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Chapter 23
Electric Potential
Copyright © 2009 Pearson Education, Inc.
Units of Chapter 23
• Electric Potential Energy and Potential
Difference
• Relation between Electric Potential and
Electric Field
• Electric Potential Due to Point Charges
• Potential Due to Any Charge Distribution
• Equipotential Surfaces
• Electric Dipole Potential
Copyright © 2009 Pearson Education, Inc.
Units of Chapter 23
• E Determined from V
• Electrostatic Potential Energy; the Electron
Volt
Copyright © 2009 Pearson Education, Inc.
23-1 Electrostatic Potential Energy
and Potential Difference
The electrostatic force is
conservative – potential
energy can be defined.
Change in electric potential
energy is negative of work
done by electric force:
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ConcepTest 23.1 Electric Potential Energy
A proton and an electron are in
a constant electric field created
by oppositely charged plates.
You release the proton from
the positive side and the
electron from the negative side.
Which has the larger
acceleration?
1) proton
2) electron
3) both feel the same acceleration
4) neither – there is no acceleration
5) they feel the same magnitude
acceleration but opposite
direction
Electron
electron
-
+

E
Proton
proton
ConcepTest 23.1 Electric Potential Energy
A proton and an electron are in
a constant electric field created
by oppositely charged plates.
You release the proton from
the positive side and the
electron from the negative side.
Which has the larger
acceleration?
1) proton
2) electron
3) both feel the same acceleration
4) neither – there is no acceleration
5) they feel the same magnitude
acceleration but opposite
direction
Since F = ma and the electron is much less
Electron
electron
-
massive than the proton, the electron
experiences the larger acceleration.
+

E
Proton
proton
23-1 Electrostatic Potential Energy
and Potential Difference
Electric potential is defined as potential
energy per unit charge:
Unit of electric potential: the volt (V):
1 V = 1 J/C.
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23-1 Electrostatic Potential Energy
and Potential Difference
Only changes in potential can be measured,
allowing free assignment of V = 0:
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23-1 Electrostatic Potential Energy
and Potential Difference
Analogy between gravitational and electrical
potential energy:
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23-1 Electrostatic Potential Energy
and Potential Difference
Conceptual Example 23-1: A negative charge.
Suppose a negative charge, such as an electron, is
placed near the negative plate at point b, as shown
here. If the electron is free to move, will its electric
potential energy increase or decrease? How will
the electric potential change?
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23-1 Electrostatic Potential Energy
and Potential Difference
Electrical sources
such as batteries and
generators supply a
constant potential
difference. Here are
some typical potential
differences, both
natural and
manufactured:
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23-1 Electrostatic Potential Energy
and Potential Difference
Example 23-2: Electron in CRT.
Suppose an electron in a
cathode ray tube is
accelerated from rest
through a potential
difference Vb – Va = Vba =
+5000 V. (a) What is the
change in electric potential
energy of the electron? (b)
What is the speed of the
electron (m = 9.1 × 10-31 kg)
as a result of this acceleration?
Copyright © 2009 Pearson Education, Inc.
ConcepTest 23.2 Work and Potential Energy
Which group of charges took more work to bring together
from a very large initial distance apart?
+1
d
+2
+1
d
+1
Both took the same amount of work.
d
d
+1
ConcepTest 23.2 Work and Potential Energy
Which group of charges took more work to bring together
from a very large initial distance apart?
+1
d
+2
d
+1
+1
Both took the same amount of work.
The work needed to assemble
a collection of charges is the
same as the total PE of those
charges:
Q1Q 2
PE  k
r
added over
all pairs
d
d
+1
For case 1: only 1 pair
(2)(1)
2
PE  k
k
d
d
For case 2: there are 3 pairs
(1)(1)
1
PE  3k
 3k
d
d
23-2 Relation between Electric
Potential and Electric Field
The general relationship
between a conservative force
and potential energy:
Substituting the
potential difference
and the electric field:
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23-2 Relation between Electric
Potential and Electric Field
The simplest case is a uniform field:
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23-2 Relation between Electric
Potential and Electric Field
Example 23-3: Electric field
obtained from voltage.
