Chapter 21
Electric Potential
Topics:
• 
• 
• 
• 
Conservation of energy
Work and Delta PE
Electric potential energy
Electric potential
Sample question:
Shown is the electric potential measured on the surface of a patient.
This potential is caused by electrical signals originating in the beating
heart. Why does the potential have this pattern, and what do these
measurements tell us about the heart s condition?
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Slide 21-1
Chapter 21 Key Energy Equations
Key Energy Equations from Physics 151 and Ch. 21 so far
!
! ! !
Definition of Work Work!!W = F!i!!r = F !r cos "
Where ! = angle between the vectors
Work done by a conservative force (Fg, Fs, & Fe) We = !"U e
Also work done by conservative force is path independent
Conservation of Energy Equation
(can ignore Ug and Us unless they are relevant)
Ki +
!
U i + " Esys = K f +
different !types
!
U f + "Eth
different !types
Electric Energy – Special Cases (Similar equations for gravity)
qq
2 Point Charges U = k 1 2
e
r12
Charge in a
Uniform E-field
!
!
!
!U e = "We = " %& Fe # !r cos $ '( = " q E !r cos $
Note: in both cases of Electric Energy must assume where Ue = 0
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Slide 21-16
Chapter 21 Key Equations (3)
Key Points about Electric Potential
Electric Potential is the Electric Energy per Charge
V=
Ue
!!
qtest
!V =
!U e
W
=" e
qtest
qtest
Electric Potential increases as you approach positive source
charges and decreases as you approach negative source
charges (source charges are the charges generating the electric
field)
A line where Delta V= 0 V is an equipotential line
(The electric force does zero work on a test charge that moves
on an equipotential line and Delta PEe= 0 J)
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Slide 21-16
Electric Potential and E-Field for Three Important Cases
For a point charge
q
1 q
V=K =
r 4!" 0 r
For very large charged plates, must use
!
!
!
!
!
! !
!PEe
We
Fe !i!!r
qtest E!i!!r
!
!V =
="
="
="
= " E!i!!r = " E !r cos #
qtest
qtest
qtest
qtest
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Slide 21-25
The superposition principle and the V field
due to multiple charges
• where Q , Q , Q , … are the source charges
1
2
3
(including their signs) creating the field and r1, r2, r3,
… are the distances between the source charges
and the location where we are determining the V
field.
• So Electric Potentials (V) add just like Electric Potential Energies
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Voting question
Which of the following statements is incorrect?
• The E field is a vector.
• The electric potential is a scalar.
• The E field describes an interaction between
charges.
• The electric potential energy describes an
interaction between charges.
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Quantitative Example
•  Suppose that the heart's dipole charges −Q and +Q are separated
by distance d. Write an expression for the V field due to both
charges at point A, a distance d to the right of the +Q charge.
1.  Simplify and diagram.
2.  Represent mathematically.
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Example Problem
For the situation shown in the figure, find
A.  The potential at points a and b.The potential difference between
a and b.
B.  The potential energy of a proton at a and b.
C.  The speed at point b of a proton that was moving to the right at
point a with a speed of 4.0 x 105 m/s.
D.  The speed at point a of a proton that was moving to the left at
point b with a speed of 4.0 x 105 m/s.
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Slide 21-22
Particles in a potential difference
• A positively charged object accelerates from
regions of higher electric potential toward
regions of lower potential (like an object
falling to lower elevation in Earth's
gravitational field).
• A negatively charged particle tends to do the
opposite, accelerating from regions of lower
potential toward regions of higher potential. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
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Slide 21-15
A Topographic Map
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Slide 21-12
Topographic Maps
1. Describe the region
represented by this map.
2. Describe the directions a
ball would roll if placed at
positions A – D.
3. If a ball were placed
at location D and
another ball were placed
at location C and both were
released,
which would have the greater acceleration?
Which has the greater potential energy when released?
Which will have a greater speed when at the bottom of the hill?
4. What factors does the speed at the bottom of the hill depend on? What factors
does the acceleration of the ball depend on?
5. Is it possible to have a zero acceleration, but a non-zero height? Is it possible
to have a zero height, but a non-zero acceleration?
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Slide 21-16
Equipotential surfaces: Representing the V field
•  The lines represent surfaces of constant electric potential V,
called equipotential surfaces. •  The surfaces are spheres (they look like circles on a twodimensional page).
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Equipotential surfaces and E field
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Contour maps: An analogy for
equipotential surfaces
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Equipotential Maps (Contour Maps)
1. Describe the charges that
could create equipotential lines
such as those shown above.
2. Describe the forces a proton
would feel at locations A and B.
3.  Describe the forces an
electron would feel at locations
A and B 4. Where could an electron be
placed so that it would not
move?
