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9/10/12
Example: Electric Potential Energy
Summary
What is the change in electrical potential energy of a released electron in
the atmosphere when the electrostatic force from the near Earth’s
electric field (directed downward) causes the electron to move vertically
upwards through a distance d?
ΔU = −W
2.  Work done by a constant force on a particle
undergoing displacement:
Key Idea:

E

Fe
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
d
1
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V (R) =
kQ
R
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Chapter 23
•  Electrostatic Potential Energy of a system of fixed point
charges is equal to the work that must be done by an
external agent to assemble the system, bringing each
charge in from an infinite distance.
1 N qn
V (r) = ∑ Vn (r) =
∑
4πε o n =1 rn
n =1
q3 =+31 nC
+
+
+
+
Note:
A positively charged particle produces a
positive electric potential.
A negatively charged particle produces
a negative electric potential.
N
+
+
R
r
Find the Potential at the center of the square.
+
+
+
V(r)
kQ
R
Potential due to a Group of Point Charges
q1 = +12 nC
+
kq
r
 
W = F•d


 
F = qE
W = qE • d = −qEd cosθ = −qEd cos180 = qEd
ΔU = −W = −qEd
Electric potential decreases as electron rises.
+
V(r)
3.  Electrostatic Force and Electric Field are
related:
Key Idea:
+
Electric Potential for + charge
1.  U of the electron is related to the work done
on it by the electric field:
Key Idea:
Electric Potential for positively
charged spherical conductor
Point 1
q2 = -24 nC
Point 2
q1
-
+
q3 =+17 nC
3
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Example: Three Point Charges
Example: Electrostatic Potential Energy
Point 1
Point 2
q1
q2
q3
Point 3
r1,3
Point 1
r2,3
q1
q2
Point 2
r1,2
If q1 & q2 have the same sign, U is positive because positive work by an
external agent must be done to push against their mutual repulsion.
If q1 & q2 have opposite signs, U is negative because negative work by an
external agent must be done to work against their mutual attraction.
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Electrostatic Potential Energy
Electric field always points in the direction of
steepest descent of V (steepest slope) and its
magnitude is the slope.
Potential from a Negative
Point Charge
kq2 q1 kq3q1 kq3q2
+
+
r1, 2
r1,3
r2,3
Potential from a Positve
Point Charge
V(r )
x
The electrostatic potential energy of a
system of point charges is the work needed
to bring the charges from an infinite
separation to their final position
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Calculate Electric Field from the Potential
•  We can conclude that the total work
required to assemble the three charges is
the electrostatic potential energy U of
the system of three point charges:
U = Wtotal = W2 + W3 =
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y
y
7
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-V(r )
x
8
2
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Calculating the Electric Field from the Potential
Field
Calculating the Electric Field from the Potential Field
If we can get the potential by integrating the electric field:


⎛ ∂V ˆ ∂V
∂V
E = −∇V = − ⎜
i+
ĵ +
⎝ ∂x
∂y
∂z
In the direction of
steepest descent
∂V
∂V
∂V
, Ey = −
, and Ez = −
∂x
∂y
∂z
Ex = −
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Example: Calculating the Electric Field from the
Potential Field
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Potential due to a Group of Point Charges
r=r
What is the electric field at any point on the central
axis of a uniformly charged disk given the potential?
Given: V =
⎞
k̂ ⎟
⎠
V (r) = −
∫
 
E • dl
r=∞
σ
( z 2 + R 2 − z)
2ε 0
r1
q1
X
r2
q2
Ex =
Ey =
q3
r3
r4
q4
Ez =
N
V (r) = ∑ Vn (r) =
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n =1
1 N qn
∑
4πε o n =1 rn
12
3
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Charge Densities
Potential from a Continuous Charge Distribution
total charge
Q
small pieces
of charge
dq
Line
of
charge:
 = charge per unit length [C/m]
dq
=
 dx
Surface of charge:  = charge per unit area [C/m2]
dq =
dA
Cylinder: dq = σ rdθ dz
Sphere:
dq = σ r 2 sin θ dθ dφ
Volume of Charge:  = charge per unit volume [C/m3]
dq = dV
Cylinder: dq = ρrdrdθ dz
Sphere:
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Calculate Potential on the central axis of a
charged ring
dq = ρr 2 dr sin θ dθ dφ
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Calculate Potential on the central axis of a
charged disk
dq = σ A = σ 2π a da
V = ∫k
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dq
r
#
16
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Calculate Potential on the central axis of a
charged disk (another way)
Calculate Potential due to an infinite sheet
 
V = − ∫ E • dl
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
From Lecture 3:
⎤
σ ⎡
z
Ez =
⎢1 − 2
⎥
2
2ε o ⎣
z +R ⎦
#
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 
dV = − E • dl
dV = −2π kσ iˆ • (dx iˆ + dx ĵ + dx k̂)
x
V = − ∫ 2π kσ dx = −2π kσ x + C 0
x
0
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Equipotentials
E due to an infinite line charge
Definition: locus of points with the same potential.
• General Property: The electric field is always
perpendicular to an equipotential surface.
Corona discharge around a high voltage power line,
which roughly indicates the electric field lines.
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Equipotentials: Examples
Point charge
q
V (r) = k
r
infinite positive
charge sheet
Equipotential Lines on a Metal Surface
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
electric dipole
–
–
–
–
–
–
–
–
–
–
–
–
–
Locally E⊥ =
σ
ε0
Gauss: E|| = 0
at electrostatic
equilibrium
in electrostatic equilibrium
all of this metal is an equipotential;
i.e., it is all at the same voltage
V (x) = −2π kσ x + Vo
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Summary
Potential inside & outside a conducting sphere
•  If you know the functional behavior of the potential V
at any point, you can calculate the electric field.
•  The electric potential for a continuous charge
distribution can be calculated by breaking the
distribution into tiny pieces of dq and then integrating
over the whole distribution.
•  Finally no work needs to be done if you move a charge
on an equipotential, since it would be moving
perpendicular to the electric field.
Vref = 0
at r = ∞.
•  The charge concentrates on a conductor on surfaces
with smallest radius of curvature.
The electric field is zero inside a conductor.
The electric potential is constant inside a conductor.
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