Two parallel plates are
charged to produce a potential
difference of 50 V. If the
separation between the plates
is 0.050 m, calculate the
magnitude of the electric field
in the space between the
plates.
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23-2 Relation between Electric
Potential and Electric Field
Example 23-4:
Charged conducting
sphere.
Determine the
potential at a distance
r from the center of a
uniformly charged
conducting sphere of
radius r0 for (a) r > r0,
(b) r = r0, (c) r < r0. The
total charge on the
sphere is Q.
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23-2 Relation between Electric
Potential and Electric Field
The previous example
gives the electric
potential as a function of
distance from the surface
of a charged conducting
sphere, which is plotted
here, and compared with
the electric field:
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23-3 Electric Potential Due to Point
Charges
To find the electric potential due to a point
charge, we integrate the field along a field line:
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23-3 Electric Potential Due to Point
Charges
Setting the potential to zero at r = ∞ gives the
general form of the potential due to a point
charge:
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23-3 Electric Potential Due to Point
Charges
Example: Work required to bring two positive
charges close together.
What minimum work must be done by an
external force to bring a proton q = 1.60×10-19
C from a great distance away (take r = ∞) to a
point 1.60×10-15 m from another proton?
Copyright © 2009 Pearson Education, Inc.
23-3 Electric Potential Due to Point
Charges
Example: Work required to bring two positive
charges close together.
What minimum work must be done by an
external force to bring a proton q = 1.60×10-19
C from a great distance away (take r = ∞) to a
point 1.60×10-15 m from another proton?
W = ke2/r = (9×109) (1.6×10-19)2/ (1×10-15)
= 2.3×10-13 J = 2.3×10-13 J
Copyright © 2009 Pearson Education, Inc.
ConcepTest 23.3 Electric Potential
1) V > 0
What is the electric
potential at point B?
2) V = 0
3) V < 0
A
B
ConcepTest 23.3 Electric Potential
1) V > 0
What is the electric
potential at point B?
2) V = 0
3) V < 0
Since Q2 and Q1 are equidistant
from point B, and since they have
equal and opposite charges, the
total potential is zero.
Follow-up: What is the potential
at the origin of the x y axes?
A
B
23-3 Electric Potential Due to Point
Charges
Example 23-7: Potential above two charges.
Calculate the electric potential (a) at point A in
the figure due to the two charges shown.
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23-4 Potential Due to Any Charge
Distribution
The potential due to an arbitrary charge
distribution can be expressed as a sum or
integral (if the distribution is continuous):
or
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23-4 Potential Due to Any Charge
Distribution
Example 23-8:
Potential due to a
ring of charge.
A thin circular ring of
radius R has a
uniformly distributed
charge Q. Determine
the electric potential
at a point P on the
axis of the ring a
distance x from its
center.
Copyright © 2009 Pearson Education, Inc.
23-4 Potential Due to Any Charge
Distribution
Example 23-9:
Potential due to a
charged disk.
A thin flat disk, of
radius R0, has a
uniformly
distributed charge
Q. Determine the
potential at a point
P on the axis of the
disk, a distance x
from its center.
Copyright © 2009 Pearson Education, Inc.
ConcepTest 23.5 Equipotential Surfaces
Which of these configurations gives V = 0 at all points on the x axis?
+2mC
+1mC
+2mC
+1mC
x
-1mC
-2mC
+2mC
-2mC
x
-2mC
-1mC
1)
x
2)
4) all of the above
+1mC
-1mC
3)
5) none of the above
ConcepTest 23.5 Equipotential Surfaces
Which of these configurations gives V = 0 at all points on the x axis?
