1. Electromagnetic Field and Waves
Gi-Dong Lee
Outline:
Electrostatic Field
Magnetostatic Field
Maxwell Equation
Electromagnetic Wave Propagation

2. Vector Calculus
Basic mathematical tool for electromagnetic field solution and understanding.

3. Line, Surface and Volume Integral
Line Integral :
Circulation of A around L
( )
Perfect circulation :
Surface Integral :
Path L
Net outward flux of A

4. Del operator : Volume Integral : Gradient Divergence Curl
Laplacian of scalar

5. dV = potential difference btw the scalar field V
Gradient of a scalar → V1 V2 
dV = potential difference btw the scalar field V

6. Divergence, Gaussian’s law
It is a scalar field

7. Curl, Stoke’s theorem ds
Closed path L

8. Practical solution method
Laplacian of a scalar
Practical solution method

9. Classification of the vector field

10. Electrostatic Fields
Time-invariant electric field in free space

11. Coulomb’s law and field intensity Experimental law
Coulomb’s law in a point charge Q1 Q2
Vector Force F12 or F21 Q1 Q2
F21
F12 r1 r2

12. E : Field intensity to the normalized charge (1)
Electric Field E r r’ 1 Q R
E : Field intensity to the normalized charge (1)

13. Electric Flux density D
Flux density D is independent on the material property (0)
Maxwell first equation from the Gaussian’s law

14. From the Gaussian’s law
From this
From the Gaussian’s law

15. In case of a normalized charge Q
Electric potential
Electric Field can be obtained by charge distribution and electric potential E A Q B
In case of a normalized charge Q
+ : work from the outside
- : work by itself

16. Second Maxwell’s Equ. From E and V
Absolute potential E r
O : origin point 
Q=1
Second Maxwell’s Equ. From E and V

17. Relationship btn. E and V
Second Maxwell’s Equ
Relationship btn. E and V
3,4,5 : EQUI-POTENTIAL LINE E 3 4 5

18. Energy density We

19. E field in material space ( not free space)
Conductor
Non conductor
Insulator
Dielctric material
Material can be classified by conductivity 
 << 1 : insulator
 >> 1 : conductor (metal :  )
Middle range of  : dielectric

20. Convection current ( In the case of insulator)
Current related to charge, not electron
Does not satisfy Ohm’s law

21. Conduction current (current by electron : metal)

22. Displacement can be occurred
Polarization in dielectric + -
After field is induced
Displacement can be occurred
Equi-model -Q +Q
Dipole moment
Therefore, we can expect strong electric field in the dielectric
material, not current

23. Multiple dipole moments- +
0 : permittivity of free space
: permittivity of dielectric
r : dielectric constant

24. Linear, Isotropic and Homogeneous dielectric
D  E : linear or not linear
When (r) is independent on its distance r
: homogeneous
When (r) is independent on its direction
: isotropic  anisotropic (tensor form)

25. Continuity equation
Qinternal
time

26. Boundary condition Dielectric to dielectric boundary
Conductor to dielectric boundary
Conductor to free space boundary

27. Poisson eq. and Laplacian
Practical solution for electrostatic field

28. 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

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

30. 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

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

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

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

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

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

36. 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

37. 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

38. Slant loop   an B F0 

39. 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

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

41. Material in B field B

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

43. Magnetic energy

44. 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

45. 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

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

47. Stationary loop, time-varying B field

48. Time-varying loop and static B field

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

50. 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

51. Displacement current density
Therefore,
Displacement current density

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

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

54.  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

55. Relationship btn. A and V ?

56. From coupled wave eq.
Uncoupled wave eq.

57. 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 

58. 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

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

60. 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

61. Waves in general form
Sourceless
u : Wave velocity

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

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

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

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

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

67. Propagation constant and H field

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

69. EM wave in free space

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

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