Chapter 22 Electric Potential (Voltage)
Question 29.6 Work and Electric Potential II
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2
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P
4
Question 29.5 Work and Electric Potential I
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P
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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 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 (Voltage)
•  Electric potential, or voltage, is defined as the electric
potential energy per unit charge a test particle would have
if it were located at a position
•  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.
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
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.
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.
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)
Voltage of a point charge
•  The point-charge field varies with
position, so we need to integrate:
•  Taking the zero of potential at infinity
and letting
gives
kq
VB = V (�r) =
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)?
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 = −
•  For a continuous charge distribution,
��
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
CT 29.12c
At which labeled point is voltage highest?
B
A
D
E
C
Potential of charged sphere
•  Outside sphere, electric field is identical to that of a point
charge.
•  What is V for (r<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
�
0
2π
kλa dθ
r(θ, x, y, z)
•  Very hard integral in general! If P
is on x axis, then r is independent
of θ.
•  Integrating the potentials of charged rings gives the
potential of a uniformly charged disk:
V (x) =
�
dQ
k
=
r
�
0
a
λ2πr dr
k √
x2 + r2
•  This result reduces to the infinite-sheet potential close to the
disk, and the point-charge potential far from the disk.
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)
Conductors
•  There’s no electric field inside a conductor in
electrostatic equilibrium.
•  And even at the surface there’s no field
component parallel to the surface.
•  Therefore it takes no work to move charge
inside or on the surface of a conductor in
electrostatic equilibrium.
A conductor in electrostatic equilibrium is an
equipotential.
•  The electric field must be perpendicular to the
surface of a conductor (in electrostatic
equilibrium
Determining E from V
•  Voltage can be determined if electric field is known
•  Can electric field be determined if voltage is known?
� · ∆�r
•  For a very small displacement, ∆V = −E
•  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:
Use this to determine E(x):