PHYSICS
FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E
Chapter 26 Lecture
RANDALL D. KNIGHT
Chapter 26 Potential and Field
IN THIS CHAPTER, you will learn how the electric
potential is related to the electric field.
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Chapter 26 Preview
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Chapter 26 Preview
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Chapter 26 Preview
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Chapter 26 Preview
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Chapter 26 Preview
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Chapter 26 Content, Examples, and
QuickCheck Questions
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Connecting Potential and Field
The figure
shows the
four key ideas
of force, field,
potential
energy, and
potential.
We know, from Chapters 9 and 10, that force and
potential energy are closely related.
The focus of this chapter is to establish a similar
relationship between the electric field and the
electric potential.
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Finding the Potential from the Electric Field
The potential difference between two points in space is
where s is the position along a line from point i to point f.
We can find the potential difference between two points if
we know the electric field.
Thus a graphical interpretation of the equation above is
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Example 26.1 Finding the Potential
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Example 26.1 Finding the Potential
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Tactics: Finding the Potential From the Electric
Field
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Finding the Potential of a Point Charge
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Example 26.2 The Potential of a Parallel-Plate
Capacitor
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Example 26.2 The Potential of a Parallel-Plate
Capacitor
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Example 26.2 The Potential of a Parallel-Plate
Capacitor
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Finding the Electric Field from the Potential
The figure shows
two points i and f
separated by a
small distance Δs.
The work done by the electric
field as a small charge q
moves from i to f is W = FsΔs =
qEsΔs.
The potential difference
between the points is
The electric field in the s-direction
is Es = –ΔV/Δs. In the limit Δs → 0:
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Finding the Electric Field from the Potential:
Quick Example
Suppose we knew the potential of a point charge to be
V = q/4π 0r but didn’t remember the electric field.
Symmetry requires that the field point straight outward
from the charge, with only a radial component Er.
If we choose the s-axis to be in the radial direction,
parallel to E, we find
This is, indeed, the well-known electric field of a point
charge!
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Example 26.3 The Electric Field of a Ring of
Charge
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Example 26.3 The Electric Field of a Ring of
Charge
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Example 26.4 Finding E From the Slope of V
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Example 26.4 Finding E From the Slope of V
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Example 26.4 Finding E From the Slope of V
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The Geometry of Potential and Field
In three dimensions, we can find the electric field
from the electric potential as
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Kirchhoff’s Loop Law
For any path that
starts and ends at
the same point:
The sum of all the
potential differences
encountered while
moving around a loop
or closed path is zero.
This statement is
known as Kirchhoff’s
loop law.
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A Conductor in Electrostatic Equilibrium
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A Conductor in Electrostatic Equilibrium
When a conductor is in
equilibrium:
• All excess charge sits on the
surface.
• The surface is an equipotential.
• The electric field inside is zero.
• The external electric field is A corona discharge
perpendicular to the surface occurs at pointed metal
tips where the electric
at the surface.
field can be very strong.
• The electric field is strongest
at sharp corners of the
conductor’s surface.
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A Conductor in Electrostatic Equilibrium
The figure shows a
negatively charged
metal sphere near a
flat metal plate.
Since a conductor
surface must be an
equipotential, the
equipotential surfaces
close to each electrode
roughly match the
shape of the electrode.
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Sources of Electric Potential
A separation of charge
creates an electric
potential difference.
Shuffling your feet on
the carpet transfers
electrons from the
carpet to you, creating
a potential difference
between you and other
objects in the room.
This potential difference
can cause sparks.
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Van de Graaff Generator
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Charge escalator model of a battery
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Batteries and emf
emf is the work done per charge to pull positive and
negative charges apart.
In an ideal battery, this work creates a potential
difference ∆Vbat = between the positive and negative
terminals.
This is called the terminal voltage.
A battery constructed to
have an emf of 1.5 V
creates a 1.5 V potential
difference between its
positive and negative
terminals.
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Batteries in Series
The total potential difference of
batteries in series is simply the
sum of their individual terminal
voltages:
Flashlight batteries are placed
in series to create twice the
potential difference of one
battery.
For this flashlight:
ΔVseries = ΔV1 + ΔV2
= 1.5 V + 1.5 V
= 3.0 V
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Capacitance and Capacitors
The figure shows two
arbitrary electrodes
charged to ±Q.
There is a potential
difference ΔVC that is
directly proportional to Q.
The ratio of the charge Q to the potential difference ΔVC is
called the capacitance C:
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Capacitance and Capacitors
Capacitance is a purely geometric property of two electrodes
because it depends only on their surface area and spacing.
The SI unit of capacitance is the farad:
The charge on the capacitor plates is directly
proportional to the potential difference between the
plates:
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Capacitance and Capacitors
Capacitors are important
elements in electric circuits.
