1. Electromagnetic waves

2. Wave in an ideal dielectric
We will derive the wave equations for electric and magnetic field from Maxwell’s equations.
It will be a proof that the electromagnetic field is a wave.
Assumptions:
Considered area is unlimited
Considered medium is linear, homogenous and isotropic
γ=0, it means a medium is lossless
There are no currents and no charges in the considered area

3. We have a set of four differential equations with two unknown vector functions.
In a general case we have to find 6 unknown scalar functions – components of searched vectors.

4. Assumptions: μ=const., ε=const., γ=0, ρ=0
Let’s calculate the rotation of both sides
of both equations

5. From vector identity:

6. =0 =0

7. We have received two wave equations.
Two vector functions E(P,t) i H(P,t) which satisfy this equations form electromagnetic wave.
Attention:
This means there might exist such solutions of wave equations which don’t satisfy Maxwell’s equations.
Solutions E(P,t) and H(P,t) must satisfy condition coming from Maxwell’s equations.

8. Let’s use the x,y,z coordinates system:
Vector Laplasian:
Each of components satisfies scalar wave equation.

10. Then the wave equation is:
Let’s consider only one of the field components, but received conclusions will be general.
Assumption:
The field changes occur in one direction only, let’s assume it is z direction.
It means that all derivatives with respect of x and y equal zero.
Plane wave
Then the wave equation is:

11. The solution of this equation has a form of:
When the shape of function f depends of the wave generator.
a velocity of the wave propagation

12. There are equations of uniform motion.
Let’s assume f1=const. and f2=const.
What does result from this assumption?
There are equations of uniform motion.
Points in which f1 has a constant value move in positive direction of the z axis.
f1 – a wave moving ahead called ORIGINAL WAVE
Points in which f2 has a constant value move in negative direction of the z axis.
f2 – a wave moving back called RETURNED WAVE

13. What is the result of the fact that the changes occur in z direction only (in the direction of wave propagation)?

14. Ex,Ey,Hx,Hy are functions of
After integration we receive Ez=const. i Hz=const.
Ez=0
Hz=0
The field is produced by the changes.
Constancy is contradictory to the assumption of field existence.
CONCLUSION:
Considered wave doesn’t have components in direction
of propagation.
Ex,Ey,Hx,Hy are functions of
z-vt, time and variable z.

15. Ex=f1(z-vt), Ey=f2(z-vt), Hx=f3 (z-vt), Hy=f4 (z-vt)
Def.: A wave is called transverse when wave vector doesn’t have components in direction of wave propagation.
This wave is marked TEM – transverse electromagnetic
Ex,Ey,Hx,Hy - are solutions of independent wave equations. Moreover, they have to satisfy Maxwell’s equations.
Let’s mark:
Ex=f1(z-vt), Ey=f2(z-vt),
Hx=f3 (z-vt), Hy=f4 (z-vt)

17. f1 f4 f3 f2
integration
Constants of integration are omitted, because they don’t satisfy
the assumption of wave existence.

18. After substitution f1, f2, f3, f4 :x y z x y z Hy v Hx v Ex
-Ey

19. - The wave admitance of medium
- The wave impedance of the medium
- The wave admitance of medium
The impedance is the ratio of the values
of mutually perpendicular components
of electric and magnetic wave vectors.

20. The scalar product:
Scalar product =0 when vectors are perpendicular.

21. Twierdzenie Poyntinga
Poynting’s theorem
Poynting’s theorem determines the law of power conservation
in area V in which electric and magnetic fields exist.
Assumptions:
A wave propagates in homogenous medium
ε=const μ=const γ=const

22. Maxwell’s equations:
Let’s subtract both sides of both equations

23. From vector identity:
we receive
We integrate this equation over the volume V.
The area V is bounded by the surface S.

24. Using Gauss’s theorem:
where S – the surface bounding the area V and vector dS is directed along the external normal.

25. Let’s assume that any kind of energy is transformed into electric energy in a part of the volume V.
An electric energy sources are situated in the considered area.

26. Let’s rearrange the terms:

27. After substitutions:
Volume density of the generated power
Power generated in area V
Volume density of the power connected with the transformation of electric energy into the heat
Power connected with the transformation of electric energy into the heat in area V.

28. Volume density of electric field energy
Volume density of magnetic field energy
Electromagnetic field energy in area V
A derivative of energy is the power of electromagnetic field in area V.

