Analogy between gravitational potential energy near
the Earth, and electrical potential energy near a
charged plate.
8. Electric potential energy
gravitational field g
• analogy with gravitational potential energy
• electrostatic forces are conservative
• potential energy of a collection of
charges
M
_
Earth
path “s” of test charge q0
ri
q0
ds
Q
For the Coulomb force due to a point charge
r r kQ
E ( s ) = 2 rˆ
r
rf
_
Physics 1E03 Lecture 8
sf
W=
∫
_
_
si
2
rf
r
−1 r
1 1
kQq0
dr
rˆ • (dr rˆ + ds|| ) = kQq0 ∫ 2 = kQq0 [ ]rif = kQq0 ( − )
r2
r
r
r
rf
i
ri
motions perpendicular
to force give no work
net work depends
only on initial and
final radii.
Since the work doesn’t depend on the path q0 takes, only its
initial and final positions, electrostatic forces are conservative.
The work done by a conservative force can be written as a
potential energy U, using the conservation of mechanical
energy.
WC = U i − U f
r
r
ds = dr rˆ + ds||
component in
radial direction
_
UE = QEd
1
Recall: the work done by a conservative force on an object
depends only on the initial and final positions of the object,
not on the path the object follows.
step along path
_
The concept of electrostatic potential energy
can be more precisely defined.
Concept 8.1: Because electrostatic forces are
conservative, electric potential energy can be
defined.
si
_
charged plane
Ug= mgd
Physics 1E03 Lecture 8
Force q0E at
each point on path
+Q
d
Serway and Beichner
Sections 25.1, 25.3
sf
r r
r
W = ∫ F ( s ) • ds
electric field E
U=
kQq0
r
potential energy of two point charges
separated by distance r.
It is a scalar, not a vector.
tangential
component
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Quiz of concept 8.1
Two small objects of mass m1 and m2
hold excess charges of +Q1 and
+Q2. They are held a distance L
apart and then released. If the
gravitational force is much less than
the electrostatic force, and can be
neglected, the ratio of their final
velocities v1/v2 is proportional to
L
m1,+Q1
m2,+Q2
Any charge distribution can rbe written as
r a sum over
qi (ri )
individual point charges Q ( r ) =
∑
i
For a test charge q0 near this charge distribution
a) the ratio of masses m2/m1
b) the ratio of charges Q1/Q2
c) the ratio of their initial potential
energies U1/U2
d) the root of the ratio of their initial
potential energies, [U1/U2]1/2
U = kq0 ∑
i
qi
, where ri is the distance between q 0 and q i .
ri
Remember: Potential energy is a scalar, not a vector, so U
is a simple sum of numbers.
Physics 1E03 Lecture 8
Quiz of concept 8.2
Three charges of equal size are held
at the corners of an equilateral
triangle. A test charge placed at
any position has potential energy
U and feels force F. Compared
to placing the test charge at
position A,
Concept 8.2: The electric potential energy of
a test charge near any charge distribution can
be defined.
kQq0
Near a single point charge,
U=
r
5
A
B
6
Concept 8.3: The total electrical potential
energy of a collection of charges is the sum of
the potential energy of each charge in the
presence of all the other charges.
5
+q
+q
Physics 1E03 Lecture 8
-q
If there are N charges,
For the first charge there are N-1
pair interactions giving potential energy.
For the second charge there are N-2
pair interactions giving potential energy,
since one pair has already been counted.
a) |F| and U are larger at point B
b) |F| and U are smaller at point B
c) |F| is larger and U is smaller at
point B
d) |F| is smaller and U is larger at
point B
4
1
3
2
Etc., etc.
Physics 1E03 Lecture 8
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The are ½ N(N-1) pairs to sum over to get the total potential
energy. An easy way to calculate it correctly is to “assemble”
the charges one at a time, bringing them all from infinity where
Physics 1E03 Lecture 8
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the potential energy is zero.
2
Quiz of concept 8.3
The electrostatic potential energy of two objects
with equal excess charges q, which are
separated by distance L, is U.
How much work must be done to
assemble four such charged objects in
a square of side L.
a) 2U
c) 5U
b) 4U
d) (4+21/2)U
Summary
• Electric potential energy is another example of
a potential energy (like spring and gravitational)
• For a test charge q0
L
q
U = kq0 ∑ i , whereri is the distancebetweenq0 and qi .
i ri
L
• Add up the potential energy from each pair of
charges only once.
L
Practice problems: Chapter 25, #19, 27, 32, 33
Next lecture: read sections 25.1-.4, 25.6
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