1. Magnetostatic Fields Electrostatic field : stuck charge distribution
E, D field to H, B field
Moving charge (velocity = const)
Bio sarvart’s law and Ampere’s circuital law

2. Independent on material property
Bio-Savart’s law  I dl 
H field R
Experimental eq.
Independent on material property

3. The direction of dH is determined by right-hand rule
Independent on material property
Current is defined by Idl (line current)
Kds (surface current)
Jdv (volume current)
Current element I K

4. By applying the Stoke’s theorem
Ampere’s circuital law I H dl
I enc : enclosed by path
By applying the Stoke’s theorem

5. Magnetic flux density From this Magnetic flux line always has
same start and end point

6. Electric flux line always start isolated (+) pole to isolated (-) pole :
Magnetic flux line always has same start and end point : no isolated poles

7. Maxwell’s eq. For static EM field
Time varient system

8. Magnetic scalar and vector potentials
Vm : magnetic scalar potential
It is defined in the region that J=0
A : magnetic vector potential

9. Magnetic force and materials Magnetic forceQ E B u Q
Fm : dependent on charge velocity
does not work (Fm dl = 0) only rotation
does not make kinetic energy of charges change

10. Magnetic torque and moment
Lorentz force
Magnetic torque and moment
Current loop in the magnetic field H
D.C motor, generator
Loop//H  max rotating power

11. Slant loop   an B F0 

12. A bar magnet or small current loop
Magnetic dipole
A bar magnet or small current loop N S m I m
A bar magnet
A small current loop

13. Magnetization in material
Similar to polarization in dielectric material
Atom model (electron+nucleus) Ib B
Micro viewpoint
Ib : bound current in atomic model

14. Material in B field B

15. Magnetic boundary materials
Two magnetic materials
Magnetic and free space boundary

16. Magnetic energy

17. Maxwell equations Maxwell equations
In the static field, E and H are independent on each other, but interdependent in the dynamic field
Time-varying EM field : E(x,y,z,t), H(x,y,z,t)
Time-varying EM field or waves : due to accelated charge or time varying current

18. Electric field can be shown by emf-produced field
Faraday’s law
Time-varying magnetic field could produce electric current
Electric field can be shown by emf-produced field

19. Motional EMFs
E and B are related
B(t):time-varying I E

20. Stationary loop, time-varying B field

21. Time-varying loop and static B field

22. Time-varying loop and time-varyinjg B field

23. Displacement current → Maxwell’s eq. based on Ampere’s
circuital law for time-varying field
In the static field
In the time-varying field :
density change is supposed to be changed

24. Displacement current density
Therefore,
Displacement current density

25. Maxwell’s Equations in final forms
Point form
Integral form
Gaussian’s law
Nonexistence of
Isolated M charge
Faraday’s law
Ampere’s law

26. In the tme-varying field ?
Time-varying potentials
stationary E field
In the tme-varying field ?

27.  Coupled wave equation
Poisson’s eqation in time-varying field
poisson’s eq. in stationary field
poisson’s eq. in time-varying field ?
 Coupled wave equation

28. Relationship btn. A and V ?

29. From coupled wave eq.
Uncoupled wave eq.

30. Explanation of phasor Z
Time-harmonic fields
Fields are periodic or sinusoidal with time →
Time-harmonic solution can be practical because most of waveform can be decomposed with sinusoidal ftn by fourier transform. Im Re  
Explanation of phasor Z
Z=x+jy=r 

31. Phasor form If A(x,y,z,t) is a time-harmonic field
Phasor form of A is As(x,y,z)
For example, if

32. Maxwell’s eq. for time-harmonic EM field
Point form
Integral form

33. EM wave propagation Most important application of Maxwell’s equation
→ Electromagnetic wave propagation
First experiment → Henrich Hertz
Solution of Maxwell’s equation, here is
General case

34. Waves in general form
Sourceless
u : Wave velocity

35. Special case : time-harmonic
Solution of general Maxwell’s equation
Special case : time-harmonic

36. Solution of general Maxwell’s equation
A, B : Amplitude
t - z : phase of the wave
: angular frequency
 : phase constant or wave number

37. Plot of the wave E A
/2 
3/2 z A
T/2 T
3T/2 t

38. EM wave in Lossy dielectric material
Time-harmonic field

39. Propagation constant and E field
If z-propagation and
only x component of Es

40. Propagation constant and H field

41. E field plot of example x z
t=t0
t=t0+t

42. EM wave in free space

43. E field plot in free spacex z ak aE aH y
TEM wave
(Transverse EM)
Uniform plane wave
Polarization : the direction of E field

44. Reference
Matthew N. O. Sadiku, “Elements of electromagnetic” Oxford University Press,1993
Magdy F. Iskander, “Electromagnetic Field & Waves”, prentice hall