They come in a variety of The keys on most computer
sizes and shapes. keyboards are capacitor
switches. Pressing the key
pushes two capacitor plates
closer together, increasing
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Example 26.6 Charging a Capacitor
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Charging a Capacitor
The figure shows a
capacitor just after it
has been connected
to a battery.
Current will
flow in this manner
for a nanosecond or
so until the capacitor
is fully charged.
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Charging a Capacitor
The figure shows
a fully charged
capacitor.
Now the system
is in electrostatic
equilibrium.
Capacitance always refers to the charge per voltage
on a fully charged capacitor.
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Combinations of Capacitors
In practice, two or more capacitors are sometimes
joined together.
The circuit diagrams below illustrate two basic
combinations: parallel capacitors and series
capacitors.
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Capacitors Combined in Parallel
Consider two capacitors C1
and C2 connected in parallel.
The total charge drawn from
the battery is Q = Q1 + Q2.
In figure (b) we have replaced
the capacitors with a single
“equivalent” capacitor:
Ceq = C1 + C2
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Capacitors Combined in Parallel
If capacitors C1, C2, C3, … are in parallel, their
equivalent capacitance is:
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Capacitors Combined in Series
Consider two capacitors
C1 and C2 connected in
series.
The total potential
difference across
both capacitors is
ΔVC = ΔV1 + ΔV2.
The inverse of the
equivalent
capacitance is
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Capacitors Combined in Series
If capacitors C1, C2, C3, …
are in series, their equivalent
capacitance is
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The Energy Stored in a Capacitor
The figure shows a
capacitor being charged.
As a small charge dq is
lifted to a higher
potential, the potential
energy of the capacitor
increases by
The total energy
transferred from the
battery to the capacitor is
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The Energy Stored in a Capacitor
Capacitors are important elements in electric circuits
because of their ability to store energy.
The charge on the two plates is ±q and this charge
separation establishes a potential difference ΔV = q/C
between the two electrodes.
In terms of the capacitor’s potential difference, the
potential energy stored in a capacitor is
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The Energy Stored in a Capacitor
A capacitor can be
charged slowly but then
can release the energy
very quickly.
An important medical
application of capacitors
is the defibrillator.
A heart attack or a serious injury can cause the heart to
enter a state known as fibrillation in which the heart
muscles twitch randomly and cannot pump blood.
A strong electric shock through the chest completely
stops the heart, giving the cells that control the heart’s
rhythm a chance to restore the proper heartbeat.
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Example 26.8 Storing Energy in a Capacitor
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Example 26.8 Storing Energy in a Capacitor
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The Energy in the Electric Field
The energy density of an electric field, such as the
one inside a capacitor, is:
The energy density
has units J/m3.
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Dielectrics
The figure shows a
parallel-plate capacitor
with the plates
separated by a vacuum.
When the capacitor is
fully charged to voltage
(ΔVC)0, the charge on
the plates will be ±Q0,
where Q0 = C0(ΔVC)0.
In this section the
subscript 0 refers to a
vacuum-filled capacitor.
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Dielectrics
Now an insulating
material is slipped
between the
capacitor plates.
An insulator in an
electric field is called
a dielectric.
The charge on the
capacitor plates does
not change (Q = Q0).
However, the voltage
has decreased:
ΔVC < (ΔVC)0
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Dielectrics
The figure shows
how an insulating
material becomes
polarized in an
external electric field.
The insulator as a
whole is still neutral,
but the external
electric field
separates positive
and negative charge.
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Dielectrics
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Dielectrics
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Dielectrics
We define the dielectric constant:
The dielectric constant, like density or specific heat, is a
property of a material.
Easily polarized materials have larger dielectric constants
than materials not easily polarized.
Vacuum has κ = 1 exactly.
Filling a capacitor with a dielectric increases the
capacitance by a factor equal to the dielectric
constant:
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Dielectrics
The production of a practical capacitor, as shown, almost
always involves the use of a solid or liquid dielectric.
All materials have a maximum electric
field they can sustain without
breakdown—the production of a spark.
The breakdown electric field
of air is about 3 × 106 V/m.
A material’s maximum
sustainable electric field is
called its dielectric
strength.
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Dielectrics
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Example 26.9 A Water-Filled Capacitor
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Example 26.9 A Water-Filled Capacitor
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Example 26.9 A Water-Filled Capacitor
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Example 26.9 A Water-Filled Capacitor
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Example 26.10 Energy Density of a Defibrillator
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Example 26.10 Energy Density of a Defibrillator
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Example 26.10 Energy Density of a Defibrillator
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Example 26.10 Energy Density of a Defibrillator
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Chapter 26 Summary Slides
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General Principles
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General Principles
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General Principles
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Important Concepts
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Important Concepts
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Applications
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Applications
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