Andres La Rosa
Portland State University
Lecture Notes
PH-212
ELECTRIC POTENTIAL-ENERGY (U)
and
the ELECTRIC POTENTIAL (V)
U
V
in units of Joules
in units of
Joule/Coulomb = Volt
Electric potential difference
between two points A and B:
VA - VB
Interpretation of the
electric potential-energy
Consider an electric charge Q. The charge creates an electric field in the
surrounding region
Consider an electric test charge qo, which is initially located far away (at
infinity) from the charge Q
qo
Q
Point
charge
We want to bring qo from "infinity" to a place located at a distance r from Q
Q
Point
qo
charge
Question:
How much work does an external force Fext have to do to bring qo from
infinity to a place located at a distance r from Q , at constant velocity?
ds is the differential
displacement vector
qo
Q
Point
charge
Fext
External
force
Electrical
force
Constant velocity implies:
magnitude of
the electrical force
(Coulomb force )
=
qo
Q
Point
charge
magnitude of
the external force
r
Wext
Magnitude
of the
external
force
Wext
=
Magnitude of the
vector
displacement
=U
Definition of the ELECTRIC POTENTIAL
ENERGY U of a system formed by the
point-charges Q and qo
Interpretation: U=U(r) is the energy deposited by
the external agent into the system formed by Q
and qo , in order to place these two charges
(initially separated by an infinite distance) to a
distance r from each other.
Units of work: Joule
Unit of U :
Joule
Electrical potential energy of this
system is:
qo
Q
Point
U=
r
charge
Notice the greater the charge qo the greater the value of U
Definition of the
electric potential V
=
For the particular case of a point-charge Q,
we have:
Q
Point
charge
r
Electric potential established by
the charge Q at the position P
(located at a distance r from Q)
is:
General working procedure to obtain the electric potential:
Checkpoints
Q
qo
qo
Positive
+
+
-
POTENTIAL DIFFERENCE
Given a charge Q,
The electric potential at A is calculated as follows,
Q
Point
charge
VA =
Electric potential at B is calculated as follows,
B
Q
Point
charge
rB
VB =
Definition
VB - VA =
Electrical potential difference between the points Bfinal
and Ainitial
In the example above, the path joining the points A and B was along the
radial direction (with center at the charge q).
It turns out that, for arbitrary locations of the points A and B, the potential
difference VB -VA does not depend on the particular path that joins A and B.
This is shown below.
Particular path
from A to B
rB
E
r
q
Point
charge
dr
rA
εo rB
rA
C
The result above indicates that the potential difference VB -VA does
not depend on the particular path joining the points A and B.
q
The integral renders the
same value whether we
we choose path
I, II, or III
Relationship between the ELECTRIC
POTENTIAL and the ELECTRIC FIELD
Q
Arbitrary charge distribution
VB - VA =
VB - VA =
VB - VA = -
E
Electric-potential
difference existent
between the points B
and A
Electric potential energy U
stored in a system composed by several
point-charges
Exercise
1
4πε0
1
4πε0
1
4πε0
Electric potential V
established by a system composed of several
point-charges
qo
A
Electric potential V
established by a system composed of several
point-charges
A
A
A
Example: Electric potential V at a point located
along the axis perpendicular to a charged
ring
A
A
A
V(z) = (1/ 4πεo) Q / ( R2 + z2)1/2
Alternative calculation (without using integrals)
A
r1A
r1A
But notice
r2A
r3A
r1A=r2A= r3A
r1A
V(z) = (1/ 4πεo) Q / ( R2 + z2)1/2
Example: Electric potential V at a point located
along the axis perpendicular to a
uniformly charged disk
The total electric potential is obtained by adding the potentials
established by all the individual strips of radius r, from r=0 to r=R
Electric potential produced by the
disk at the coordinate z along a
perpendicular axis that passes
through the center point of the disk
Electric potential energy:
Electric potential:
U
V
Joules
First, let's recall that,
W=
Work done by the force F
W =
W =
W = Kf - Ki
Work done by the
force F
In this chapter, the force is of
electrical origin
Relationship between
Work done by the electrical (Coulomb)
conservative force
We also know that:
Work done by a external force
where
ΔWext = -ΔW
ΔWext = ΔU
ΔU is the change in potential energy.
From the last two expressions we obtain,
ΔW = - ΔU
Since V = U/qo, we obtain
Example
Answer: VP = 4.5 x 105 Volts
To answer this question, let's place a (positive) test
charge at P and let's find out how to move it around in
such a way that the external work (done by us) is zero.
VP = 4.5 x 105 Volts
dR
qo
Notice, if the displacement
dR is along the
circumference of radius 2
cm, then
ΔV = - E .dR = 0
Q
Hence, there is not change in
the electric potential along
the circumference of radius 2
cm
VP =VQ
+
+
+
+
+
+
+
-
V
r