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![❑ Third, from Gauss’s law in electric field
ׯ
𝒔
ഥ
𝑫. 𝒅ത
𝒔 = 𝑸 = 𝒗
𝝆𝒗 𝒅𝒗
❑ Applying divergence theorem, we get
ර
𝒔
ഥ
𝑫. 𝒅ത
𝒔 = න
𝒗
𝛁 ∙ ഥ
𝑫 𝒅𝒗 = න
𝒗
𝝆𝒗 𝒅𝒗
❑ Two volume integrals are equal only if their integrands are
equal,
𝛁 ∙ ഥ
𝑫 = 𝝆𝒗
❑ Fourth, from Gauss’s law for magnetic field [the magnetic flux
lines are a closed loop]
ර
𝒔
ഥ
𝑩. 𝒅ത
𝒔 = 𝟎
❑ Applying divergence theorem, we get
𝒗
𝛁 ∙ ഥ
𝑩 𝒅𝒗 = 𝟎 OR 𝛁 ∙ ഥ
𝑩 = 𝟎](https://image.slidesharecdn.com/2electromagneticwavesandpropagationlecturespart11-251014090444-844389de/85/2_Electromagnetic-waves-and-propagation-Lectures_part1_1-pdf-25-320.jpg)
2_Electromagnetic waves and propagation Lectures_part1_1.pdf
- 1. LECTURES ON ELECTROMAGNETIC WAVES ANDWAVE PROPAGATION Assoc. Prof. Yasser Mahmoud Madany Senior Member, IEEE URSI Senior Member Founder and Chair of the IEEE Egypt AP-S/MTT-S Joint Chapter Founder and Counselor of the IEEE AL Ryada Student Branch
- 2. PART ONE Maxwell’s Equationsand Electromagnetic Waves
- 3. INTRODUCTION ❑ Static electricfield has applications in: ❖ Cathode ray oscilloscopes for deflecting charged particles. ❖ Ink jet printers to increase the speed of printing and improve print quality. ❑ Static magnetic field has applications in: ❖ Magnetic separators to separate magnetic materials from non-magnetic materials. ❖ Cyclotrons for imparting high energy to charged particles.
- 4. INTRODUCTION ❑ Time-varying fieldsconstitute electromagnetic (EM) waves which have wide applications in all the communications, radars and also in Bio-medical engineering. ❑ EM waves are produced by time-varying currents. ❑ In brief, it may be noted that: ❖ Static charges produce electrostatic fields. ❖ Steady currents (DC currents) produce magneto-static fields. ❖ Static magnetic charges (magnetic dipoles) also produce magneto-static fields. ❖ Time-varying currents produce EM waves or EM fields.
- 5. THE FIELDS ഥ 𝑬,ഥ 𝑯, ഥ 𝑫 AND ഥ 𝑩 IN STATIC FORM Coordinate / Field Cartesian Cylindrical Spherical Cartesian Time- varying Electric Field, ഥ 𝑬 ഥ 𝑬(𝒙, 𝒚, 𝒛) ഥ 𝑬(𝝆, ∅, 𝒛) ഥ 𝑬(𝒓, 𝜽, ∅) ഥ 𝑬(𝒕, 𝒙, 𝒚, 𝒛) Magnetic Field, ഥ 𝑯 ഥ 𝑯(𝒙, 𝒚, 𝒛) ഥ 𝑯(𝝆, ∅, 𝒛) ഥ 𝑯(𝒓, 𝜽, ∅) ഥ 𝑯(𝒕, 𝒙, 𝒚, 𝒛) Electric Flux Density, ഥ 𝑫 ഥ 𝑫(𝒙, 𝒚, 𝒛) ഥ 𝑫(𝝆, ∅, 𝒛) ഥ 𝑫(𝒓, 𝜽, ∅) ഥ 𝑫(𝒕, 𝒙, 𝒚, 𝒛) Magnetic Flux Density, ഥ 𝑩 ഥ 𝑩(𝒙, 𝒚, 𝒛) ഥ 𝑩(𝝆, ∅, 𝒛) ഥ 𝑩(𝒓, 𝜽, ∅) ഥ 𝑩(𝒕, 𝒙, 𝒚, 𝒛)
- 6. EQUATION OF CONTINUITYFOR TIME-VARYING FIELDS ❑ Consider a closed surface enclosing a charge 𝑸 and there exists an outward flow of current given by 𝑰 = ර 𝒔 ҧ 𝑱. 𝒅ത 𝒔 ❑ The previous equation known as equation of continuity in integral form, where 𝑰 is the current flowing away through a closed surface (A). ҧ 𝑱 is the conduction current density (A/m2). 𝒅ത 𝒔 is the differential area on the surface whose direction is always outward and normal to the surface.