5. At which point is the magnitude of the electric field the greatest?
6. Is it possible to have a zero electric field, but a non-zero electric potential?
7. Is it possible to have a zero electric potential, but a non-zero electric field?
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Slide 21-16
E-field lines and Equipotential lines
E-field Lines
•  Go from + charges to - charges
•  Perpendicular at surface of conductor or charged surface
•  E-field in stronger where E-field lines are closer together
•  More charge means more lines
Equipotential Lines
•  Parallel to conducting surface
•  Perpendicular to E-field lines
•  Near a charged object, that charges influence is greater, then blends as
you to from one to the other
•  E-field is stronger where Equipotential lines are closer together
•  Spacing represents intervals of constant Delta V
•  Higher potential as you approach a positive charge; lower potential as you
approach a negative charge
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Slide 21-16
Connecting Potential and Field
Note: Since the E-field everywhere inside a conductor is 0,
V = constant
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Slide 21-31
Polling Question
The electric field
A. 
B. 
C. 
D. 
is always perpendicular to an equipotential surface.
is always tangent to an equipotential surface.
always bisects an equipotential surface.
makes an angle to an equipotential surface that depends
on the amount of charge.
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Slide 21-12
Answer
4.  The electric field
A. 
B. 
C. 
D. 
is always perpendicular to an equipotential surface.
is always tangent to an equipotential surface.
always bisects an equipotential surface.
makes an angle to an equipotential surface that depends
on the amount of charge.
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Slide 21-13
3D view
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Slide 21-16
Graphical Representations of Electric Potential
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Slide 21-13
Reading Quiz
3.  The electric potential inside a parallel-plate capacitor
A. 
B. 
C. 
D. 
is constant.
increases linearly from the negative to the positive plate.
decreases linearly from the negative to the positive plate.
decreases inversely with distance from the negative
plate.
E.  decreases inversely with the square of the distance from
the negative plate.
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Slide 21-10
Answer
3.  The electric potential inside a parallel-plate 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.
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Slide 21-11
The Potential Inside a Parallel-Plate Capacitor
Uelec
Q
V=
= Ex =
x
q
!0 A
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Slide 21-25
Example Problem
Source charges create the electric potential shown below.
A.  Rank the Electric Fields at
points A, B, C, and D
B.  Rank the Electric Potentials
at points A, B, C, and D
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Slide 21-33
Example Problem
Source charges create the electric
potential shown.
A.  What is the potential at point
A? At which point, A, B, or C,
does the electric field have its
largest magnitude?
B.  Is the magnitude of the electric
field at A greater than, equal
to, or less than at point D?
C.  What is the approximate magnitude of the electric field at
point C?
D.  What is the approximate direction of the electric field at
point C?
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Slide 21-33
Example Problem
A proton is released from rest at point a. It then travels past point
b. What is its speed at point b?
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Slide 21-23
Assembling a square of charges
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Slide 21-16
Analyzing a square of charges
Energy to Assemble
Wme = Delta UE = UEf - UEi
(UEi = 0 J)
UEf = q1Vnc@1 + q2V1@2 + q3V12@3 + q4V123@4
V123@4 = V1@4 +V2@4 + V3@4
Energy to move
(Move 2q from Corner to Center)
Wme = Delta UE = UEf - UEi
= q2qV123@center - q2qV123@corner
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Slide 21-16
Example Problem
A parallel-plate capacitor is held at a potential difference of 250 V.
A proton is fired toward a small hole in the negative plate with a
speed of 3.0 x 105 m/s. What is its speed when it emerges through
the hole in the positive plate? (Hint: The electric potential outside of
a parallel-plate capacitor is zero).
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Slide 21-26
Example Problem
What is Q2?
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Slide 21-35
Deriving a relation between the E field and ΔV
•  We attach a small object with
charge +q to the end of a very thin
wooden stick and place the charged
object and stick in the electric field
produced by the plate.
•  The only energy change is the
system's electric potential energy,
because the positively charged
object moves farther away from the
positively charged plate.
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Deriving a relation between the E field andΔV
•  Applying the generalized work-energy equation, we get:
•  Equivalently, the component of the E field along the line
connecting two points on the x-axis is the negative change
of the V field divided by the distance between those two
points:
Ex = - Delta V / Delta x
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Conceptual Exercise
•  Can you think of locations relative to charge
distributions where:
1.  The V field at a particular location is zero but
the E field is not?
2.  The E field is zero but the V field is not zero?
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Testing the relation between the E field and ΔV
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Electric field of a charged conductor
•  Free electrons in a conductor are quickly redistributed
until equilibrium is reached, at which point the E field
inside the conductor and parallel to its surface becomes
zero.
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Electric field outside a charged conductor
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Grounding
• Grounding discharges an
object made of
conducting material by
connecting it to Earth. • Electrons will move
between and within the
spheres until the V field
on the surfaces of and
within both spheres
achieves the same value.
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Uncharged conductor in an electric field: Shielding
•  The free electrons inside the object become redistributed
due to electric forces, until the E field within the conducting
object is reduced to zero.
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Uncharged conductor in an electric field: Shielding
•  The interior is protected from the external field—an effect called
shielding.
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