1. Chapter 22: Electric Potential
Electric Potential Energy
Review of work, potential and kinetic energy
Consider a force acts on a particle moving from point a to point b.
The work done by the force WAB is given by:
If the force is conservative, namely when the work done by the force
depends only on the initial and final position of the particle but not on
the path taken along the particle’s path, the work done by the force F
can always be expressed in terms of a potential energy U.

2. Electric Potential Energy
Review of work, potential and kinetic energy
In case of a conservative force, the work done by the force can be
expressed in terms of a potential energy U:
The change in kinetic energy DK of a particle during any displacement is
equal to the total work done on the particle:
If the force is conservative, then

3. Electric Potential Energy
Electric potential energy in a uniform field
Consider a pair of charged parallel metal plates that generate a uniform
downward electric field E and a test charge q0 >0 A + + + + + + + + + + d
conservative
force - - - - - - - - - - B
the force is in the same direction as the
net displacement of the test charge
In general a force is a vector: m g Fg
Note that this force is similar to the force due to gravity:

4. Electric Potential Energy
Electric potential energy in a uniform field (cont’d)
In analogy to the gravitational force, a potential can be defined as:
When the test charge moves from height ya to height yb , the work done
on the charge by the field is given by:
U increases (decreases) if the test charge moves in the direction
opposite to (the same direction as) the electric force
DUAB <0
DUAB>0
DUAB >0
DUAB<0 A B A B + + - - + B + A - B - A

5. Electric Potential Energy
Electric potential energy of two point charges
The force on the test charge at a distance r b rb
The work done on the test charge q0 r a ra + q

6. Electric Potential Energy
Electric potential energy of two point charges (cont’d)
In more general situation
tangent to the path B r A
Natural and consistent definition of the
electric potential

7. Electric Potential Energy
Electric potential energy of two point charges (cont’d)
Definition of the electric potential energy
Reference point of the electric potential energy
Potential energy is always defined relative to a reference point
where U=0. When r goes to infinity, U goes to zero. Therefore
r= is the reference point. This means U represents the work
to move the test charge from an initial distance r to infinity.
If q and q0 have the same sign, this work is POSITIVE ; otherwise
it is NEGATIVE. U U
qq0>0
qq0<0

8. Electric Potential Energy
Electric potential energy with several point charges
A test charge placed in electric field by several particles
Electric potential energy to assemble particles in a configuration

9. Electric Potential - + + q1=-e q2=+e q3=+e x=0 x=a x=2a
Example : A system of point charges
q1=-e
q2=+e
q3=+e - + +
x=0
x=a
x=2a
Work done to take q3 from x=2a to x=infinity
Work done to take q1,q2 and q3 to infinity

10. Electric Potential Energy
Two interpretations of electric potential energy
Work done by the electric field on a charged particle moving in the field
Work done by the electric force when the particle moves from A to B
Work needed by an external force to move a charged particle
slowly from the initial to the final position against the electric force
Work done by the external force when the particle moves from B to A

11. Electric Potential
Electric potential or potential
Electric potential V is potential energy per unit charge
1 V = 1 volt = 1 J/C = 1 joule/coulomb
potential of A with respect to B
work done by the electric force
when a unit charge moves from
A to B
work needed to move a unit
charge slowly from b to a
against the electric force

12. Electric Potential Electric potential due to a single point charge
Electric potential or potential (cont’d)
Electric potential due to a single point charge
Electric potential due to a collection of point charges
Electric potential due to a continuous distribution of charge

13. Electric Potential
From E to V
Sometimes it is easier to calculate the potential from the known
electric field
The unit of electric field can be expressed as:
1 V/m = 1 volt/meter = 1 N/C = 1 newton / coulomb

14. Electric Potential
Example :
Replace R with r

15. Electric Potential
Example: q1
m, q0 q2 + - A B
= 0

16. Electric Potential
Unit: electron volt (useful in atomic & nuclear physics)
Consider a particle with charge q moves from a point where the potential
is VA to a point where it is VB , the change in the potential energy U is:
If the charge q equals the magnitude e of the electron charge
1.602 x C and the potential difference VAB= 1 V, the change
in energy is:
meV, keV, MeV, GeV, TeV,…

17. Calculating Electric Potential
Example: A charged conducting sphere
Using Gauss’s law we calculated the electric
field.
Now we use this result to calculate the potential
and we take V=0 at infinity. + + + + + R + + + E
the same as the potential
due to a point charge r V
inside of the conductor
E is zero. So the potential
stays constant and is
the same as at the surface r

18. Equipotential Surface
An equipotential surface is a 3-d surface on which the electric potential V
is the same at every point
No point can be at two different potentials, so equipotential surfaces for
different potentials can never touch or intersect
Because potential energy does not change as a test charge moves over an
equipotential surface, the electric field can do no work
E is perpendicular to the surface at every point
Field lines and equipotential surfaces are always mutually perpendicular

19. Equipotential Surface
Examples of equipotential surface

20. Equipotential Surface
Equipotentials and conductors
E = 0 everywhere inside a conductor
- At any point just inside the conductor the component of E tangent to
the surface is zero
- The tangential component of E is also zero just outside the surface
If it were not, a charge could move around a
rectangular path partly inside and partly outside
and return to its starting point with a net amount
of work done on it.
vacuum
conductor
When all charges are at rest, the electric field just outside a conductor must
be perpendicular to the surface at every point
When all charges are at rest, the surface of a conductor is always an
equipotential surface

21. Equipotential Surface
Equipotentials and conductors (cont’d)
Consider a conductor with a cavity without any charge inside the cavity
- The conducting cavity surface is an equipotential surface A
Take point P in the cavity at a different potential and it is on a
different equipotential surface B
The field goes from surface B to A or A to B
Draw a Gaussian surface which surrounds the surface B inside
cavity
Guassian surface
equipotential
surface through P B A P
The net flux that goes through this Gaussian surface is not zero
because the electric field is perpendicular to the surface
Gauss’s law says this flux is zero as there is no charge inside
Then the surfaces A and B are at the same potential
conductor
surface of cavity
In an electrostatic situation, if a conductor contains a cavity and if no
charge is present inside the cavity, there can be no net charge anywhere
on the surface of the cavity

22. Equipotential Surface
Electrostatic shielding

23. Potential Gradient Potential difference and electric field

24. Potential Gradient E from V Gradient of a function f
Potential gradient (cont’d)
E from V
Gradient of a function f
If E is radial with respect to
a point or an axis

25. Potential Gradient
Potential gradient (cont’d)

26. Exercises
Exercise 1

27. Exercises
Exercise 1 (cont’d)

28. Exercises
Exercise 1 (cont’d)

29. Exercises
Exercise 2

30. Exercises
Exercise 3

31. Exercises
Exercise 4

32. Exercises
Exercise 4 (cont’d)

33. Exercises
Exercise 4 (cont’d)

34. Exercises
Exercise 4 (cont’d)

35. Exercises r r Q -Q line charge density l
Exercise 5: An infinite line charge + a conducting cylinder Q -Q
Outer metal braid r r
Signal wire
line charge density l