- 7. ❑ As thereis outward flow of current, there will be a decrease the rate of charge of which is given by 𝑰 = −𝒅𝑸 𝒅𝒕 where 𝑸 is the enclosed charge (C). ❑ From the principle of conservative of charge, we have 𝑰 = ර 𝒔 ҧ 𝑱. 𝒅ത 𝒔 = −𝒅𝑸 𝒅𝒕 ❑ From divergence theorem, we have ර 𝒔 ҧ 𝑱. 𝒅ത 𝒔 = න 𝒗 𝛁. ҧ 𝑱 𝒅𝒗
- 8. ❑ Hence, න 𝒗 𝛁. ҧ 𝑱𝒅𝒗 = −𝒅𝑸 𝒅𝒕 ❑ By definition, 𝑸 = න 𝒗 𝝆𝒗 𝒅𝒗 where 𝝆𝒗 is the volume charge density (C/m3). ❑ So, න 𝒗 𝛁. ҧ 𝑱 𝒅𝒗 = න 𝒗 −𝝏𝝆𝒗 𝝏𝒕 𝒅𝒗 = න 𝒗 − ሶ 𝝆𝒗 𝒅𝒗 ❑ Two volume integrals are equal only if their integrands are equal. So, the equation of continuity in point form is 𝛁 ∙ ҧ 𝑱 = −𝝏𝝆𝒗 𝝏𝒕 = − ሶ 𝝆𝒗
- 9. MAXWELL’S EQUATIONS FORTIME-VARYING FIELDS ❑ Maxwell’s equations in differential form are given by 𝛁 × ഥ 𝑯 = 𝝏ഥ 𝑫 𝝏𝒕 + ҧ 𝒋 = ሶ ഥ 𝑫 + ҧ 𝒋 (1) 𝛁 × ഥ 𝑬 = −𝝏ഥ 𝑩 𝝏𝒕 = − ሶ ഥ 𝑩 (2) 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗 (3) 𝛁 ∙ ഥ 𝑩 = 𝟎 (4) where ഥ 𝑯 is the magnetic field strength (A/m). ഥ 𝑫 is the electric flux density (C/m2).
- 10. MAXWELL’S EQUATIONS FORTIME-VARYING FIELDS where: ሶ ഥ 𝑫 = 𝝏ഥ 𝑫 𝝏𝒕 is the displacement electric current density (A/m2). ҧ 𝑱 is the conduction current density (A/m2). ഥ 𝑬 is the electric field strength (V/m). ഥ 𝑩 is the magnetic flux density (wb/m2) or (Tesla). ሶ ഥ 𝑩 = 𝝏ഥ 𝑩 𝝏𝒕 is the time-derivative of magnetic flux density (wb/m2 . sec). Also, ሶ ഥ 𝑩 is the magnetic current density (V/m2) or (Tesla/sec). 𝝆𝒗 is the volume charge density (C/m3)
- 11. MAXWELL’S EQUATIONS FORTIME-VARYING FIELDS ❑ Maxwell’s equations in integral form are given by ׯ 𝑳 ഥ 𝑯. 𝒅ഥ 𝑳 = ׯ 𝒔 ሶ ഥ 𝑫 + ҧ 𝒋 . 𝒅ത 𝒔 (1) ׯ 𝑳 ഥ 𝑬. 𝒅ഥ 𝑳 = − ׯ 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔 (2) ׯ 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = 𝒗 𝝆𝒗 𝒅𝒗 (3) ׯ 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎 (4) where 𝒅ഥ 𝑳 is the differential length. 𝒅ത 𝒔 is the differential surface area whose direction is always outward and normal to the surface.
