Chapter 22 Electric Potential (Voltage)
Question 29.5 Work and Electric Potential I
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
Electric potential energy
•  Recall how a conservative force is related to the
potential energy associated with that force:
•  The electric potential energy is the potential energy due
to the electric force, which can be expressed in terms
of the electric field.
•  If location A is chosen to be the zero point, then the
electric potential energy at location B (which we now
call r) is given by
Potential energy of particle
is a scalar function of
space.
Consider uniform electric field (say inside a parallel capacitor)
If a proton is taken from location B to location C, how
does its potential energy change?
1.  it decreases
2.  it increases
3.  it doesn’t change
Suppose a proton is released from rest just below the
top (positive) plate of an parallel plate capacitor with
an electric field strength E = 100 N/C. If the distance
between the plates is d = 3 mm, how fast is it moving
when it hits the bottom (negative) plate?
Electric Potential energy of two Point
Charges
•  What is the change in potential
energy of the test charge as it goes
from position a to position b?
� B
�
� · dr
∆UAB = −
q0 E
A
∆UAB = −
�
rb
ra
kq
q0 2 dr
r
kq0 q rb
q0 q
q0 q
∆UAB =
| ra = k
−k
r
rb
ra
•  If we let ra be infinity (the zero point),
and rb an arbitrary distance, then
q0 q
U (r) = k
r
Example
Rutherford scattering. A helium nucleus of mass 4 mp is
emitted with an initial speed of v0 = 4.9 x 105 m/s
towards a gold nucleus of charge q2 = 79 e. What is
the minimum distance between the two particles (assume
the gold nucleus doesn’t move)?
Electric Potential (Voltage)
•  Electric potential, or voltage, at a point in space is
defined as the electric potential energy per unit charge
associated with a test charge at that point.
U (�r)
V (�r) =
q
•  Potential energy deals with the energy of a particle. Voltage
deals with all locations in space (no particle needs to be there).
•  Analogous to how a particle experiences a force, but an electric
field can exist at any point in space.
•  Unit of electric potential is the volt (V). 1 V = 1 J/C.
Potential Difference (Voltage Difference)
•  Voltage difference is defined as
•  Because the electrostatic field
is conservative, it doesn’t
matter what path is taken
between those points.
•  In a uniform field, the potential
difference becomes
� · ∆r
�
∆VAB = −E
Clicker Question
In a parallel plate capacitor,
the electric field is uniform
and is directed from the
positive plate to the
negative plate.
An electron goes from
location A to location C.
Which statement is true?
A)  The electron goes from a high
voltage to a lower voltage.
B)  The electron goes from a low
voltage to a higher voltage.
C)  The voltage is the same at both
locations.
Clicker question
•
The figure shows three straight paths AB of the same
length, each in a different electric field. Which one of
the three has the largest magnitude of a voltage
difference between the two points?
A.  (a)
B.  (b)
C.  (c)
Millikan’s Oil Drop Experiment
•  Charged oil droplets made to levitate inside capacitor
•  Measure voltage difference across plates
•  Release and measure terminal velocity (which gives droplet
radius/mass)
•  Determine net charge on droplet.
Voltage of a point charge
•  Recall the potential energy of two
point charges:
q0 q
U (r) = k
r
•  Thus the voltage a distance r from
the charge q is given by
q
V (r) = k
r
•  (there is no test charge anymore)
Voltage due to a charge distribution
•  If the electric field of the charge distribution is known,
the voltage can be found by integration.
•  Alternatively, the voltage can be found by summing
point-charge potentials:
•  For discrete point charges,
V (P ) = −
�
� · d�r = −
E
� ��
i
�i
E
�
· d�r = −
kqi
V P =∑
ri
i
•  For a continuous charge distribution,
()
k dq
V (P) = ∫
.
r
��
i
� i · d�r =
E
�
i
Vi (P )
Clicker Question
Two identical positive charges of charge Q are a distance d
apart. What is the voltage at the midway point between
the charges?
a)
b)
c)
d)
e)
k Q/d
2 k Q/d
4 k Q/d
8 k Q/d
0
Clicker question
Location P is equidistant from the two charges of an electric
dipole. The voltage at P is
a)  positive
b)  zero
c)  negative
Question 29.10 Hollywood Square
Four point charges are
arranged at the corners of
a square. Find the
electric field E and the
potential V at the center
of the square.