+2mC
+1mC
+2mC
+1mC
x
-1mC
-2mC
+2mC
-2mC
x
-2mC
-1mC
1)
x
2)
4) all of the above
+1mC
-1mC
3)
5) none of the above
Only in case (1), where opposite charges lie
directly across the x axis from each other, do
the potentials from the two charges above the
x axis cancel the ones below the x axis.
23-5 Equipotential Surfaces
An equipotential is a line
or surface over which the
potential is constant.
Electric field lines are
perpendicular to
equipotentials.
The surface of a conductor
is an equipotential
(E// =0 → ∂V/∂s// = 0)
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23-5 Equipotential Surfaces
Equipotential surfaces are always
perpendicular to field lines; they are
always closed surfaces (unlike field lines,
which begin and end on charges).
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23-5 Equipotential Surfaces
A gravitational analogy to equipotential surfaces
is the topographical map – the lines connect
points of equal gravitational potential (altitude).
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23-6 Electric Dipole Potential
The potential due to an
electric dipole is just the sum
of the potentials due to each
charge, and can be calculated
exactly. For distances large
compared to the charge
separation:
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23-7 E Determined from V
If we know the field, we can determine the
potential by integrating. Inverting this
process, if we know the potential, we can
find the field by differentiating:
This is a vector differential equation;
here it is in component form:
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23-7 E Determined from V
Example 23-11: E for ring and disk.
Use electric potential to determine the electric
field at point P on the axis of (a) a circular ring
of charge and (b) a uniformly charged disk.
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23-8 Electrostatic Potential Energy;
the Electron Volt
The potential energy of a charge in an
electric potential is U = qV. To find the electric
potential energy of two charges, imagine
bringing each in from infinitely far away. The
first one takes no work, as there is no field.
To bring in the second one, we must do work
due to the field of the first one; this means
the potential energy of the pair is:
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23-8 Electrostatic Potential Energy;
the Electron Volt
One electron volt (eV) is the energy gained by
an electron moving through a potential
difference of one volt:
1 eV = 1.6 × 10-19 J.
The electron volt is often a much more
convenient unit than the joule for measuring
the energy of individual particles.
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23-8 Electrostatic Potential Energy;
the Electron Volt
Example 23-12: Disassembling a hydrogen
atom.
Calculate the work needed to “disassemble” a
hydrogen atom. Assume that the proton and
electron are initially separated by a distance
equal to the “average” radius of the hydrogen
atom in its ground state, 0.529 × 10-10 m, and
that they end up an infinite distance apart from
each other.
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Back to the two protons
W = ke2/r = (9×109) (1.6×10-19)2/ (1×10-15)
= 2.3×10-13 J = (2.3×10-13 J)/(1.6×10-19J/eV)
= 1.4×106 eV = 1.4 MeV
eV convenient for Atomic/Molecular Physics
MeV convenient for Nuclear Physics
GeV convenient of High Energy Physics
A = Angstrom = 10-10 m for AMO Physics & Chem
fm = Fermi = femtometer = 10-15 m for NP & HEP
o
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ConcepTest 23.7 Work and Electric Potential
1) P  1
Which requires the most work,
to move a positive charge from
P to points 1, 2, 3 or 4 ? All
points are the same distance
from P.
2) P  2
3) P  3
4) P  4
5) all require the same
amount of work
3
2
1
P

E
4
ConcepTest 23.7 Work and Electric Potential
Which requires the most work,
to move a positive charge from
P to points 1, 2, 3 or 4 ? All
points are the same distance
from P.
For path #1, you have to push the
positive charge against the E field,
which is hard to do. By contrast,
path #4 is the easiest, since the
field does all the work.
1) P  1
2) P  2
3) P  3
4) P  4
5) all require the same
amount of work
3
2
1
P

E
4
Summary of Chapter 23
• Electric potential is potential energy per
unit charge:
• Potential difference between two points:
• Potential of a point charge:
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 23
• Equipotential: line or surface along which
potential is the same.
• Electric dipole potential is proportional to 1/r2.
• To find the field from the potential:
Copyright © 2009 Pearson Education, Inc.