- 12. CONVERSION OF DIFFERENTIALFORM OF MAXWELL’S EQUATIONS TO INTEGRAL FORM ❑ Consider the first Maxwell’s equation 𝛁 × ഥ 𝑯 = ሶ ഥ 𝑫 + ҧ 𝒋 ❑ Take surface integral on both sides ׯ 𝒔 𝛁 × ഥ 𝑯 . 𝒅ത 𝒔 = ׯ 𝒔 ሶ ഥ 𝑫 + ҧ 𝒋 . 𝒅ത 𝒔 ❑ Applying Stokes’ theorem, we can write ර 𝒔 𝛁 × ഥ 𝑯 . 𝒅ത 𝒔 = ර 𝑳 ഥ 𝑯. 𝒅ത 𝑳 ❑ Therefore, the first low ර 𝑳 ഥ 𝑯. 𝒅ത 𝑳 = ර 𝒔 ሶ ഥ 𝑫 + ҧ 𝒋 . 𝒅ത 𝒔
- 13. ❑ Consider thesecond Maxwell’s equation 𝛁 × ഥ 𝑬 = − ሶ ഥ 𝑩 ❑ Take surface integral on both sides ׯ 𝒔 𝛁 × ഥ 𝑬 . 𝒅ത 𝒔 = − ׯ 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔 ❑ Applying Stokes’ theorem, we can write ර 𝒔 𝛁 × ഥ 𝑬 . 𝒅ത 𝒔 = ර 𝑳 ഥ 𝑬. 𝒅ത 𝑳 ❑ Therefore, the second low ර 𝑳 ഥ 𝑬. 𝒅ത 𝑳 = − ර 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔
- 14. ❑ Consider thethird Maxwell’s equation 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗 ❑ Take volume integral on both sides න 𝒗 𝛁 ∙ ഥ 𝑫 𝒅𝒗 = න 𝒗 𝝆𝒗 𝒅𝒗 ❑ Applying divergence theorem, we can write න 𝒗 𝛁 ∙ ഥ 𝑫 𝒅𝒗 = ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 ❑ Therefore, the third low ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = න 𝒗 𝝆𝒗 𝒅𝒗
- 15. ❑ Consider thefourth Maxwell’s equation 𝛁 ∙ ഥ 𝑩 = 𝟎 ❑ Take volume integral on both sides න 𝒗 𝛁 ∙ ഥ 𝑩 𝒅𝒗 = 𝟎 ❑ Applying divergence theorem, we can write න 𝒗 𝛁 ∙ ഥ 𝑩 𝒅𝒗 = ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 ❑ Therefore, the fourth low ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎
- 16. MAXWELL’S EQUATIONS FORSTATIC FIELDS 𝛁 × ഥ 𝑯 = ҧ 𝒋 ⇔ ර 𝑳 ഥ 𝑯. 𝒅ത 𝑳 = ර 𝒔 ҧ 𝒋. 𝒅ത 𝒔 𝛁 × ഥ 𝑬 = 𝟎 ⇔ ර 𝑳 ഥ 𝑬. 𝒅ത 𝑳 = 𝟎 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗 ⇔ ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = න 𝒗 𝝆𝒗 𝒅𝒗 𝛁 ∙ ഥ 𝑩 = 𝟎 ⇔ ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎
