1. Chapter 5 Magnetostatics
5.1 The Lorentz Force Law
5.2 The Biot-Savart Law
5.3 The Divergence and Curl of
5.4 Magnetic Vector Potential

2. 5.1.1 Magnetic Fields Charges induce electric field Test charge
Source charges
Test charge

3. 5.1.1

4. 5.1.2 Magnetic Force Lorentz Force Law Ex.1 Cyclotron motion moment
cyclotron frequency
relativistic cyclotron frequency
microwave
relativistic electron cyclotron maser
light
laser
EM wave

5. 5.1.2 (2) ~1960 EM : maser [ 1959 J.Schneider ; A.V. Gaponov]
ES : space [1958 R.Q. Twiss ]
(1976) K.R. Chu & J.L. Hirshfield : physics in gyrotron/plasma
1978 C.S. Wu & L.C. Lee : EM in space ( )
1986 K.R. Chen & K.R. Chu : ES in gyrotron
relativistic ion cyclotron instability
1993 K.R. Chen
ES in Lab. plasma [fusion (
EM ? Lab. & space plasmas ? )]

6. 5.1.2 (3)
Ex.2
Cycloid Motion
assume

7. 5.1.2 (4)

8. 5.1.2 (5)

9. 5.1.2 (6)
Magnetic forces do not work

10. 5.1.3 Currents
The current in a wire is the charge per unit time passing
a given point.
Amperes 1A = 1 C/S
The magnetic force on a segment of current-carrying wire

11. 5.1.3 (2) surface current density
the current per unit length-perpendicular-to-flow
(mobile)
The magnetic force on a surface current is

12. 5.1.3 (3) volume current density
The current per unit area-perpendicular-to-flow
The magnetic force on a volume current is

13. 5.1.3 (4)
Ex. 3
Sol.
Ex. 4
(a) what is J ?
(uniform I)
Sol.

14. 5.1.3 (5)
(b) For J = kr, find the total current in the wire.
Sol.

15. 5.1.3 (6)
relation?
(charge
conservation)
Continuity equation

16. 5.2.1 Steady Currents
Stationary charges constant electric field: electrostatics
Steady currents constant magnetic field: magnetostatics
No time dependence

17. 5.2.2 The Magnetic Field of a Steady Current
Biot-Savart Law: for a steady line current
Permeability of free space
Biot-Savart Law for surface currents
Biot-Savart Law for volume currents
for a moving point charge

18. 5.2.2 (2)
Solution:
In the case of an infinite wire,

19. 5.2.2 (3) The field at (2) due to is Force? The force at (2) due to is
The force per unit length is

20. 5.2.2 (4) 2

21. 5.3.1 an example: Straight-Line Currents

22. 5.3.2 The Divergence and Curl of
Biot-savart law

23. 5.3.2
for steady current
To where
Ampere’s law in differential form

24. 5.3.3 Applications of Ampere’s Law
Ampere’s Law in differential form
Ampere’s Law in integral form
Electrostatics: Coulomb Gauss
Magnetostatics: Bio-Savart Ampere
The standard current configurations which can be handled by
Ampere's law:
Infinite straight lines
Infinite planes
Infinite solenoids
Toroid

25. 5.3.3 (2)
Ex.7
symmetry ?
Ex.8

26. 5.3.3 (3)
Ex.9
loop 1.
loop 2.

27. 5.3.3 (4)
Ex.10
Solution:

28. 5.3.3 (5)

29. 5.4.1 The Vector Potential E.S. : M.S. : a constant-like vector
function
Gauge transformation
is a vector potential in magnetostatics
If there is that , can we find a function to obtain with

30. 5.4.1 (2)
Ampere’s Law if

31. 5.4.1 (3) Example 11 A spiring sphere Solution:
For surface integration over easier

32. 5.4.1 (4)

33. 5.4.1 (5)

34. 5.4.1 (6)
if R > S
if R < S

35. 5.4.1 (7)

36. 5.4.1 (8)
Note:
is uniform inside the spherical shell

37. 5.4.1 (9) =

38. 5.4.2 Summary and Magnetostaic
Boundary Conditions

39. 5.4.2 (2)

40. 5.4.2 (3)

41. 5.4.3 Multipole Expansion of the Vector Potential
line
current =0
monopole
dipole

42. 5.4.3(2)

43. 5.4.3(3)
Ex. 13

44. 5.4.3(4) Field of a “pure” magnetic dipole
Field of a “physical” magnetic dipole