1. Chapter 10 Potentials and Fields
10.1 The Potential Formulation
10.2 Continuous Distributions
10.3 Point Charges

2. 10.1 The Potential Formulation
Scalar and vector potentials
Gauge transformation
Coulomb gauge and Lorentz gauge

3. 10.1.1 Scalar and Vector Potentials
field formulism or
potential formulism
Maxwell’s eqs

4. (2)

5. (3)
Ex.10.1
where k is a constant,
Sol:
for

6. (4)

7. Gauge transformation

8. 10.1.2 (2) When The fields are independent of the gauges.
(note: physics is independent of the coordinates.)

9. 10.1.3 Coulomb gauge and Lorentz gauge
Potential formulation
Sources:
Coulomb gauge:
easy to solve
difficult to solve

10. 10.1.3 (2) Lorentz gauge: inhomogeneous wave eq. the d’Alembertion
wave equation
[Note:Since is with ,the potentials with both
and are solutions.]

11. 10.1.3 (3) Gauge transformation Coulomb gauge : If you have a and ,
Find ,
Then, you have a solution and

12. 10.1.3 (4) Lorentz gauge : If you have a set of and , and Find ,
Then ,you have a set if solutions and , and

13. 10.2 Continuous Distributions :
With the Lorentz gauge ,
where
In the static case

14. 10.2 (2) For nonstatic case, the above solutions only valid when
for , and due to and ,
where is the retarded time. Because the message of
the pensence of and must travel a distance
the delay is ; that is ,
(Causality)

15. 10.2 (3)
The solutions of retarded potentials for nonstatic sources are
Proof:

16. 10.2 (4)
The same procedure is for proving

17. 10.2 (5)
Example 10.2
Solution:

18. 10.2 (6)

19. 10.2 (7)
Note:

20. 10.2 (8)
recover the static case

21. 10.3 Point Charges 10.3.1 Lienard-Wiechert Potentials
The Fields of a Moving Point Charge

22. 10.3.1 Lienard-Wiechert potentials
Consider a point charge q moving on a trajectory
retarded position
location of the observer at time t
Two issues
There is at most one point on the trajectory communicating
with at any time t.
Suppose there are two points:
Since q can not move at the speed of light,
there is only one point at meet.

23. (2)
the point chage
due to Doppler –shift effect as the point charge is considered as
an extended charge.
Proof.
consider the extended charge has a length L as a train
(a) moving directly to the observer
time for the light to arrive the observer. E F

24. 10.3.1 (3) (b)moving with an angle to the observer actual volume
The apparent volume

25. (4)
Lienard-Wiechert Potentials for a moving point charge

26. (5)
Example 10.3 q
Solution: 1
consider

27. (6) 1

28. 10.3.2 The Fields of a Moving Point Charge
Lienard-Weichert potentials:
Math., Math., and Math,…. are in the following:

29. (2) =

30. (4)
Prob.10.17

31. (3) = =

32. 10.3.2 (5) define Electrostatic field if generalized Coulomb field
radiation field or acceleration
field dominates at large R if
Electrostatic field

33. (6)

34. (7)

35. 10.3.2 (8) The force on a test charge Q with velocity due to a moving
charge q with velocity is
Where

36. (9)
Example 10.4 q
Solution:
Ex.10.3
Prob.10.14

37. (10)
Coulomb`s law
“Biot-savart Law for a point charge.”