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EE3321 ELECTROMAGENTIC FIELD THEORY

EE3321 ELECTROMAGENTIC FIELD THEORY. Week 9 Maxwell’s Equations. Notation. Summary of Fields. Summary of Fields. James C. Maxwell. Demonstrated that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field. Behavior of EM Field.

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EE3321 ELECTROMAGENTIC FIELD THEORY

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  1. EE3321 ELECTROMAGENTIC FIELD THEORY Week 9 Maxwell’s Equations

  2. Notation

  3. Summary of Fields

  4. Summary of Fields

  5. James C. Maxwell • Demonstrated that electricity, magnetism, and light are all manifestations of the same phenomenon: the electromagnetic field

  6. Behavior of EM Field • Electric charges generate fields • Charges generate electric fields • Moving charges generate magnetic fields • Fields interact with each other • changing electric field acts like a current, generating vortex of magnetic field • changing magnetic field induces (negative) vortex of electric field • Fields act upon charges • electric force: same direction as electric field • magnetic force: perpendicular both to magnetic field and to velocity of charge • Electric charges move in space

  7. Conceptual Map

  8. Maxwell’s Equations • Gauss’ Law for Electricity • Gauss’ Law for Magnetism • Faraday’s Law of Induction • Ampere’s Law

  9. Ampere’s Law Correction Term • Integral Form • Differential form

  10. EM Wave Equation • B and E must obey the same relationship

  11. Exercise • Show that E = Eocos (ωt - kz) axsatisfies the wave equation

  12. Wavelength • Frequency f (cycles per second or Hz) • Wavelength λ (meter) • Speed of propagation c = f λ Distance (meters)

  13. EM Spectrum

  14. Exercise • Determine the frequency of an EM wave with a wavelength of • 1000 m (longwave) • 30 m (shortwave) • 1 cm (microwave) • 500 nm (green light)

  15. Example of Application: RADAR

  16. Example of Application: LASER

  17. Harmonic Fields E = Eocos (ωt - kR) aE H = Hocos(ωt - kR) aH where A is the amplitude t is time ω is the angular frequency 2πf k is the wave number 2π/λ aE is the direction of the electric field aH is the direction of the magnetic field R is the distance traveled

  18. Complex Notation • Euler’s Formula A e+jφ= Acos(φ) + jAsin(φ) A cos(φ) = Re {Ae+jφ} A sin(φ) = Im {Ae+jφ} A e-jφ= A cos(φ) - jA sin(φ) Imaginary Real unit circle

  19. Exercise • Show that A cos(φ) = ½ Ae+jφ + ½ Ae-jφ jA sin(φ) = ½ Ae+jφ - ½ Ae-jφ

  20. Phasors • Complex field E = Eo exp (jωt) exp(jψ) aE • Phasor convention • E = Eo exp(jψ) aE

  21. Sum of Phasors • The frequency must be the same

  22. Plane Wave • The plane wave has a constant value on the plane normal to the direction of propagation • The spacing between planes is the wavelength

  23. Plane Wave • The magnetic field H is perpendicular to the electric field E • The vector product E x H is in the direction of the propagation of the wave.

  24. Direction of Propagation • The wave vector is normal to the wave front and its length is the wavenumber |k| = 2π/λ

  25. Exercise • A plane wave propagates in the direction k = 2ax + 1ay + 0.5az • Determine the following: • wavelength (m) • frequency (Hz)

  26. Diffraction • A plane wave becomes cylindrical when it goes through a slit • The wave fronts have the shape of aligned cylinders

  27. Isotropic Radiation • A spherical wave can be visualized as a series of concentric sphere fronts

  28. Example: Dipole Radiation

  29. Power Density • Poynting Vector (Watts/m2) S = ½ E x H*

  30. Power Density • Poynting Vector (Watts/m2) S = ½ E x H* • For plane waves S = |E|2/ 2η • Electromagnetic (Intrinsic) Impedance

  31. Exercise • A plane wave propagating in the +x direction is described by E = 1.00 e –jkzaxV/m  H = 2.65 e –jkzaymA/m • Determine the following: • Direction of propagation • Intrinsic impedance • Power density

  32. Homework • Read Chapter Sections 7-1, 7-2, 7-6 • Solve Problems 7.1 7.3, 7.25, 7.30, and 7.33

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