Fall 2008Physics 231 Lecture 3-1
Electric Potential Energy
and
Electric Potential
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Lecture 3-
Electric Potential Energy
and
Electric Potential
Lecture 3-
Energy Considerations
When a force, F, acts on a particle, work is done on the
particle in moving from point
a^
to point
b
∫^
→
b a b a^
ld
F
W
r
r
If the force is a conservative, then the work done can be expressed in terms of a change in potential energy
(^
)^
U
U
U
W
a b
b a^
→
Also if the force is conservative, the total energy of the particle remains
constant
b
b
a
a^
PE
KE
PE
KE
Lecture 3-
Electric Potential Energy
The work done by the force is the same asthe change in the particle’s potential energy
(^
)^
U
U
U
W
a
b
b a^
→
(^
) a
b
uniform
b a
a
b^
y
y
qE
sd
F
U
U
−^
∫^
r
r
The work done only depends upon the change
in position
Lecture 3-
Electric Potential Energy
General Points
1) Potential Energy
increases
if the particle
moves in the direction
opposite
to the force on it
Work will have to be done by anexternal agent for this to occur
and
2) Potential Energy
decreases
if the particle
moves in the
same
direction as the force on it
Lecture 3-
Potential Energy of Two Point Charges
The work done is notdependent upon the pathtaken in getting frompoint
a
to point
b r d F
r r^ ⋅
The work done is related tothe component of the forcealong the displacement
Fall 2008
Lecture 3-
Potential Energy
Looking at the work done we notice thatthere is the same
functional
at points a and
b and that we are taking the difference
⎞ ⎟⎟ ⎠
⎛^ ⎜⎜ ⎝
−
= →
b a
ba
r r qq
W^
1 1 4
(^0) ε^0 π
We define this functional to be the potential energy
qq r
U^
0 0
επ
The signs of the charges areincluded in the calculation The potential energy is taken to be zero when the twocharges are infinitely separated
Fall 2008
Lecture 3-
Case 1: Potential Energy of one charge
with respect to others
Given several charges, q
…q 1
, in placen
Now a test charge, q
, is brought into 0
position Work must be done against theelectric fields of the original charges This work goes into the potential energy of q
0
We calculate the potential energy of q
with respect to each of 0
the other charges and thenJust sum the individual potential energies
=∑^ i^
i i
q^
qq r
PE
0 0 (^14)
0
επ
Remember - Potential Energy is a Scalar
Lecture 3-
Case 2: Potential Energy of a System of Charges^ Start by putting first charge in position
No work is necessary to do this Next bring second charge into place
Now work is done by the electric field of the firstcharge. This work goes into the potential energybetween these two charges. Now the third charge is put into place
Work is done by the electric fields of the two previouscharges. There are two potential energy terms for thisstep. We continue in this manner until all the charges are in place
∑ =^
<^ ji
ji ji
system
qq r
PE
0 4
1 επ
The total potential is thengiven by
Fall 2008
Lecture 3- Example 2
Two test charges are brought separately to thevicinity of a positive charge
Q
q A r Q
B
Q
2 q
2 r
Charge +
q^ is brought to pt A, a
distance
r^
from
Q
Charge +
q^ is brought to pt B,
a distance 2
r^ from
Q
(a)
U
A U
B
I) Compare the potential energy of
q^
( U
) to that of 2A
q^ (
U B
)
The potential energy of
q^
is proportional to
/ r
The potential energy of 2
q^ is proportional to
Q
(2 q
)/(
r ) =
/ r
Therefore, the potential energies
U
andA
U
are EQUAL!!!B
Fall 2008
Lecture 3- Example 3
(a)
(b)
(c)
II) Suppose charge 2
q^ has mass
m
and is released from rest
from the above position (a distance 2
r^ from
Q
). What is its
velocity
v f
as it approaches
r^
=^ ∞
?
Qq mr
v^ f
0 (^14) πε =^
Qqmr
v^ f
0 (^12) πε =^
(^0) = vf
The principle at work here is
CONSERVATION OF ENERGY
.
Initially:
The charge has no kinetic energy since it is at rest.The charge does have potential energy (electric) =
U
.B
Finally:
The charge has no potential energy (
U^
∝^
1/ R
)
The charge does have kinetic energy =
KE
KE UB
=^
2
0
1 2 2
) (^2) ( (^14)
f mv q r Q^
=
πε
Qqmr
v^ f
0
2
(^1) = 2 πε
Lecture 3-
Electric Potential
We define the term to the right of the summation asthe electric potential at point
a ∑
i^
i i
a^
q r
Potential
0 4
1 επ
Like energy, potential is a
scalar
We define the potential of a given point charge asbeing
q r
V
Potential
0 4
=
This equation has the convention that the potentialis zero at infinite distance
Lecture 3-
Electric Potential
The potential at a given point
Represents the potential energy that a positiveunit charge would have, if it were placed at thatpoint
It has units of
joules coulomb
=^
Energycharge
Volts
Lecture 3-
Example 4
E
A B^
C
Points A, B, and C lie ina uniform electric field. What is the potential difference between points A and B? Δ
V AB
= V
- V B
A
a)^
Δ
V AB
>^ 0
b)
Δ
V AB
=^ 0
c)^
Δ
V AB
<
0 The electric field,
E
, points in the direction of decreasing
potential Since points A and B are in the same relative horizontallocation in the electric field there is on potential differencebetween them
Lecture 3-
Example 5
E
A B^
C
Points A, B, and C lie ina uniform electric field. Point C is at a higher potential than point A.
True
False
As stated previously the electric field points in the direction of decreasing
potential
Since point C is further to the right in the electric field andthe electric field is pointing to the right, point C is at a lowerpotential The statement is therefore false