactions
Slide 28-67
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Problem-Solving Strategy: Conservation of
Energy in Charge Interactions
Slide 28-68
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Example 28.6 Moving Through a Potential
Difference
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Example 28.6 Moving Through a Potential
Difference
Slide 28-70
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Example 28.6 Moving Through a Potential
Difference
Slide 28-71
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Example 28.6 Moving Through a Potential
Difference
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Example 28.6 Moving Through a Potential
Difference
Slide 28-73
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The Electric Field Inside a Parallel-Plate Capacitor
This is a review of Chapter 26.
Slide 28-74
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The Electric Potential Inside a Parallel-Plate
Capacitor
The electric potential inside a parallel-plate capacitor is
where s is the distance from the negative
electrode.
The potential difference ∆VC, or “voltage” between
the two capacitor plates is
Slide 28-75
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Units of Electric Field
If we know a capacitor’s voltage ∆V and the distance between the plates d, then the electric field strength within the capacitor is:
This implies that the units of electric field are volts per meter, or V/m.
Previously, we have been using electric field units of
newtons per coulomb.
In fact, as you can show as a homework problem, these units are equivalent to each other:
1 N/C = 1 V/mSlide 28-76
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The Electric Potential Inside a Parallel-Plate
Capacitor
Slide 28-77
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The Electric Potential Inside a Parallel-Plate
Capacitor
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Two protons, one after the
other, are launched from point
1 with the same speed. They
follow the two trajectories
shown. The protons’ speeds at
points 2 and 3 are related by
QuickCheck 28.9
A. v2 > v3.
B. v2 = v3.
C. v2 < v3.
D. Not enough information to compare their speeds.
Slide 28-79
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Two protons, one after the
other, are launched from point
1 with the same speed. They
follow the two trajectories
shown. The protons’ speeds at
points 2 and 3 are related by
QuickCheck 28.9
A. v2 > v3.
B. v2 = v3.
C. v2 < v3.
D. Not enough information to compare their speeds.
Slide 28-80
Energy conservation
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The Parallel-Plate Capacitor
The figure shows the contour lines of the electric potential and the electric field vectors inside a parallel-plate capacitor.
The electric field vectors are perpendicular to the equipotential surfaces.
The electric field points in the direction of decreasing potential.
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The Zero Point of Electric Potential
Where you choose V = 0 is arbitrary. The three contour maps below represent the same physical situation.
Slide 28-82
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Example 28.8 The Force on an Ion
Slide 28-83
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Example 28.8 The Force on an Ion
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Example 28.8 The Force on an Ion
Slide 28-85
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Example 28.8 The Force on an Ion
Slide 28-86
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The Electric Potential of a Point Charge
Let q in the figure be the source charge, and let a second charge q', a distance r
away, probe the electric potential of q.
The potential
energy of the two point charges is
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The electric potential due to a point charge q is
The potential extends through all of space, showing the influence of charge q, but it weakens with distance as 1/r.
This expression for V assumes that we have chosen V = 0 to be at r = ∞.
The Electric Potential of a Point Charge
Slide 28-88
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What is the ratio VB/VA of the
electric potentials at the two
points?
QuickCheck 28.10
A. 9.
B. 3.
C. 1/3.
D. 1/9.
E. Undefined without knowing the charge.
Slide 28-89
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What is the ratio VB/VA of the
electric potentials at the two
points?
QuickCheck 28.10
A. 9.
B. 3.
C. 1/3.
D. 1/9.
E. Undefined without knowing the charge.
Slide 28-90
Potential of a point charge decreases
inversely with distance.
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Example 28.9 Calculating the Potential of a
Point Charge
Slide 28-91
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Example 28.9 Calculating the Potential of a
Point Charge
Slide 28-92
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Example 28.9 Calculating the Potential of a
Point Charge
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The Electric Potential of a Point Charge
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The Electric Potential of a Point Charge
Slide 28-95
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The Electric Potential of a Charged Sphere
Outside a uniformly charged sphere of radius R, the electric
potential is identical to that of a point charge Q at the center.
where r > R. If the potential at the surface V0
is known, then the potential at r > R is:
A plasma ball consists of a small metal ball charged to a potential of about 2000 V inside a hollow glass sphere filled with low-pressure neon gas. The high voltage of the ball creates “lightning bolts” between the ball and the glass sphere.
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An electron follows the trajectory shown from point 1 to point 2. At point 2,
QuickCheck 28.11
A. v2 > v1.
B. v2 = v1.
C. v2 < v1.
D. Not enough information to compare the speeds at these points.
Slide 28-97
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An electron follows the trajectory shown from point 1 to point 2. At point 2,
QuickCheck 28.11
A. v2 > v1.
B. v2 = v1.
C. v2 < v1.
D. Not enough information to compare the speeds at these points.
Slide 28-98
Increasing PE (becoming less
negative) so decreasing KE
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Example 28.10 A Proton and a Charged Sphere
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Example 28.10 A Proton and a Charged Sphere
Slide 28-100
VISUALIZE
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Example 28.10 A Proton and a Charged Sphere
Slide 28-101
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Example 28.10 A Proton and a Charged Sphere
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The Electric Potential of Many Charges The electric potential V at a point in space is the sum of
the potentials due to each charge:
where ri is the distance from charge qi to the point in space where the potential is being calculated.
The electric potential, like the electric field, obeys
the principle of superposition.
Slide 28-103
The Electric Potential of Many Charges
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The Electric Potential of an Electric Dipole
Slide 28-104
The Electric Potential of an Electric Dipole
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The Electric Potential of a Human Heart Electrical activity
within the body can be monitored by measuring
equipotential lines on the skin.
The equipotentials near the heart are a slightly distorted but
recognizable electric dipole.
Slide 28-105
The Electric Potential of a Human Heart
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At the midpoint between these two equal but opposite charges,
QuickCheck 28.12
A. E = 0; V = 0.
B. E = 0; V > 0.
C. E = 0; V < 0.
D. E points right; V = 0.
E. E points left; V = 0.
Slide 28-106
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At the midpoint between these two equal but opposite charges,
QuickCheck 28.12
A. E = 0; V = 0.
B. E = 0; V > 0.
C. E = 0; V < 0.
D. E points right; V = 0.
E. E points left; V = 0.
Slide 28-107
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At which point or points is the electric potential zero?
QuickCheck 28.13
A. B. C. D.
E. More than one of these.
Slide 28-108
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At which point or points is the electric potential zero?
QuickCheck 28.13
A. B. C. D.
E. More than one of these.
Slide 28-109
V = 0 V = 0
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Example 28.11 The Potential of Two Charges
Slide 28-110
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Example 28.11 The Potential of Two Charges
Slide 28-111
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Example 28.11 The Potential of Two Charges
Slide 28-112
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Problem-Solving Strategy: The Electric
Potential of a Continuous Distribution of Charge
Slide 28-113
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Problem-Solving Strategy: The Electric
Potential of a Continuous Distribution of Charge
Slide 28-114
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Example 28.12 The Potential of a Ring of Charge
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Example 28.12 The Potential of a Ring of Charge
Slide 28-116
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Example 28.12 The Potential of a Ring of Charge
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Example 28.12 The Potential of a Ring of Charge
Slide 28-118
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Example 28.12 The Potential of a Ring of Charge
Slide 28-119
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Chapter 28 Summary Slides
Slide 28-120
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General Principles
Slide 28-121
General Principles
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General Principles
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General Principles
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General Principles
Slide 28-124