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
Magnitude of the
vector
displacement
Energy deposited by the external
agent into the system formed by Q
and qo , in order to place these two
charges a distance r from each
other.
Wext
Units of work: Joule
Unit of U :
Joule
Electrical energy of this
system is:
qo
Q
Point
r
charge
Notice the greater the charge qo the greater the 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
Positive
POTENTIAL DIFFERENCE
Given a charge Q,
Electric potential at A
Q
Point
charge
VA =
Electric potential at B
Q
Point
charge
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
1
4πε0
1
4πε0
1
4πε0
A
A
A
A
A
A
A
A
r1A
r1A
But notice
r1A
r2A
r1A=r2A= r3A
r3A
V
r