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Chapter 21
Electric Potential
Topics:
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• 
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• 
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Conservation of energy
Work and Delta PE
Electric potential energy
Electric potential
Contour Maps
E-Field and Equipotential
Conductors & Fields
Capacitance
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 => Wext = - We
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: The angle is between electric force and the displacement
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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 Ue= 0 J)
For multiple source charges
VPOI = V1@POI + V2@POI + …
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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
!
!
!
!
!
! !
!U e
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
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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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
3D view
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Slide 21-16
Graphical Representations of Electric Potential
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Slide 21-13
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
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Slide 21-31
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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Connecting Potential and Field
E = Delta V / d
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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
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