- 17. CHARACTERISTICS OF FREESPACE Parameter Symbol Value Relative permittivity 𝜺𝒓 1 Relative permeability 𝝁𝒓 1 Conductivity 𝝈 0 Conduction current density ҧ 𝑱 0 Volume charge density 𝝆𝒗 0 Intrinsic or characteristic impedance 𝜼 𝟏𝟐𝟎 𝝅 𝑶𝑹 𝟑𝟕𝟕
- 18. MAXWELL’S EQUATIONS FORFREE SPACE 𝛁 × ഥ 𝑯 = ሶ ഥ 𝑫 ⇔ ර 𝑳 ഥ 𝑯. 𝒅ത 𝑳 = ර 𝒔 ሶ ഥ 𝑫. 𝒅ത 𝒔 𝛁 × ഥ 𝑬 = − ሶ ഥ 𝑩 ⇔ ර 𝑳 ഥ 𝑬. 𝒅ത 𝑳 = − ර 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔 𝛁 ∙ ഥ 𝑫 = 𝟎 ⇔ ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = 𝟎 𝛁 ∙ ഥ 𝑩 = 𝟎 ⇔ ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎
- 19. MAXWELL’S EQUATIONS FORSTATIC FIELDS IN FREE SPACE 𝛁 × ഥ 𝑯 = 𝟎 ⇔ ර 𝑳 ഥ 𝑯. 𝒅ത 𝑳 = 𝟎 𝛁 × ഥ 𝑬 = 𝟎 ⇔ ර 𝑳 ഥ 𝑬. 𝒅ത 𝑳 = 𝟎 𝛁 ∙ ഥ 𝑫 = 𝟎 ⇔ ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = 𝟎 𝛁 ∙ ഥ 𝑩 = 𝟎 ⇔ ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎
- 20. PROOF OF MAXWELL’SEQUATIONS ❑ First, from Ampere’s circuital law , we have 𝛁 × ഥ 𝑯 = ҧ 𝒋 ❑ Take dot product on both sides 𝛁. 𝛁 × ഥ 𝑯 = 𝛁. ҧ 𝒋 ❑ As the divergence of curl of a vector is zero, 𝛁. ҧ 𝒋 = 𝟎 ❑ But the equation of continuity in point form 𝛁 ∙ ҧ 𝑱 = −𝝏𝝆𝒗 𝝏𝒕 = − ሶ 𝝆𝒗 ❑ This means that if 𝛁 × ഥ 𝑯 = ҧ 𝒋 is true, we get 𝛁. ҧ 𝒋 = 𝟎. ❑ As the equation of continuity is more fundamental, Ampere’s circuital law should be modified.
- 21. ❑ Modifying Ampere’scircuital law , we have 𝛁 × ഥ 𝑯 = ҧ 𝒋 + ഥ 𝑭 ❑ Take dot product on both sides 𝛁. 𝛁 × ഥ 𝑯 = 𝛁. ҧ 𝒋 + 𝛁. ഥ 𝑭 ❑ As the divergence of curl of a vector is zero, 𝛁. ҧ 𝒋 + 𝛁. ഥ 𝑭 = 𝟎 ❑ Substituting the value of 𝛁. ҧ 𝒋 from the equation of continuity in the previous equation, 𝛁 ∙ ഥ 𝑭 + − ሶ 𝝆𝒗 = 𝟎 OR 𝛁 ∙ ഥ 𝑭 = ሶ 𝝆𝒗
- 22. ❑ But thepoint form of Gauss’s law 𝛁. ഥ 𝑫 = 𝝆𝒗 OR 𝛁. ሶ ഥ 𝑫 = ሶ 𝝆𝒗 ❑ Then, 𝛁. ഥ 𝑭 = 𝛁. ሶ ഥ 𝑫 ❑ The divergence of two vectors are equal only if the vectors are identical. ഥ 𝑭 = ሶ ഥ 𝑫 ❑ Substituting the value of ഥ 𝑭, 𝛁 × ഥ 𝑯 = ሶ ഥ 𝑫 + ҧ 𝒋
