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ELECTRIC POTENTIAL-ENERGY (U) and the ELECTRIC POTENTIAL (V)
U V
Andres La Rosa Lecture Notes Portland State University PH-212
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
Q
Q
Q
ds is the differential displacement vector
qo
qo
Consider an electric charge Q. The charge creates an electric field in the surrounding regionConsider an electric test charge qo, which is initially located far away (at infinity) from the charge Q
We want to bring qo from "infinity" to a place located at a distance r from Q
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?
Fext
External force
Electrical force
qo
Point charge
Point charge
Point charge
Q
Wext
Wext
qo
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.
Units of work: Joule Unit of U : Joule
r
Magnitude of the vector displacement
Magnitude of the external force
Constant velocity implies: magnitude of magnitude of the electrical force = the external force (Coulomb force )
Point charge
Q qo
Definition of the electric potential V
For the particular case of a point-charge Q, we have:
Electrical energy of this system is:
Notice the greater the charge qo the greater the U
Q
r
r
Electric potential established by the charge Q at the position P (located at a distance r from Q) is:
Point charge
Point charge
Positive
General working procedure to obtain the electric potential:
Checkpoints
POTENTIAL DIFFERENCE
Q
Given a charge Q,
Electric potential at A
Electric potential at B
Definition
Q
Electrical potential difference between the points Bfinal and Ainitial
VB - VA =
Point charge
Point charge
VA =
VB =
q Point
charge
rB
rA
r
εo rB rA
rBB
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.
B
r
Particular path from A to B
E
dr C
The integral renders the same value whether we we choose path I, II, or III
q
The result above indicates that the potential difference VB -VA does not depend on the particular path joining the points A and B.
Relationship between the ELECTRIC POTENTIAL and the ELECTRIC FIELD
Electric-potential difference existent between the points B and A
Q
Arbitrary charge distribution
VB - VA = -
VB - VA =
VB - VA =
E
1 4πε0
1 4πε0
1 4πε0
A
A
A
A
AA
A
A
r1A
r1A r2A r3A
r1A=r2A= r3A But notice
r1A
�
r
V
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