Physics 2112
Unit 6: Electric Potential
Today’s Concept:
Electric Potential
(Defined in terms of Path Integral of Electric Field)
Unit 6, Slide 1
Big Idea
Last time we defined the electric potential energy of charge q in an electric field:
b
 
 
   F  dl    qE  dl
b
U ab
a
a
The only mention of the particle was through its charge q.
We can obtain a new quantity, the electric potential, which is a PROPERTY OF
THE SPACE, as the potential energy per unit charge.
 
U ab

   E  dl
q
a
b
Vab
Note the similarity to the definition of another quantity which is also a PROPERTY

OF THE SPACE, the electric field.
 F
E
q
Unit 6, Slide 2
Example 6.2a (Potential from Field)
A +8uC charge is placed in
| E | = 6 N/C a downwards pointing
electric field with a
magnitude of 6N/C. An
outside force moves the
q=+8uC
charge up a distance of
0.4m from point 1 to point 2.
a) What is the force on this charge?
b) How much work was done by the outside force during
this move?
Example 6.2.b (Potential from Field)
A +8uC charge is placed in
| E | = 6 N/C a downwards pointing
electric field with a
magnitude of 6N/C. An
outside force moves the
q=+8uC
charge up a distance of
0.4m from point 1 to point 2.
c) How much work was done by the electric field during
this move?
d) What is the change in the electrical potential energy
of the particle?
Unit 6, Slide 4
Example 6.2.c (Potential from Field)
A +8uC charge is placed in
| E | = 6 N/C a downwards pointing
electric field with a
magnitude of 6N/C. An
outside force moves the
q=+8uC
charge up a distance of
0.4m from point 1 to point 2.
e) What is the difference in electrical potential between
points 1 and 2?
f) What is the electrical potential at point 2?
Unit 6, Slide 5
Electric Potential from E field
Consider the three points A, B, and C located in a region of constant electric
field as shown.
D
Dx
What is the sign of VAC  VC  VA ?
A) VAC < 0
B) VAC  0
C) VAC > 0
Choose a path (any will do!)
  C 
  E  dl   E  dl
D
VAC
A
D
 
 0   E  dl   Ex < 0
C
VAC
D
Unit 6, Slide 6
CheckPoint: Zero Electric Field
Suppose the electric field is zero in a certain region of space. Which of the following
statements best describes the electric potential in this region?
A. The electric potential is zero everywhere in this region.
B. The electric potential is zero at least one point in this region.
C. The electric potential is constant everywhere in this region.
D. There is not enough information given to distinguish which of the above
answers is correct.
Remember the definition
 
  E  dl
B
VAB
A
Electricity & Magnetism Lecture 6, Slide 7
Example 6.2 (V from point charges)
+5nc
B
A
+Q
10cm
-5nc
10cm
Q
x
10cm
What is the electrical potential at
points A and B?
Define V = 0 at r =
(standard)

Unit 6, Slide 8
Example 6.3 (V from point charges)
C
+5nc
-5nc
10cm
Q
+Q
10cm
x
20cm
What is the electrical potential at
point C?
Define V = 0 at r =
(standard)

Unit 6, Slide 9
E from V
If we can get the potential by integrating the electric field:
 
   E  dl
b
Va b
a
We should be able to get the electric field by differentiating the potential?


E  V
In Cartesian coordinates:
Ex  
V
dx
Ey  
V
dy
V
Ez  
dz
Electricity & Magnetism Lecture 6, Slide 10
Example 6.4 (E-field above a ring of charge)
What is the electrical
potential, V, a distance y
above the center of ring
of uniform charge Q and
radius a? (Assume V = 0
at y =
)
P
y
x

y
a
What is the electrical
field, E, at that point?
Unit 2, Slide 11
CheckPoint: Spatial Dependence of Potential 1
The electric potential in a certain region is plotted in the following graph
At which point is the magnitude of the
E-FIELD greatest?
A. A
B. B
C. C
D. D
Electricity & Magnetism Lecture 6, Slide 12
CheckPoint: Spatial Dependence of Potential 2
The electric potential in a certain region is plotted in the following graph
At which point is the direction of the Efield along the negative x-axis?
A. A
B. B
C. C
D. D
Electricity & Magnetism Lecture 6, Slide 13
Example 6.5 (V near line of charge)
An infinitely long solid insulating cylinder of radius a = 4.1 cm
is positioned with its symmetry axis along the z-axis as
shown. The cylinder is uniformly charged with a charge
density ρ = 27.0 μC/m3. Concentric with the cylinder is a
cylindrical conducting shell of inner radius b = 19.9 cm, and
outer radius c = 21.9 cm. The conducting shell has a linear
charge density λ = -0.36μC/m.
What is V(P) – V(R), the potential
difference between points P and R?
Point P is located at
(x,y) = (46.0 cm, 46.0 cm).
Point R is located at
(x,y) = (0 cm, 46.0 cm).
Unit 2, Slide 14
Example 6.5 (V for charges)
cross-section
a4
a3
+Q
a2
a1
+q
Point charge q at center of concentric
conducting spherical shells of radii a1, a2, a3, and
a4. The inner shell is uncharged, but the outer
shell carries charge Q.
What is V as a function of r?
metal
metal
Main Idea:
 Charges q and Q will create an E field throughout space
 
 V (r )   E  d 

r
r0
Plan:
 Spherical symmetry: Use Gauss’ Law to calculate E everywhere
 Integrate E to get V
Electricity & Magnetism Lecture 6, Slide 15
Equipotentials
In previous example,
all these points had
same V, same
electrical potential
Line is called “equipotenial”
or “a line of equipotential”.
Unit 2, Slide 16
Topographic Map
Lines on topo map
are lines of “equal
gravitational
potential”
Closer the lines are,
the steeper the hill
Gravity does no
work when you walk
along a brown line.
Equipotentials
Equipotentials produced
by a point charge
V  0
Welec  0
 
 E  dl  0
Equipotentials always perpendicular to field lines.
SPACING of the equipotentials indicates STRENGTH of the E field.
Electricity & Magnetism Lecture 6, Slide 18
CheckPoint: Electric Field Lines 1
The field-line representation of the E-field in a certain region in space is shown below.
The dashed lines represent equipotential lines.
At which point in space is the E-field the
weakest?
A. A
B. B
C. C
D. D
Electricity & Magnetism Lecture 6, Slide 19
CheckPoint: Electric Field Lines 2
The field-line representation of the E-field in a certain region in space is shown below.
The dashed lines represent equipotential lines.
Compare the work done moving a negative charge
from A to B and from C to D. Which one requires
more work?
A. More work is required to move a negative
charge from A to B than from C to D
B. More work is required to move a negative
charge from C to D than from A to B
C. The same amount of work is required to
move a negative charge from A to B as to
move it from C to D
D. Cannot determine without performing the
calculation
Electricity & Magnetism Lecture 6, Slide 20