- 23. ❑ Second, accordingto Faraday’s law 𝒆𝒎𝒇 = − 𝒅∅ 𝒅𝒕 , ∅ ∶ 𝒎𝒂𝒈𝒏𝒆𝒕𝒊𝒄 𝒇𝒍𝒖𝒙 𝒊𝒏 𝒘𝒃 ❑ By definition, 𝒆𝒎𝒇 = ර 𝑳 ഥ 𝑬. 𝒅ഥ 𝑳 ❑ Then, ර 𝑳 ഥ 𝑬. 𝒅ഥ 𝑳 = − 𝒅∅ 𝒅𝒕 ❑ But, ∅ = ׯ 𝒔 ഥ 𝑩. 𝒅ത 𝒔 OR − 𝒅∅ 𝒅𝒕 = − ׯ 𝒔 𝝏ഥ 𝑩 𝝏𝒕 . 𝒅ത 𝒔 ❑ So, ර 𝑳 ഥ 𝑬. 𝒅ഥ 𝑳 = − ර 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔
- 24. ❑ Applying Stokes’theorem, we get ර 𝒔 𝛁 × ഥ 𝑬 . 𝒅ത 𝒔 = ර 𝑳 ഥ 𝑬. 𝒅ഥ 𝑳 ❑ Hence, ර 𝒔 𝛁 × ഥ 𝑬 . 𝒅ത 𝒔 = − ර 𝒔 ሶ ഥ 𝑩. 𝒅ത 𝒔 ❑ Two surface integrals are equal only if their integrands are equal, 𝛁 × ഥ 𝑬 = − ሶ ഥ 𝑩
- 25. ❑ Third, fromGauss’s law in electric field ׯ 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = 𝑸 = 𝒗 𝝆𝒗 𝒅𝒗 ❑ Applying divergence theorem, we get ර 𝒔 ഥ 𝑫. 𝒅ത 𝒔 = න 𝒗 𝛁 ∙ ഥ 𝑫 𝒅𝒗 = න 𝒗 𝝆𝒗 𝒅𝒗 ❑ Two volume integrals are equal only if their integrands are equal, 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗 ❑ Fourth, from Gauss’s law for magnetic field [the magnetic flux lines are a closed loop] ර 𝒔 ഥ 𝑩. 𝒅ത 𝒔 = 𝟎 ❑ Applying divergence theorem, we get 𝒗 𝛁 ∙ ഥ 𝑩 𝒅𝒗 = 𝟎 OR 𝛁 ∙ ഥ 𝑩 = 𝟎
- 26. SINUSOIDAL TIME-VARYING FIELDS ❑In practice, electric and magnetic fields vary sinusoidally. ❑ Any periodic variation can be described in terms of sinusoidal variations. ❑ The fields can be represented by ෩ ഥ 𝑬 = 𝑬𝒎 𝐜𝐨𝐬 𝝎𝒕 OR ෩ ഥ 𝑬 = 𝑬𝒎 𝒄𝒐𝒔 𝝎𝒕 where 𝝎 : Angular frequency, 𝝎 = 𝟐𝝅𝒇 𝒇 : Frequency variation of the field. 𝑬𝒎 : The maximum field strength.
- 27. ❑ It isalso possible to represent the fields using the phasor notation. ❑ The time-varying field ෩ ഥ 𝑬(𝒓, 𝒕) is related to phasor field ഥ 𝑬(𝒓) as ෩ ഥ 𝑬(𝒓, 𝒕) = 𝑹𝒆 ഥ 𝑬 𝒓 𝒆𝒋𝝎𝒕 OR ෩ ഥ 𝑬(𝒓, 𝒕) = 𝑰𝒎 ഥ 𝑬 𝒓 𝒆𝒋𝝎𝒕 where 𝝎 : Angular frequency.