29. Poynting’s vector direction determines the direction
It has a physical sense of surface density of the power.
Poynting’s vector flux.
Physically it represents the power flux through the surface S.
Poynting’s vector direction determines the direction
of the power flow.

30. V The flux of the Poynting vector through the surface S
is called the power flux or the power radiated through this surface.
The power radiated through the bound
of the area V. V
Input
power
Output

31. Poynting’s theorem:
The power produced in any area V is equal to the sum of:
the power transformed into heat
the power gathered in electromagnetic field in this area
the power radiated through the bound of V
This theorem expresses the law of energy conservation
in electromagnetic field.

32. Let’s write this law as a power balance:
POWER RADIATED
POWER GATHERED AND LOST
POWER GENERATED

33. electromagnetic field
Harmonic
electromagnetic field

34. COMPLEX VECTORS Time vector W(t) has sinusoidal components:
Def. The field is harmonic when vectors describing this field are sinusoidally changing in time.
Time vector W(t) has sinusoidal components:

35. Def. Complex vector has complex components which are complex values of the time vector components.
Relations between components of the time vector and complex vector.

36. The measure of the complex vector is its NORM
Relation between the time vector and complex vector:
The measure of the complex vector is its NORM
Def.
The norm of complex vector is expressed as:

37. MAXWELL’s EQUATIONS
IN COMPLEX FORM

38. - complex vectors
where
under assumptoion:

39. Maxwell’s equations in complex form:
and

40. Considered area is:
linear
isotropic
uniform
There are no charges in considered area.

41. Г - complex coefficient – propagation constant
Complex equivalents of wave equations
They are called the Helmholtz wave equations
Г - complex coefficient – propagation constant

42. There are complex equations for components

43. where The solution of Helmholtz’s equation will be analogical
for each component
where
-propagation constant
damping constant
phase constant

44. Physical interpretation of
and
Let’s consider the case
Assumption:

45. Plane wave E2 E1 Assumptions:
Plane wave equations in time and in complex forms
Assumptions:
Let’s consider one of the components, eg. Ex E1 E2

46. E1(t)
E2(t)

47. A. Original wave B. Returned wave Wave length:
An equation of the wave motion:
A. Original wave
B. Returned wave
Wave length:

48. Wave amplitude Wave amplitude decreases with
A. Original wave
B. Returned wave
Wave amplitude decreases with
z axis, it means that amplitude is damped in direction of wave propagation
Wave amplitude decreases with
–z axis, it means that amplitude is damped in direction of wave propagation (-z)

49. From Maxwell’s equation:

50. H1 H2

51. Wave impedance of the medium in which the wave propagates
There is no returned wave, when the medium is unlimited

52. Plane wave in ideal dielectric

53. ! Assumption: there is no returned wave The wave is undamped
Ex and Hy are in phase because Zf is real number

54. Energy divides into halves for electric and magnetic fields.
Let’s compare the density of energy in electric and magnetic fields:
Energy divides into halves for electric and magnetic fields.

55. Plane wave in ideal conductor

56. Damping constant and phase constant are equal in ideal conductor.

57. Argument of wave impedance equals to difference of initial phases of electric and magnetic fields.

58. Electric field leads magnetic field by 45
A damping is very large in conductor.
The wave amplitude is almost entirely damped on a way equal one wave length.
times
After passing one wave length the wave in conductor fades.

59. Let’s compare energy densities in electric and magnetic fields:

60. Because
the average value
for the period
where
when
In ideal conductor whole energy is included in magnetic field.

61. Polarisation of harmonic wave

62. x y z Hy Ex v Hx
-Ey

63. Now let’s consider relations between
Ex z Ey
and
Hx z Hy

64. maximum values initial phases of components Ex and Ey
The end of the vector E draws the line which shape depends on
maximum values
initial phases
of components Ex and Ey

65. ( ) Eliptic polarisation+ - = 90 2 , 10
sin 7 z t E y x j
z=const.
Eliptic polarisation
Components have different amplitudes and different phases

66. ( ) z=const. Eliptic polarisation in the axis x and y directions+ - = 90 2 , 10
sin 7 z t E y x
z=const. r
Eliptic polarisation
in the axis x and y directions
Amplitudes of components are different and phases differ by 90

67. Amplitudes of components are equal and phases differ by 90°
Circular polarisation
Amplitudes of components are equal and phases differ by 90°

68. Linear polarisation
Amplitudes are different, but the phases are the same.