1) E = 0
V=0
2) E = 0
V≠0
3) E ≠ 0
V≠0
4) E ≠ 0
V=0
5) E = V regardless of the value
-Q
+Q
-Q
+Q
Clicker Question
•  A solid sphere of radius R has a UNIFORM charge
density per unit volume ρ and net charge Q. The voltage
at the center of the sphere is
1.  V = k Q/R
2.  V < kQ/R
3.  V > kQ/R
Maximum voltage of a Van de Graaff
generator.
•  Molecules in air get ionized for electric fields greater than
roughly Emax = 3 x 106 V/m. What is the maximum
voltage of a charged sphere of radius R=0.2 m?
Voltage due to a charged ring
•  For a uniformly charged ring of
total charge Q, integration gives
the potential
� on the ring axis:
V =
k dq
r
V (x, y, z) =
dq = λadθ
�
2π
0
kλa dθ
r(θ, x, y, z)
•  Very hard integral in general! If P
is on x axis, then r only depends on
x and a.
On x axis:
V ( x) = ∫
k dq k
= ∫ dq =
r
r
kQ
x2 + a2
Voltage due to a long charged wire
•  Find the voltage a distance r from a very long line of
charge with linear charge density λ and radius R
•  since this is an infinitely extended object, we can’t use infinity
as a zero point. Instead, let’s say V(r=R) = 0.
Equipotentials
•  An equipotential is a surface on which the potential
(voltage) is constant.
•  In two-dimensional drawings, we
represent equipotentials by curves similar
to the contours of height on a map.
•  The electric field is always perpendicular to the
equipotentials. (∆V = −E
� · ∆�s = 0)
Clicker question
•
The figure shows cross sections through two
equipotential surfaces. In both diagrams the potential
difference between adjacent equipotentials is the same.
Which of these two could represent the field of a point
charge?
A.  (a)
B.  (b)
C.  neither (a) nor (b)
Conductors
•  There’s no electric field inside a
conductor in electrostatic equilibrium.
•  At the surface there’s no parallel
component of the electric field.
•  Therefore in electrostatic equilibrium,
the entire conductor is at the same
potential
Determining E from V
•  Voltage can be determined if electric field is known
•  Can electric field be determined if voltage
� is known?
� · d�r = −E
� · ∆�r
E
•  For a very small displacement, ∆V = −
•  Suppose
∆�r = ∆x î
� · ∆�r = Ex ∆x
Then E
∆V
∂V
Ex = −
=−
∆x
∂x
Can do the same thing in other direction:
� = −∇V = −
E
�
∂V
∂V
∂V
î +
ĵ +
k̂
∂x
∂y
∂z
�
The derivatives here are partial derivatives,
expressing the variation with respect to one
variable alone.
(gradient of V)
•  For which region is the magnitude of the electric field the
highest?
1.
2.
3.
4.
200 V
180 V
A
160 V
140 V
B
120 V
9
Distance (cm)
8
100 V
7
C
6
5
4
80 V
D
3
2
1
1
2
3 4 5 6
Distance (cm)
7
8
9
10
A
B
C
D
CT 29.13b
What is the approximate magnitude of the
electric field at point A?
(Each equipotential line is 2 m from the nearestneighbor equipotential.)
A) 0.1 Volts/m
B) 0.2 Volts/m
C) 1.6 Volts/m
D) 0.7 Volts/m
E) None of these
A
0V
-1.4V
-1.8V
-2.1V
Example: Electric field along axis of charged
Ring:
•  Recall that the voltage due to a charged ring is:
V ( x) = ∫
k dq k
= ∫ dq =
r
r
kQ
x2 + a2
Use this to determine E(x):