- 28. MAXWELL’S EQUATIONS INPHASOR FORM ❑ Consider the first Maxwell’s equation 𝛁 × ෩ ഥ 𝑯 = ෩ሶ ഥ 𝑫 + ሚҧ 𝒋 ❑ If ෩ ഥ 𝑯 = 𝑹𝒆 ഥ 𝑯𝒆𝒋𝝎𝒕 ෩ ഥ 𝑫 = 𝑹𝒆 ഥ 𝑫𝒆𝒋𝝎𝒕 ෨ҧ 𝑱 = 𝑹𝒆 ҧ 𝑱𝒆𝒋𝝎𝒕 Then 𝛁 × 𝑹𝒆 ഥ 𝑯𝒆𝒋𝝎𝒕 = 𝝏 𝝏𝒕 𝑹𝒆 ഥ 𝑫𝒆𝒋𝝎𝒕 + 𝑹𝒆 ҧ 𝑱𝒆𝒋𝝎𝒕 ❑ Interchanging the operation of taking the real part, we get 𝑹𝒆 𝛁 × ഥ 𝑯 − 𝒋𝝎ഥ 𝑫 − ҧ 𝑱 𝒆𝒋𝝎𝒕 = 𝟎 ∴ 𝛁 × ഥ 𝑯 = 𝒋𝝎ഥ 𝑫 + ҧ 𝑱
- 29. ❑ Consider thesecond Maxwell’s equation 𝛁 × ෩ ഥ 𝑬 = − ෩ሶ ഥ 𝑩 ❑ If ෩ ഥ 𝑬 = 𝑹𝒆 ഥ 𝑬𝒆𝒋𝝎𝒕 ෩ ഥ 𝑩 = 𝑹𝒆 ഥ 𝑩𝒆𝒋𝝎𝒕 ❑ Then 𝛁 × 𝑹𝒆 ഥ 𝑬𝒆𝒋𝝎𝒕 = − 𝝏 𝝏𝒕 𝑹𝒆 ഥ 𝑩𝒆𝒋𝝎𝒕 ❑ Interchanging the operation of taking the real part, we get 𝑹𝒆 𝛁 × ഥ 𝑬 + 𝒋𝝎ഥ 𝑩 𝒆𝒋𝝎𝒕 = 𝟎 ∴ 𝛁 × ഥ 𝑬 = −𝒋𝝎ഥ 𝑩
- 30. ❑ Consider thethird Maxwell’s equation 𝛁 ∙ ෩ ഥ 𝑫 = 𝝆𝒗 ❑ If ෩ ഥ 𝑫 = 𝑹𝒆 ഥ 𝑫𝒆𝒋𝝎𝒕 ❑ Then 𝛁 ∙ 𝑹𝒆 ഥ 𝑫𝒆𝒋𝝎𝒕 = 𝝆𝒗 ❑ Interchanging the operation of taking the real part, we get 𝑹𝒆 𝛁 ∙ ഥ 𝑫 𝒆𝒋𝝎𝒕 = 𝝆𝒗 ∴ 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗
- 31. ❑ Consider thefourth Maxwell’s equation 𝛁 ∙ ෩ ഥ 𝑩 = 𝟎 ❑ If ෩ ഥ 𝑩 = 𝑹𝒆 ഥ 𝑩𝒆𝒋𝝎𝒕 ❑ Then 𝛁 ∙ 𝑹𝒆 ഥ 𝑩𝒆𝒋𝝎𝒕 = 𝟎 ❑ Interchanging the operation of taking the real part, we get 𝑹𝒆 𝛁 ∙ ഥ 𝑩 𝒆𝒋𝝎𝒕 = 𝟎 ∴ 𝛁 ∙ ഥ 𝑩 = 𝟎
- 32. ❑ In summary,Maxwell’s equations in phasor form are as follows: 𝛁 × ഥ 𝑯 = 𝒋𝝎ഥ 𝑫 + ҧ 𝑱 𝛁 × ഥ 𝑬 = −𝒋𝝎ഥ 𝑩 𝛁 ∙ ഥ 𝑫 = 𝝆𝒗 𝛁 ∙ ഥ 𝑩 = 𝟎
- 33. The Part (1)will be continued … Thanks