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
